Traveling Waves

  • The Sine Wave is the simplest of all possible waves.  A periodic wave is one in which the shape of the wave is repeated "periodically" - at regular fixed intervals.
  • Five basic properties which describe periodic waves.
  • Wavelength  (lambda)  -  Distance after which the wave begins to repeat (Units: metres).
  • Frequency (f)  -  Number of waves passing a fixed point in one second  (Units: Hertz).
  • Wave period (T)  -  Time taken for one wave to pass a given point (Units: seconds).
  • Wave velocity (v)  -  Distance travelled by the wave per second also called the phase velocity (Units: m/s).
  • Amplitude (y m )  -  Maximum displacement of particle which comprise the wave from their equilibrium position  (Units: metres).
  • Frequency, wavelength and velocity are related by,

tw1

  • There are two basic types of traveling waves.
  • TRANSVERSE   :  Motion of the constituent particles is at right angles to the wave direction, e.g. waves on a string, "stadium" wave, electromagnetic waves.
  • Comparison of SHM and Waves
  • The amplitude of the SHM of the particles which comprise a Sine Wave is the same as the amplitude of the wave.
  • The frequency of the SHM of the particles which comprise a Sine Wave is the same as the frequency of the wave.
  • Particles which comprise the wave typically do not move at the speed of the wave, e.g. molecules of air do not move at the speed of sound  in air.
  • Miscellaneous important facts :
  • A wave "travels" from A to B.  The particles that comprise the wave do not move from A to B, they oscillate about their fixed (equilibrium) points.
  • A wave transmits energy from one point to another.
  • Transmission of Electromagnetic waves does not require a medium - there are no "particles".  These waves are comprosed of oscillating electric and magneic fields and are quite happy to propagate in a vacuum.
  • Water waves appear to be transverse  -  boats bob up and down due to water waves.  However,  a  detailed study shows that the molecules of water actually perform a circular motion, which can be considered as a combination of transverse and longitudinal wave motion.

trvwfig1

  • Initial conditions

tw8

  • Wave Velocity on a Stretched String

tw9

Dr. C. L. Davis Physics Department University of Louisville email : [email protected]  

  • 16.2 Mathematics of Waves
  • Introduction
  • 1.1 The Scope and Scale of Physics
  • 1.2 Units and Standards
  • 1.3 Unit Conversion
  • 1.4 Dimensional Analysis
  • 1.5 Estimates and Fermi Calculations
  • 1.6 Significant Figures
  • 1.7 Solving Problems in Physics
  • Key Equations
  • Conceptual Questions
  • Additional Problems
  • Challenge Problems
  • 2.1 Scalars and Vectors
  • 2.2 Coordinate Systems and Components of a Vector
  • 2.3 Algebra of Vectors
  • 2.4 Products of Vectors
  • 3.1 Position, Displacement, and Average Velocity
  • 3.2 Instantaneous Velocity and Speed
  • 3.3 Average and Instantaneous Acceleration
  • 3.4 Motion with Constant Acceleration
  • 3.5 Free Fall
  • 3.6 Finding Velocity and Displacement from Acceleration
  • 4.1 Displacement and Velocity Vectors
  • 4.2 Acceleration Vector
  • 4.3 Projectile Motion
  • 4.4 Uniform Circular Motion
  • 4.5 Relative Motion in One and Two Dimensions
  • 5.2 Newton's First Law
  • 5.3 Newton's Second Law
  • 5.4 Mass and Weight
  • 5.5 Newton’s Third Law
  • 5.6 Common Forces
  • 5.7 Drawing Free-Body Diagrams
  • 6.1 Solving Problems with Newton’s Laws
  • 6.2 Friction
  • 6.3 Centripetal Force
  • 6.4 Drag Force and Terminal Speed
  • 7.2 Kinetic Energy
  • 7.3 Work-Energy Theorem
  • 8.1 Potential Energy of a System
  • 8.2 Conservative and Non-Conservative Forces
  • 8.3 Conservation of Energy
  • 8.4 Potential Energy Diagrams and Stability
  • 8.5 Sources of Energy
  • 9.1 Linear Momentum
  • 9.2 Impulse and Collisions
  • 9.3 Conservation of Linear Momentum
  • 9.4 Types of Collisions
  • 9.5 Collisions in Multiple Dimensions
  • 9.6 Center of Mass
  • 9.7 Rocket Propulsion
  • 10.1 Rotational Variables
  • 10.2 Rotation with Constant Angular Acceleration
  • 10.3 Relating Angular and Translational Quantities
  • 10.4 Moment of Inertia and Rotational Kinetic Energy
  • 10.5 Calculating Moments of Inertia
  • 10.6 Torque
  • 10.7 Newton’s Second Law for Rotation
  • 10.8 Work and Power for Rotational Motion
  • 11.1 Rolling Motion
  • 11.2 Angular Momentum
  • 11.3 Conservation of Angular Momentum
  • 11.4 Precession of a Gyroscope
  • 12.1 Conditions for Static Equilibrium
  • 12.2 Examples of Static Equilibrium
  • 12.3 Stress, Strain, and Elastic Modulus
  • 12.4 Elasticity and Plasticity
  • 13.1 Newton's Law of Universal Gravitation
  • 13.2 Gravitation Near Earth's Surface
  • 13.3 Gravitational Potential Energy and Total Energy
  • 13.4 Satellite Orbits and Energy
  • 13.5 Kepler's Laws of Planetary Motion
  • 13.6 Tidal Forces
  • 13.7 Einstein's Theory of Gravity
  • 14.1 Fluids, Density, and Pressure
  • 14.2 Measuring Pressure
  • 14.3 Pascal's Principle and Hydraulics
  • 14.4 Archimedes’ Principle and Buoyancy
  • 14.5 Fluid Dynamics
  • 14.6 Bernoulli’s Equation
  • 14.7 Viscosity and Turbulence
  • 15.1 Simple Harmonic Motion
  • 15.2 Energy in Simple Harmonic Motion
  • 15.3 Comparing Simple Harmonic Motion and Circular Motion
  • 15.4 Pendulums
  • 15.5 Damped Oscillations
  • 15.6 Forced Oscillations
  • 16.1 Traveling Waves
  • 16.3 Wave Speed on a Stretched String
  • 16.4 Energy and Power of a Wave
  • 16.5 Interference of Waves
  • 16.6 Standing Waves and Resonance
  • 17.1 Sound Waves
  • 17.2 Speed of Sound
  • 17.3 Sound Intensity
  • 17.4 Normal Modes of a Standing Sound Wave
  • 17.5 Sources of Musical Sound
  • 17.7 The Doppler Effect
  • 17.8 Shock Waves
  • B | Conversion Factors
  • C | Fundamental Constants
  • D | Astronomical Data
  • E | Mathematical Formulas
  • F | Chemistry
  • G | The Greek Alphabet

Learning Objectives

By the end of this section, you will be able to:

  • Model a wave, moving with a constant wave velocity, with a mathematical expression
  • Calculate the velocity and acceleration of the medium
  • Show how the velocity of the medium differs from the wave velocity (propagation velocity)

In the previous section, we described periodic waves by their characteristics of wavelength, period, amplitude, and wave speed of the wave. Waves can also be described by the motion of the particles of the medium through which the waves move. The position of particles of the medium can be mathematically modeled as wave function s , which can be used to find the position, velocity, and acceleration of the particles of the medium of the wave at any time.

A pulse can be described as wave consisting of a single disturbance that moves through the medium with a constant amplitude. The pulse moves as a pattern that maintains its shape as it propagates with a constant wave speed. Because the wave speed is constant, the distance the pulse moves in a time Δ t Δ t is equal to Δ x = v Δ t Δ x = v Δ t ( Figure 16.8 ).

Modeling a One-Dimensional Sinusoidal Wave using a Wave Function

Consider a string kept at a constant tension F T F T where one end is fixed and the free end is oscillated between y = + A y = + A and y = − A y = − A by a mechanical device at a constant frequency. Figure 16.9 shows snapshots of the wave at an interval of an eighth of a period, beginning after one period ( t = T ) . ( t = T ) .

Notice that each select point on the string (marked by colored dots) oscillates up and down in simple harmonic motion, between y = + A y = + A and y = − A , y = − A , with a period T . The wave on the string is sinusoidal and is translating in the positive x -direction as time progresses.

At this point, it is useful to recall from your study of algebra that if f ( x ) is some function, then f ( x − d ) f ( x − d ) is the same function translated in the positive x -direction by a distance d . The function f ( x + d ) f ( x + d ) is the same function translated in the negative x -direction by a distance d . We want to define a wave function that will give the y -position of each segment of the string for every position x along the string for every time t .

Looking at the first snapshot in Figure 16.9 , the y -position of the string between x = 0 x = 0 and x = λ x = λ can be modeled as a sine function. This wave propagates down the string one wavelength in one period, as seen in the last snapshot. The wave therefore moves with a constant wave speed of v = λ / T . v = λ / T .

Recall that a sine function is a function of the angle θ θ , oscillating between + 1 + 1 and −1 −1 , and repeating every 2 π 2 π radians ( Figure 16.10 ). However, the y -position of the medium, or the wave function, oscillates between + A + A and − A − A , and repeats every wavelength λ λ .

To construct our model of the wave using a periodic function, consider the ratio of the angle and the position,

Using θ = 2 π λ x θ = 2 π λ x and multiplying the sine function by the amplitude A , we can now model the y -position of the string as a function of the position x :

The wave on the string travels in the positive x -direction with a constant velocity v , and moves a distance vt in a time t . The wave function can now be defined by

It is often convenient to rewrite this wave function in a more compact form. Multiplying through by the ratio 2 π λ 2 π λ leads to the equation

The value 2 π λ 2 π λ is defined as the wave number . The symbol for the wave number is k and has units of inverse meters, m −1 : m −1 :

Recall from Oscillations that the angular frequency is defined as ω ≡ 2 π T . ω ≡ 2 π T . The second term of the wave function becomes

The wave function for a simple harmonic wave on a string reduces to

where A is the amplitude, k = 2 π λ k = 2 π λ is the wave number, ω = 2 π T ω = 2 π T is the angular frequency, the minus sign is for waves moving in the positive x -direction, and the plus sign is for waves moving in the negative x -direction. The velocity of the wave is equal to

Think back to our discussion of a mass on a spring, when the position of the mass was modeled as x ( t ) = A cos ( ω t + ϕ ) . x ( t ) = A cos ( ω t + ϕ ) . The angle ϕ ϕ is a phase shift, added to allow for the fact that the mass may have initial conditions other than x = + A x = + A and v = 0 . v = 0 . For similar reasons, the initial phase is added to the wave function. The wave function modeling a sinusoidal wave, allowing for an initial phase shift ϕ , ϕ , is

is known as the phase of the wave , where ϕ ϕ is the initial phase of the wave function. Whether the temporal term ω t ω t is negative or positive depends on the direction of the wave. First consider the minus sign for a wave with an initial phase equal to zero ( ϕ = 0 ) . ( ϕ = 0 ) . The phase of the wave would be ( k x − ω t ) . ( k x − ω t ) . Consider following a point on a wave, such as a crest. A crest will occur when sin ( k x − ω t ) = 1.00 sin ( k x − ω t ) = 1.00 , that is, when k x − ω t = n π + π 2 , k x − ω t = n π + π 2 , for any integral value of n . For instance, one particular crest occurs at k x − ω t = π 2 . k x − ω t = π 2 . As the wave moves, time increases and x must also increase to keep the phase equal to π 2 . π 2 . Therefore, the minus sign is for a wave moving in the positive x -direction. Using the plus sign, k x + ω t = π 2 . k x + ω t = π 2 . As time increases, x must decrease to keep the phase equal to π 2 . π 2 . The plus sign is used for waves moving in the negative x -direction. In summary, y ( x , t ) = A sin ( k x − ω t + ϕ ) y ( x , t ) = A sin ( k x − ω t + ϕ ) models a wave moving in the positive x -direction and y ( x , t ) = A sin ( k x + ω t + ϕ ) y ( x , t ) = A sin ( k x + ω t + ϕ ) models a wave moving in the negative x -direction.

Equation 16.4 is known as a simple harmonic wave function. A wave function is any function such that f ( x , t ) = f ( x − v t ) . f ( x , t ) = f ( x − v t ) . Later in this chapter, we will see that it is a solution to the linear wave equation. Note that y ( x , t ) = A cos ( k x + ω t + ϕ ′ ) y ( x , t ) = A cos ( k x + ω t + ϕ ′ ) works equally well because it corresponds to a different phase shift ϕ ′ = ϕ − π 2 . ϕ ′ = ϕ − π 2 .

Problem-Solving Strategy

Finding the characteristics of a sinusoidal wave.

  • To find the amplitude, wavelength, period, and frequency of a sinusoidal wave, write down the wave function in the form y ( x , t ) = A sin ( k x − ω t + ϕ ) . y ( x , t ) = A sin ( k x − ω t + ϕ ) .
  • The amplitude can be read straight from the equation and is equal to A .
  • The period of the wave can be derived from the angular frequency ( T = 2 π ω ) . ( T = 2 π ω ) .
  • The frequency can be found using f = 1 T . f = 1 T .
  • The wavelength can be found using the wave number ( λ = 2 π k ) . ( λ = 2 π k ) .

Example 16.3

Characteristics of a traveling wave on a string.

Find the amplitude, wavelength, period, and speed of the wave.

  • The amplitude, wave number, and angular frequency can be read directly from the wave equation: y ( x , t ) = A sin ( k x − ω t ) = 0.2 m sin ( 6.28 m −1 x − 1.57 s −1 t ) . y ( x , t ) = A sin ( k x − ω t ) = 0.2 m sin ( 6.28 m −1 x − 1.57 s −1 t ) . ( A = 0.2 m; k = 6.28 m −1 ; ω = 1.57 s −1 ) ( A = 0.2 m; k = 6.28 m −1 ; ω = 1.57 s −1 )
  • The wave number can be used to find the wavelength: k = 2 π λ . λ = 2 π k = 2 π 6.28 m −1 = 1.0 m . k = 2 π λ . λ = 2 π k = 2 π 6.28 m −1 = 1.0 m .
  • The period of the wave can be found using the angular frequency: ω = 2 π T . T = 2 π ω = 2 π 1.57 s −1 = 4 s . ω = 2 π T . T = 2 π ω = 2 π 1.57 s −1 = 4 s .
  • The speed of the wave can be found using the wave number and the angular frequency. The direction of the wave can be determined by considering the sign of k x ∓ ω t k x ∓ ω t : A negative sign suggests that the wave is moving in the positive x -direction: | v | = ω k = 1.57 s −1 6.28 m −1 = 0.25 m/s . | v | = ω k = 1.57 s −1 6.28 m −1 = 0.25 m/s .

Significance

There is a second velocity to the motion. In this example, the wave is transverse, moving horizontally as the medium oscillates up and down perpendicular to the direction of motion. The graph in Figure 16.12 shows the motion of the medium at point x = 0.60 m x = 0.60 m as a function of time. Notice that the medium of the wave oscillates up and down between y = + 0.20 m y = + 0.20 m and y = −0.20 m y = −0.20 m every period of 4.0 seconds.

Check Your Understanding 16.3

The wave function above is derived using a sine function. Can a cosine function be used instead?

Velocity and Acceleration of the Medium

As seen in Example 16.4 , the wave speed is constant and represents the speed of the wave as it propagates through the medium, not the speed of the particles that make up the medium. The particles of the medium oscillate around an equilibrium position as the wave propagates through the medium. In the case of the transverse wave propagating in the x -direction, the particles oscillate up and down in the y -direction, perpendicular to the motion of the wave. The velocity of the particles of the medium is not constant, which means there is an acceleration. The velocity of the medium, which is perpendicular to the wave velocity in a transverse wave, can be found by taking the partial derivative of the position equation with respect to time. The partial derivative is found by taking the derivative of the function, treating all variables as constants, except for the variable in question. In the case of the partial derivative with respect to time t , the position x is treated as a constant. Although this may sound strange if you haven’t seen it before, the object of this exercise is to find the transverse velocity at a point, so in this sense, the x -position is not changing. We have

The magnitude of the maximum velocity of the medium is | v y max | = A ω | v y max | = A ω . This may look familiar from the Oscillations and a mass on a spring.

We can find the acceleration of the medium by taking the partial derivative of the velocity equation with respect to time,

The magnitude of the maximum acceleration is | a y max | = A ω 2 . | a y max | = A ω 2 . The particles of the medium, or the mass elements, oscillate in simple harmonic motion for a mechanical wave.

The Linear Wave Equation

We have just determined the velocity of the medium at a position x by taking the partial derivative, with respect to time, of the position y . For a transverse wave, this velocity is perpendicular to the direction of propagation of the wave. We found the acceleration by taking the partial derivative, with respect to time, of the velocity, which is the second time derivative of the position:

Now consider the partial derivatives with respect to the other variable, the position x , holding the time constant. The first derivative is the slope of the wave at a point x at a time t ,

The second partial derivative expresses how the slope of the wave changes with respect to position—in other words, the curvature of the wave, where

The ratio of the acceleration and the curvature leads to a very important relationship in physics known as the linear wave equation . Taking the ratio and using the equation v = ω / k v = ω / k yields the linear wave equation (also known simply as the wave equation or the equation of a vibrating string),

Equation 16.6 is the linear wave equation, which is one of the most important equations in physics and engineering. We derived it here for a transverse wave, but it is equally important when investigating longitudinal waves. This relationship was also derived using a sinusoidal wave, but it successfully describes any wave or pulse that has the form y ( x , t ) = f ( x ∓ v t ) . y ( x , t ) = f ( x ∓ v t ) . These waves result due to a linear restoring force of the medium—thus, the name linear wave equation. Any wave function that satisfies this equation is a linear wave function.

An interesting aspect of the linear wave equation is that if two wave functions are individually solutions to the linear wave equation, then the sum of the two linear wave functions is also a solution to the wave equation. Consider two transverse waves that propagate along the x -axis, occupying the same medium. Assume that the individual waves can be modeled with the wave functions y 1 ( x , t ) = f ( x ∓ v t ) y 1 ( x , t ) = f ( x ∓ v t ) and y 2 ( x , t ) = g ( x ∓ v t ) , y 2 ( x , t ) = g ( x ∓ v t ) , which are solutions to the linear wave equations and are therefore linear wave functions. The sum of the wave functions is the wave function

Consider the linear wave equation:

This has shown that if two linear wave functions are added algebraically, the resulting wave function is also linear. This wave function models the displacement of the medium of the resulting wave at each position along the x -axis. If two linear waves occupy the same medium, they are said to interfere. If these waves can be modeled with a linear wave function, these wave functions add to form the wave equation of the wave resulting from the interference of the individual waves. The displacement of the medium at every point of the resulting wave is the algebraic sum of the displacements due to the individual waves.

Taking this analysis a step further, if wave functions y 1 ( x , t ) = f ( x ∓ v t ) y 1 ( x , t ) = f ( x ∓ v t ) and y 2 ( x , t ) = g ( x ∓ v t ) y 2 ( x , t ) = g ( x ∓ v t ) are solutions to the linear wave equation, then A y 1 ( x , t ) + B y 2 ( x , t ) , A y 1 ( x , t ) + B y 2 ( x , t ) , where A and B are constants, is also a solution to the linear wave equation. This property is known as the principle of superposition. Interference and superposition are covered in more detail in Interference of Waves .

Example 16.4

Interference of waves on a string.

  • Write the wave function of the second wave: y 2 ( x , t ) = A sin ( 2 k x + 2 ω t ) . y 2 ( x , t ) = A sin ( 2 k x + 2 ω t ) .
  • Write the resulting wave function: y R ( x , t ) = y 1 ( x , t ) + y ( x , t ) = A sin ( k x − ω t ) + A sin ( 2 k x + 2 ω t ) . y R ( x , t ) = y 1 ( x , t ) + y ( x , t ) = A sin ( k x − ω t ) + A sin ( 2 k x + 2 ω t ) .
  • Find the partial derivatives: ∂ y R ( x , t ) ∂ x = + A k cos ( k x − ω t ) + 2 A k cos ( 2 k x + 2 ω t ) , ∂ 2 y R ( x , t ) ∂ x 2 = − A k 2 sin ( k x − ω t ) − 4 A k 2 sin ( 2 k x + 2 ω t ) , ∂ y R ( x , t ) ∂ t = − A ω cos ( k x − ω t ) + 2 A ω cos ( 2 k x + 2 ω t ) , ∂ 2 y R ( x , t ) ∂ t 2 = − A ω 2 sin ( k x − ω t ) − 4 A ω 2 sin ( 2 k x + 2 ω t ) . ∂ y R ( x , t ) ∂ x = + A k cos ( k x − ω t ) + 2 A k cos ( 2 k x + 2 ω t ) , ∂ 2 y R ( x , t ) ∂ x 2 = − A k 2 sin ( k x − ω t ) − 4 A k 2 sin ( 2 k x + 2 ω t ) , ∂ y R ( x , t ) ∂ t = − A ω cos ( k x − ω t ) + 2 A ω cos ( 2 k x + 2 ω t ) , ∂ 2 y R ( x , t ) ∂ t 2 = − A ω 2 sin ( k x − ω t ) − 4 A ω 2 sin ( 2 k x + 2 ω t ) .

∂ 2 y ( x , t ) ∂ x 2 = 1 v 2 ∂ 2 y ( x , t ) ∂ t 2 , − A k 2 sin ( k x − ω t ) − 4 A k 2 sin ( 2 k x + 2 ω t ) = 1 v 2 ( − A ω 2 sin ( k x − ω t ) − 4 A ω 2 sin ( 2 k x + 2 ω t ) ) , k 2 ( − A sin ( k x − ω t ) − 4 A sin ( 2 k x + 2 ω t ) ) = ω 2 v 2 ( − A sin ( k x − ω t ) − 4 A sin ( 2 k x + 2 ω t ) ) , k 2 = ω 2 v 2 , | v | = ω k . ∂ 2 y ( x , t ) ∂ x 2 = 1 v 2 ∂ 2 y ( x , t ) ∂ t 2 , − A k 2 sin ( k x − ω t ) − 4 A k 2 sin ( 2 k x + 2 ω t ) = 1 v 2 ( − A ω 2 sin ( k x − ω t ) − 4 A ω 2 sin ( 2 k x + 2 ω t ) ) , k 2 ( − A sin ( k x − ω t ) − 4 A sin ( 2 k x + 2 ω t ) ) = ω 2 v 2 ( − A sin ( k x − ω t ) − 4 A sin ( 2 k x + 2 ω t ) ) , k 2 = ω 2 v 2 , | v | = ω k .

Check Your Understanding 16.4

The wave equation ∂ 2 y ( x , t ) ∂ x 2 = 1 v 2 ∂ 2 y ( x , t ) ∂ t 2 ∂ 2 y ( x , t ) ∂ x 2 = 1 v 2 ∂ 2 y ( x , t ) ∂ t 2 works for any wave of the form y ( x , t ) = f ( x ∓ v t ) . y ( x , t ) = f ( x ∓ v t ) . In the previous section, we stated that a cosine function could also be used to model a simple harmonic mechanical wave. Check if the wave

is a solution to the wave equation.

Any disturbance that complies with the wave equation can propagate as a wave moving along the x -axis with a wave speed v . It works equally well for waves on a string, sound waves, and electromagnetic waves. This equation is extremely useful. For example, it can be used to show that electromagnetic waves move at the speed of light.

As an Amazon Associate we earn from qualifying purchases.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
  • Authors: William Moebs, Samuel J. Ling, Jeff Sanny
  • Publisher/website: OpenStax
  • Book title: University Physics Volume 1
  • Publication date: Sep 19, 2016
  • Location: Houston, Texas
  • Book URL: https://openstax.org/books/university-physics-volume-1/pages/1-introduction
  • Section URL: https://openstax.org/books/university-physics-volume-1/pages/16-2-mathematics-of-waves

© Jan 19, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Travelling Wave

When something about the physical world changes, the information about that disturbance gradually moves outwards, away from the source, in every direction. As the information travels, it travels in the form of a wave. Sound to our ears, light to our eyes, and electromagnetic radiation to our mobile phones are all transported in the form of waves. A good visual example of the propagation of waves is the waves created on the surface of the water when a stone is dropped into a lake. In this article, we will be learning more about travelling waves.

Describing a Wave

A wave can be described as a disturbance in a medium that travels transferring momentum and energy without any net motion of the medium. A wave in which the positions of maximum and minimum amplitude travel through the medium is known as a travelling wave. To better understand a wave, let us think of the disturbance caused when we jump on a trampoline. When we jump on a trampoline, the downward push that we create at a point on the trampoline slightly moves the material next to it downward too.

When the created disturbance travels outward, the point at which our feet first hit the trampoline recovers moving outward because of the tension force in the trampoline and that moves the surrounding nearby materials outward too. This up and down motion gradually ripples out as it covers more area of the trampoline. And, this disturbance takes the shape of a wave.

Following are a few important points to remember about the wave:

  • The high points in the wave are known as crests and the low points in the wave are known as troughs.
  • The maximum distance of the disturbance of the wave from the mid-point to either the top of the crest or to the bottom of a trough is known as amplitude.
  • The distance between two adjacent crests or two adjacent troughs is known as a wavelength and is denoted by 𝛌.
  • The time interval of one complete vibration is known as a time period.
  • The number of vibrations the wave undergoes in one second is known as a frequency.
  • The relationship between the time period and frequency is given as follows:
  • The speed of a wave is given by the equation

Different Types of Waves

Different types of waves exhibit distinct characteristics. These characteristics help us distinguish between wave types. The orientation of particle motion relative to the direction of wave propagation is one way the traveling waves are distinguished. Following are the different types of waves categorized based on the particle motion:

  • Pulse Waves – A pulse wave is a wave comprising only one disturbance or only one crest that travels through the transmission medium.
  • Continuous Waves – A continuous-wave is a waveform of constant amplitude and frequency.
  • Transverse Waves – In a transverse wave, the motion of the particle is perpendicular to the direction of propagation of the wave.
  • Longitudinal Waves – Longitudinal waves are the waves in which the motion of the particle is in the same direction as the propagation of the wave.

Although they are different, there is one property common between them and that is the transportation of energy. An object in simple harmonic motion has an energy of

Constructive and Destructive Interference

A phenomenon in which two waves superimpose to form a resultant wave of lower, greater, or the same amplitude is known as interference. Constructive and destructive interference occurs due to the interaction of waves that are correlated with each other either because of the same frequency or because they come from the same source. The interference effects can be observed in all types of waves such as gravity waves and light waves.

Wave Interference

According to the principle of superposition of the waves , when two or more propagating waves of the same type are incidents on the same point, the resultant amplitude is equal to the vector sum of the amplitudes of the individual waves. When a crest of a wave meets a crest of another wave of the same frequency at the same point, then the resultant amplitude is the sum of the individual amplitudes. This type of interference is known as constructive interference. If a crest of a wave meets a trough of another wave, then the resulting amplitude is equal to the difference in the individual amplitudes and this is known as destructive interference.

Stay tuned to BYJU’S to learn more physics concepts with the help of interactive videos.

Watch the video and understand longitudinal and transverse waves in detail.

travelling harmonic wave

Frequently Asked Questions – FAQs

What is a pulse wave, what are longitudinal waves, what is superposition of waves, what is electromagnetic radiation, what is constructive interference.

Quiz Image

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

Visit BYJU’S for all Physics related queries and study materials

Your result is as below

Request OTP on Voice Call

Leave a Comment Cancel reply

Your Mobile number and Email id will not be published. Required fields are marked *

Post My Comment

travelling harmonic wave

  • Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

close

previous   index   next

Analyzing Waves on a String

Michael Fowler, University of Virginia

From Newton’s Laws to the Wave Equation: a Tiny Bit of String

Everything there is to know about waves on a uniform string can be found by applying Newton’s Second Law, F → = m a → , to one tiny bit of the string.  Well, at least this is true of the small amplitude waves we shall be studying — we’ll be assuming the deviation of the string from its rest position is small compared with the wavelength of the waves being studied.  This makes the math simpler, and is an excellent approximation for musical instruments, etc.   Having said that, we’ll draw diagrams, like the one below, with rather large amplitude waves, to show more clearly what’s going on.

Click here to animate!

Let’s write down F → = m a →  for the small length of string between x and x + Δ x in the diagram above. 

Taking the string to have mass density μ  kg/m, we have m = μ Δ x .  

The forces on the bit of string (neglecting the tiny force of gravity, air resistance, etc.) are the tensions T at the two ends.  To an excellent approximation, the tension will be uniform in magnitude along the string, but the string curves if it’s waving, so the two T →   vectors at opposite ends of the bit of string do not quite cancel, this is the net force F →  we’re looking for.

Bearing in mind that we’re only interested here in small amplitude waves, we can see from the diagram (squashing it mentally in the y -direction) that both  T →  vectors will be close to horizontal, and, since they’re pointing in opposite directions, their sum — the net force F → — will be very close to vertical:

The vertical component of the tension T →  at the x + Δ x end of the bit of string is T sin θ , where θ  is the angle of slope of the string at that end. This slope is of course just d y ( x + Δ x ) / d x , or, more precisely, d y / d x = tan θ . 

However, if the wave amplitude is small, as we’re assuming, then θ  is small, and we can take tan θ = sin θ = θ , and therefore take the vertical component of the tension force on the string to be T θ = T d y ( x + Δ x ) / d x .  So the total vertical force from the tensions at the two ends becomes

F → = T ( d y ( x + Δ x ) d x − d y ( x ) d x ) ≅ T d 2 y ( x ) d x 2 Δ x

the equality becoming exact in the limit Δ x → 0 .

At this point, it is necessary to make clear that y is a function of t as well as of x : y = y ( x , t ) .  In this case, the standard convention for denoting differentiation with respect to one variable while the other is held constant (which is the case here — we’re looking at the sum of forces at one instant of time) is to replace d / d x  with  ∂ / ∂ x . 

So we should write:

F → = T ∂ 2 y ∂ x 2 . Δ x

The final piece of the puzzle is the acceleration of the bit of string: in our small amplitude approximation, it’s only moving up and down, that is, in the y -direction — so the acceleration is just ∂ 2 y / ∂ t 2 , and canceling Δ x between the mass m = μ Δ x  and  F → = T ∂ 2 y ∂ x 2 Δ x ,  F → = m a →  gives:

T ∂ 2 y ∂ x 2 = μ ∂ 2 y ∂ t 2 .

This is called the wave equation . 

It’s worth looking at this equation to see why it is equivalent to F → = m a → .  Picture the graph y = y ( x , t ) showing the position of the string at the instant t .  At the point x , the differential ∂ y / ∂ x  is the slope of the string.  The second differential,  ∂ 2 y / ∂ x 2 , is the rate of change of the slope — in other words, how much the string is curved at x .  And, it’s this curvature that ensures the T →  ’s at the two ends of a bit of string are pointing along slightly different directions, and therefore don’t cancel. This force, then, gives the mass × acceleration on the right.

Solving the Wave Equation

Now that we’ve derived a wave equation from analyzing the motion of a tiny piece of string, we must check to see that it is consistent with our previous assertions about waves, which were based on experiment and observation.  For example, we stated that a wave traveling down a rope kept its shape, so we could write y ( x , t ) = f ( x − v t ) .  Does a general function f ( x − v t )  necessarily satisfy the wave equation?  This f is a function of a single variable, let’s call it u = x − v t .  On putting it into the wave equation, we must use the chain rule for differentiation:

∂ f ∂ x = ∂ f ∂ u ∂ u ∂ x = ∂ f ∂ u ,   ∂ f ∂ t = ∂ f ∂ u ∂ u ∂ t = − v ∂ f ∂ u

and the equation becomes

T ∂ 2 f ∂ u 2 = μ v 2 ∂ 2 f ∂ u 2

so the function f ( x − v t )  will always satisfy the wave equation provided

v 2 = T μ .

All traveling waves move at the same speed — and the speed is determined by the tension and the mass per unit length.  We could have figured out the equation for v 2 dimensionally, but there would have been an overall arbitrary constant.  We need the wave equation to prove that constant is 1.

Incorporating the above result, the equation is often written:

∂ 2 y ∂ x 2 = 1 v 2 ∂ 2 y ∂ t 2 .

Of course, waves can travel both ways on a string: an arbitrary function g ( x + v t )  is an equally good solution.

The Principle of Superposition

The wave equation has a very important property: if we have two solutions to the equation, then the sum of the two is also a solution to the equation.  It’s easy to check this:

∂ 2 ( f + g ) ∂ x 2 = ∂ 2 f ∂ x 2 + ∂ 2 g ∂ x 2 = 1 v 2 ∂ 2 f ∂ t 2 + 1 v 2 ∂ 2 g ∂ t 2 = 1 v 2 ∂ 2 ( f + g ) ∂ t 2 .

Any differential equation for which this property holds is called a linear differential equation: note that a f ( x , t ) + b g ( x , t )  is also a solution to the equation if a , b are constants. So you can add together — superpose — multiples of any two solutions of the wave equation to find a new function satisfying the equation.

Harmonic Traveling Waves

Imagine that one end of a long taut string is attached to a simple harmonic oscillator, such as a tuning fork — this will send a harmonic wave down the string,

f ( x − v t ) = A sin k ( x − v t ) .

The standard notation is

f ( x − v t ) = A sin ( k x − ω t )

where of course

ω = v k .

More notation: the wavelength of this traveling wave is λ , and from the form A sin ( k x − ω t ) , at say t = 0 ,

k λ = 2 π .

At a fixed x , the string goes up and down with frequency given by sin ω t , so the frequency f in cycles per second (Hz) is

f = ω 2 π  Hz .

Now imagine you’re standing at the origin watching the wave go by.  You see the string at the origin do a complete up-and-down cycle f times per second.  Each time it does this, a whole wavelength of the wave travels by.  Suppose that at t = 0 the wave, coming in from the left, has just reached you.

Then at t = 1 second, the front of the wave will have traveled f wavelengths past you — so the speed at which the wave is traveling

v = λ f  meters per second .

Energy and Power in a Traveling Harmonic Wave

If we jiggle one end of a string and send a wave down its length, we are obviously supplying energy to the string — for one thing, as the wave moves down, bits of the string begin moving, so there is kinetic energy.  And, there’s also potential energy — remember the wave won’t go down at all unless there is tension in the string, and when the string is waving it’s obviously longer than when it’s motionless along the x -axis.  This stretching of the string takes work against the tension T equal to force times distance, in this case equal to the force T multiplied by the distance the string has been stretched.  (We assume that this increase in length is not sufficient to cause significant increase in T .  This is usually ok.)

For the important case of a harmonic wave traveling along a string, we can work out the energy per unit length exactly.  We take

y ( x , t ) = A sin ( k x − ω t ) .

If the string has mass μ  per unit length, a small piece of string of length Δ x  will have mass μ Δ x , and moves (vertically) at speed ∂ y / ∂ t , so has kinetic energy ( 1 / 2 ) μ Δ x ( ∂ y / ∂ t ) 2 , from which the kinetic energy of a length of string is

K . E . = ∫ 1 2 μ ( ∂ y ∂ t ) 2 d x .

For the harmonic wave y ( x , t ) = A sin ( k x − ω t ) ,  

K . E . = ∫ 1 2 μ A 2 ω 2 cos 2 ( k x − ω t ) d x

and since the average value cos 2 ( k x − ω t ) ¯ = 1 2 , for a continuous harmonic wave the average K . E . per unit length

K . E . ¯ / meter = 1 4 μ ω 2 A 2 .

To find the average potential energy in a meter of string as the wave moves through, we need to know how much the string is stretched by the wave, and multiply that length increase by the tension T .

Let’s start with a small length Δ x  of string, and suppose that the change in y from one end to the other is Δ y :

The string (red) is the hypotenuse of this right-angled triangle, so the amount of stretching Δ l  of this length Δ x  is how much longer the hypotenuse is than the base Δ x .

Δ l = ( Δ x ) 2 + ( Δ y ) 2 − Δ x = Δ x 1 + ( Δ y / Δ x ) 2 − Δ x .

Remembering that we’re only considering small amplitude waves, Δ y / Δ x  is going to be small, so we can expand the square root using the result

1 + x ≅ 1 + 1 2 x   for small  x

Δ l ≅ 1 2 ( Δ y / Δ x ) 2 Δ x .

To find the total stretching of a unit length of string, we add all these small stretches, taking the limit of small Δ x     's  to find

P . E . /meter = ∫ 1 2 T ( ∂ y / ∂ x ) 2 d x = ∫ 1 2 T A 2 k 2 cos 2 ( k x − ω t ) d x .

Now, just as for the kinetic energy discussed above, since cos 2 ( k x − ω t ) ¯ = 1 2 , the average potential energy per meter of string is

P . E . ¯ / meter = 1 4 T k 2 A 2 = 1 4 μ ω 2 A 2 ,  since  ω = v k  and  v 2 = T / μ .

That is to say, the average potential energy is the same as the average kinetic energy.  This is a very general result: it is true for all harmonic oscillators (excepting the case of heavy damping).

Finally, the power in a wave traveling down a string is the rate at which it delivers energy at its destination.  Adding together the kinetic and potential energy contributions above,

total energy ¯ / meter = 1 2 μ ω 2 A 2 .

Now, if the wave is traveling at v meters per second, and being totally absorbed at its destination (the end of the string) the energy delivered to that end in one second is all the energy in the last v meters of the string.  By definition, this is the power : the energy delivered in joules per second, That is,

power = 1 2 v μ ω 2 A 2 .

Standing Waves from Traveling Waves

An amusing application of the principle of superposition is adding together harmonic traveling waves moving in opposite directions to get a standing wave:

A sin ( k x − ω t ) + A sin ( k x + ω t ) = 2 A sin k x cos ω t .

You can easily check that 2 A sin k x cos ω t   is a solution to the wave equation (provided ω = v k , of course) and it is always zero at points x satisfying k x = n π , so for a string of length L , fixed at the two ends, the appropriate k are given by k L = n π .

The longest wavelength standing wave for a string of length L fixed at both ends has wavelength   λ = 2 L , and is termed the fundamental .

The x -dependence of this wave, sin k x ,  is clearly sin ( π x / L ) , so k = π / L .

The radial frequency of the wave is given by ω = v k , so ω = v π / L ,  and the frequency in cycles per second, or Hz, is

f = ω / 2 π = v / 2 L   Hz .  

(This is the same as the frequency f = v / λ  of a traveling wave having the same wavelength.)

Here’s a realization of the superposition of two traveling waves to form a standing wave using a spreadsheet:

Here the red wave is A sin ( k x − ω   t )  and moves to the right, the green A sin ( k x + ω   t )  moves to the left, the black is the sum of the two and its oscillations stay in place. 

But this represents just one instant!  To see the full development in time — which you need to do to get real insight into what’s going on — download the spreadsheet from here, then click and hold at the end of the slider bar to animate.

Exercise :  How do you think the black wave will move if the red and green have different amplitudes?  Predict it: then try it on the spreadsheet.  You might be surprised! (Try different ratios of the wave amplitudes: say, 1.1, 1.5, 2, 5.)

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Class 11 Physics (India)

Course: class 11 physics (india)   >   unit 19.

  • Standing waves on strings
  • Wavelength and frequency for a standing wave
  • Calculating frequency for harmonics of a standing wave

Standing waves review

Standing wave harmonics, common mistakes and misconceptions.

  • wavelength and frequency for a standing wave
  • calculating frequency for harmonics of a standing wave

Want to join the conversation?

  • Upvote Button navigates to signup page
  • Downvote Button navigates to signup page
  • Flag Button navigates to signup page

Good Answer

16.1 Traveling Waves

Learning objectives.

By the end of this section, you will be able to:

  • Describe the basic characteristics of wave motion
  • Define the terms wavelength, amplitude, period, frequency, and wave speed
  • Explain the difference between longitudinal and transverse waves, and give examples of each type
  • List the different types of waves

We saw in Oscillations that oscillatory motion is an important type of behavior that can be used to model a wide range of physical phenomena. Oscillatory motion is also important because oscillations can generate waves, which are of fundamental importance in physics. Many of the terms and equations we studied in the chapter on oscillations apply equally well to wave motion ( (Figure) ).

Photograph of an ocean wave.

Figure 16.2 From the world of renewable energy sources comes the electric power-generating buoy. Although there are many versions, this one converts the up-and-down motion, as well as side-to-side motion, of the buoy into rotational motion in order to turn an electric generator, which stores the energy in batteries.

Types of Waves

A wave is a disturbance that propagates, or moves from the place it was created. There are three basic types of waves: mechanical waves, electromagnetic waves, and matter waves.

Basic mechanical waves are governed by Newton’s laws and require a medium. A medium is the substance a mechanical waves propagates through, and the medium produces an elastic restoring force when it is deformed. Mechanical waves transfer energy and momentum, without transferring mass. Some examples of mechanical waves are water waves, sound waves, and seismic waves. The medium for water waves is water; for sound waves, the medium is usually air. (Sound waves can travel in other media as well; we will look at that in more detail in Sound .) For surface water waves, the disturbance occurs on the surface of the water, perhaps created by a rock thrown into a pond or by a swimmer splashing the surface repeatedly. For sound waves, the disturbance is a change in air pressure, perhaps created by the oscillating cone inside a speaker or a vibrating tuning fork. In both cases, the disturbance is the oscillation of the molecules of the fluid. In mechanical waves, energy and momentum transfer with the motion of the wave, whereas the mass oscillates around an equilibrium point. (We discuss this in Energy and Power of a Wave .) Earthquakes generate seismic waves from several types of disturbances, including the disturbance of Earth’s surface and pressure disturbances under the surface. Seismic waves travel through the solids and liquids that form Earth. In this chapter, we focus on mechanical waves.

Electromagnetic waves are associated with oscillations in electric and magnetic fields and do not require a medium. Examples include gamma rays, X-rays, ultraviolet waves, visible light, infrared waves, microwaves, and radio waves. Electromagnetic waves can travel through a vacuum at the speed of light, [latex] v=c=2.99792458\,×\,{10}^{8}\,\text{m/s}. [/latex] For example, light from distant stars travels through the vacuum of space and reaches Earth. Electromagnetic waves have some characteristics that are similar to mechanical waves; they are covered in more detail in Electromagnetic Waves in volume 2 of this text.

Matter waves are a central part of the branch of physics known as quantum mechanics. These waves are associated with protons, electrons, neutrons, and other fundamental particles found in nature. The theory that all types of matter have wave-like properties was first proposed by Louis de Broglie in 1924. Matter waves are discussed in Photons and Matter Waves in the third volume of this text.

Mechanical Waves

Mechanical waves exhibit characteristics common to all waves, such as amplitude, wavelength, period, frequency, and energy. All wave characteristics can be described by a small set of underlying principles.

The simplest mechanical waves repeat themselves for several cycles and are associated with simple harmonic motion. These simple harmonic waves can be modeled using some combination of sine and cosine functions. For example, consider the simplified surface water wave that moves across the surface of water as illustrated in (Figure) . Unlike complex ocean waves, in surface water waves, the medium, in this case water, moves vertically, oscillating up and down, whereas the disturbance of the wave moves horizontally through the medium. In (Figure) , the waves causes a seagull to move up and down in simple harmonic motion as the wave crests and troughs (peaks and valleys) pass under the bird. The crest is the highest point of the wave, and the trough is the lowest part of the wave. The time for one complete oscillation of the up-and-down motion is the wave’s period T . The wave’s frequency is the number of waves that pass through a point per unit time and is equal to [latex] f=1\text{/}T. [/latex] The period can be expressed using any convenient unit of time but is usually measured in seconds; frequency is usually measured in hertz (Hz), where [latex] 1\,{\text{Hz}=1\,\text{s}}^{-1}. [/latex]

The length of the wave is called the wavelength and is represented by the Greek letter lambda [latex] (\lambda ) [/latex], which is measured in any convenient unit of length, such as a centimeter or meter. The wavelength can be measured between any two similar points along the medium that have the same height and the same slope. In (Figure) , the wavelength is shown measured between two crests. As stated above, the period of the wave is equal to the time for one oscillation, but it is also equal to the time for one wavelength to pass through a point along the wave’s path.

The amplitude of the wave ( A ) is a measure of the maximum displacement of the medium from its equilibrium position. In the figure, the equilibrium position is indicated by the dotted line, which is the height of the water if there were no waves moving through it. In this case, the wave is symmetrical, the crest of the wave is a distance [latex] \text{+}A [/latex] above the equilibrium position, and the trough is a distance [latex] \text{−}A [/latex] below the equilibrium position. The units for the amplitude can be centimeters or meters, or any convenient unit of distance.

Figure shows a wave with the equilibrium position marked with a horizontal line. The vertical distance from the line to the crest of the wave is labeled x and that from the line to the trough is labeled minus x. There is a bird shown bobbing up and down in the wave. The vertical distance that the bird travels is labeled 2x. The horizontal distance between two consecutive crests is labeled lambda. A vector pointing right is labeled v subscript w.

Figure 16.3 An idealized surface water wave passes under a seagull that bobs up and down in simple harmonic motion. The wave has a wavelength [latex] \lambda [/latex], which is the distance between adjacent identical parts of the wave. The amplitude A of the wave is the maximum displacement of the wave from the equilibrium position, which is indicated by the dotted line. In this example, the medium moves up and down, whereas the disturbance of the surface propagates parallel to the surface at a speed v.

The water wave in the figure moves through the medium with a propagation velocity [latex] \overset{\to }{v}. [/latex] The magnitude of the wave velocity is the distance the wave travels in a given time, which is one wavelength in the time of one period, and the wave speed is the magnitude of wave velocity. In equation form, this is

This fundamental relationship holds for all types of waves. For water waves, v is the speed of a surface wave; for sound, v is the speed of sound; and for visible light, v is the speed of light.

Transverse and Longitudinal Waves

We have seen that a simple mechanical wave consists of a periodic disturbance that propagates from one place to another through a medium. In (Figure) (a), the wave propagates in the horizontal direction, whereas the medium is disturbed in the vertical direction. Such a wave is called a transverse wave . In a transverse wave, the wave may propagate in any direction, but the disturbance of the medium is perpendicular to the direction of propagation. In contrast, in a longitudinal wave or compressional wave, the disturbance is parallel to the direction of propagation. (Figure) (b) shows an example of a longitudinal wave. The size of the disturbance is its amplitude A and is completely independent of the speed of propagation v .

Figure a, labeled transverse wave, shows a person holding one end of a long, horizontally placed spring and moving it up and down. The spring forms a wave which propagates away from the person. This is labeled transverse wave. The vertical distance between the crest of the wave and the equilibrium position of the spring is labeled A. Figure b, labeled longitudinal wave, shows the person moving the spring to and fro horizontally. The spring is compressed and elongated alternately. This is labeled longitudinal wave. The horizontal distance from the middle of one compression to the middle of one rarefaction is labeled A.

Figure 16.4 (a) In a transverse wave, the medium oscillates perpendicular to the wave velocity. Here, the spring moves vertically up and down, while the wave propagates horizontally to the right. (b) In a longitudinal wave, the medium oscillates parallel to the propagation of the wave. In this case, the spring oscillates back and forth, while the wave propagates to the right.

A simple graphical representation of a section of the spring shown in (Figure) (b) is shown in (Figure) . (Figure) (a) shows the equilibrium position of the spring before any waves move down it. A point on the spring is marked with a blue dot. (Figure) (b) through (g) show snapshots of the spring taken one-quarter of a period apart, sometime after the end of` the spring is oscillated back and forth in the x -direction at a constant frequency. The disturbance of the wave is seen as the compressions and the expansions of the spring. Note that the blue dot oscillates around its equilibrium position a distance A , as the longitudinal wave moves in the positive x -direction with a constant speed. The distance A is the amplitude of the wave. The y -position of the dot does not change as the wave moves through the spring. The wavelength of the wave is measured in part (d). The wavelength depends on the speed of the wave and the frequency of the driving force.

Figures a through g show different stages of a longitudinal wave passing through a spring. A blue dot marks a point on the spring. This moves from left to right as the wave propagates towards the right. In figure b at time t=0, the dot is to the right of the equilibrium position. In figure d, at time t equal to half T, the dot is to the left of the equilibrium position. In figure f, at time t=T, the dot is again to the right. The distance between the equilibrium position and the extreme left or right position of the dot is the same and is labeled A. The distance between two identical parts of the wave is labeled lambda.

Figure 16.5 (a) This is a simple, graphical representation of a section of the stretched spring shown in (Figure)(b), representing the spring’s equilibrium position before any waves are induced on the spring. A point on the spring is marked by a blue dot. (b–g) Longitudinal waves are created by oscillating the end of the spring (not shown) back and forth along the x-axis. The longitudinal wave, with a wavelength [latex] \lambda [/latex], moves along the spring in the +x-direction with a wave speed v. For convenience, the wavelength is measured in (d). Note that the point on the spring that was marked with the blue dot moves back and forth a distance A from the equilibrium position, oscillating around the equilibrium position of the point.

Waves may be transverse, longitudinal, or a combination of the two. Examples of transverse waves are the waves on stringed instruments or surface waves on water, such as ripples moving on a pond. Sound waves in air and water are longitudinal. With sound waves, the disturbances are periodic variations in pressure that are transmitted in fluids. Fluids do not have appreciable shear strength, and for this reason, the sound waves in them are longitudinal waves. Sound in solids can have both longitudinal and transverse components, such as those in a seismic wave. Earthquakes generate seismic waves under Earth’s surface with both longitudinal and transverse components (called compressional or P-waves and shear or S-waves, respectively). The components of seismic waves have important individual characteristics—they propagate at different speeds, for example. Earthquakes also have surface waves that are similar to surface waves on water. Ocean waves also have both transverse and longitudinal components.

Wave on a String

A student takes a 30.00-m-long string and attaches one end to the wall in the physics lab. The student then holds the free end of the rope, keeping the tension constant in the rope. The student then begins to send waves down the string by moving the end of the string up and down with a frequency of 2.00 Hz. The maximum displacement of the end of the string is 20.00 cm. The first wave hits the lab wall 6.00 s after it was created. (a) What is the speed of the wave? (b) What is the period of the wave? (c) What is the wavelength of the wave?

  • The speed of the wave can be derived by dividing the distance traveled by the time.
  • The period of the wave is the inverse of the frequency of the driving force.
  • The wavelength can be found from the speed and the period [latex] v=\lambda \text{/}T. [/latex]
  • The first wave traveled 30.00 m in 6.00 s: [latex] v=\frac{30.00\,\text{m}}{6.00\,\text{s}}=5.00\frac{\text{m}}{\text{s}}. [/latex]
  • The period is equal to the inverse of the frequency: [latex] T=\frac{1}{f}=\frac{1}{2.00\,{\text{s}}^{-1}}=0.50\,\text{s}. [/latex]
  • The wavelength is equal to the velocity times the period: [latex] \lambda =vT=5.00\frac{\text{m}}{\text{s}}(0.50\,\text{s})=2.50\,\text{m}. [/latex]

Significance

The frequency of the wave produced by an oscillating driving force is equal to the frequency of the driving force.

Check Your Understanding

When a guitar string is plucked, the guitar string oscillates as a result of waves moving through the string. The vibrations of the string cause the air molecules to oscillate, forming sound waves. The frequency of the sound waves is equal to the frequency of the vibrating string. Is the wavelength of the sound wave always equal to the wavelength of the waves on the string?

The wavelength of the waves depends on the frequency and the velocity of the wave. The frequency of the sound wave is equal to the frequency of the wave on the string. The wavelengths of the sound waves and the waves on the string are equal only if the velocities of the waves are the same, which is not always the case. If the speed of the sound wave is different from the speed of the wave on the string, the wavelengths are different. This velocity of sound waves will be discussed in Sound .

Characteristics of a Wave

A transverse mechanical wave propagates in the positive x -direction through a spring (as shown in (Figure) (a)) with a constant wave speed, and the medium oscillates between [latex] \text{+}A [/latex] and [latex] \text{−}A [/latex] around an equilibrium position. The graph in (Figure) shows the height of the spring ( y ) versus the position ( x ), where the x -axis points in the direction of propagation. The figure shows the height of the spring versus the x -position at [latex] t=0.00\,\text{s} [/latex] as a dotted line and the wave at [latex] t=3.00\,\text{s} [/latex] as a solid line. (a) Determine the wavelength and amplitude of the wave. (b) Find the propagation velocity of the wave. (c) Calculate the period and frequency of the wave.

Figure shows two transverse waves whose y values vary from -6 cm to 6 cm. One wave, marked t=0 seconds is shown as a dotted line. It has crests at x equal to 2, 10 and 18 cm. The other wave, marked t=3 seconds is shown as a solid line. It has crests at x equal to 0, 8 and 16 cm.

Figure 16.6 A transverse wave shown at two instants of time.

  • The amplitude and wavelength can be determined from the graph.
  • Since the velocity is constant, the velocity of the wave can be found by dividing the distance traveled by the wave by the time it took the wave to travel the distance.
  • The period can be found from [latex] v=\frac{\lambda }{T} [/latex] and the frequency from [latex] f=\frac{1}{T}. [/latex]

Figure shows two transverse waves whose y values vary from -6 cm to 6 cm. One wave, marked t=0 seconds is shown as a dotted line. It has crests at x equal to 2, 10 and 18 cm. The other wave, marked t=3 seconds is shown as a solid line. It has crests at x equal to 0, 8 and 16 cm. The horizontal distance between two consecutive crests is labeled wavelength. This is from x=2 cm to x=10 cm. The vertical distance from the equilibrium position to the crest is labeled amplitude. This is from y=0 cm to y=6 cm. A red arrow is labeled distance travelled. This is from x=2 cm to x=8 cm.

Figure 16.7 Characteristics of the wave marked on a graph of its displacement.

  • The distance the wave traveled from time [latex] t=0.00\,\text{s} [/latex] to time [latex] t=3.00\,\text{s} [/latex] can be seen in the graph. Consider the red arrow, which shows the distance the crest has moved in 3 s. The distance is [latex] 8.00\,\text{cm}-2.00\,\text{cm}=6.00\,\text{cm}. [/latex] The velocity is [latex] v=\frac{\text{Δ}x}{\text{Δ}t}=\frac{8.00\,\text{cm}-2.00\,\text{cm}}{3.00\,\text{s}-0.00\,\text{s}}=2.00\,\text{cm/s}. [/latex]
  • The period is [latex] T=\frac{\lambda }{v}=\frac{8.00\,\text{cm}}{2.00\,\text{cm/s}}=4.00\,\text{s} [/latex] and the frequency is [latex] f=\frac{1}{T}=\frac{1}{4.00\,\text{s}}=0.25\,\text{Hz}. [/latex]

Note that the wavelength can be found using any two successive identical points that repeat, having the same height and slope. You should choose two points that are most convenient. The displacement can also be found using any convenient point.

The propagation velocity of a transverse or longitudinal mechanical wave may be constant as the wave disturbance moves through the medium. Consider a transverse mechanical wave: Is the velocity of the medium also constant?

  • A wave is a disturbance that moves from the point of origin with a wave velocity v .
  • A wave has a wavelength [latex] \lambda [/latex], which is the distance between adjacent identical parts of the wave. Wave velocity and wavelength are related to the wave’s frequency and period by [latex] v=\frac{\lambda }{T}=\lambda f. [/latex]
  • Mechanical waves are disturbances that move through a medium and are governed by Newton’s laws.
  • Electromagnetic waves are disturbances in the electric and magnetic fields, and do not require a medium.
  • Matter waves are a central part of quantum mechanics and are associated with protons, electrons, neutrons, and other fundamental particles found in nature.
  • A transverse wave has a disturbance perpendicular to the wave’s direction of propagation, whereas a longitudinal wave has a disturbance parallel to its direction of propagation.

Conceptual Questions

Give one example of a transverse wave and one example of a longitudinal wave, being careful to note the relative directions of the disturbance and wave propagation in each.

A sinusoidal transverse wave has a wavelength of 2.80 m. It takes 0.10 s for a portion of the string at a position x to move from a maximum position of [latex] y=0.03\,\text{m} [/latex] to the equilibrium position [latex] y=0. [/latex] What are the period, frequency, and wave speed of the wave?

What is the difference between propagation speed and the frequency of a mechanical wave? Does one or both affect wavelength? If so, how?

Propagation speed is the speed of the wave propagating through the medium. If the wave speed is constant, the speed can be found by [latex] v=\frac{\lambda }{T}=\lambda f. [/latex] The frequency is the number of wave that pass a point per unit time. The wavelength is directly proportional to the wave speed and inversely proportional to the frequency.

Consider a stretched spring, such as a slinky. The stretched spring can support longitudinal waves and transverse waves. How can you produce transverse waves on the spring? How can you produce longitudinal waves on the spring?

Consider a wave produced on a stretched spring by holding one end and shaking it up and down. Does the wavelength depend on the distance you move your hand up and down?

No, the distance you move your hand up and down will determine the amplitude of the wave. The wavelength will depend on the frequency you move your hand up and down, and the speed of the wave through the spring.

A sinusoidal, transverse wave is produced on a stretched spring, having a period T . Each section of the spring moves perpendicular to the direction of propagation of the wave, in simple harmonic motion with an amplitude A . Does each section oscillate with the same period as the wave or a different period? If the amplitude of the transverse wave were doubled but the period stays the same, would your answer be the same?

An electromagnetic wave, such as light, does not require a medium. Can you think of an example that would support this claim?

Storms in the South Pacific can create waves that travel all the way to the California coast, 12,000 km away. How long does it take them to travel this distance if they travel at 15.0 m/s?

Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?

[latex] 2d=vt⇒d=11.25\,\text{m} [/latex]

Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?

How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?

Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?

[latex] v=f\lambda ⇒\lambda =0.400\,\text{m} [/latex]

What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?

Radio waves transmitted through empty space at the speed of light [latex] (v=c=3.00\,×\,{10}^{8}\,\text{m/s}) [/latex] by the Voyager spacecraft have a wavelength of 0.120 m. What is their frequency?

Your ear is capable of differentiating sounds that arrive at each ear just 0.34 ms apart, which is useful in determining where low frequency sound is originating from. (a) Suppose a low-frequency sound source is placed to the right of a person, whose ears are approximately 18 cm apart, and the speed of sound generated is 340 m/s. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear? (b) Assume the same person was scuba diving and a low-frequency sound source was to the right of the scuba diver. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear, if the speed of sound in water is 1500 m/s? (c) What is significant about the time interval of the two situations?

(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

a. The P-waves outrun the S-waves by a speed of [latex] v=3.20\,\text{km/s;} [/latex] therefore, [latex] \text{Δ}d=0.320\,\text{km}. [/latex] b. Since the uncertainty in the distance is less than a kilometer, our answer to part (a) does not seem to limit the detection of nuclear bomb detonations. However, if the velocities are uncertain, then the uncertainty in the distance would increase and could then make it difficult to identify the source of the seismic waves.

A Girl Scout is taking a 10.00-km hike to earn a merit badge. While on the hike, she sees a cliff some distance away. She wishes to estimate the time required to walk to the cliff. She knows that the speed of sound is approximately 343 meters per second. She yells and finds that the echo returns after approximately 2.00 seconds. If she can hike 1.00 km in 10 minutes, how long would it take her to reach the cliff?

A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of [latex] f=25\,\text{kHz}. [/latex] The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?

  • OpenStax University Physics. Authored by : OpenStax CNX. Located at : https://cnx.org/contents/[email protected]:Gofkr9Oy@15 . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]

Footer Logo Lumen Candela

Privacy Policy

Library homepage

  • school Campus Bookshelves
  • menu_book Bookshelves
  • perm_media Learning Objects
  • login Login
  • how_to_reg Request Instructor Account
  • hub Instructor Commons
  • Download Page (PDF)
  • Download Full Book (PDF)
  • Periodic Table
  • Physics Constants
  • Scientific Calculator
  • Reference & Cite
  • Tools expand_more
  • Readability

selected template will load here

This action is not available.

Physics LibreTexts

24.1: Traveling Waves

  • Last updated
  • Save as PDF
  • Page ID 63295

  • Julio Gea-Banacloche
  • University of Arkansas via University of Arkansas Libraries

In our study of mechanics we have so far dealt with particle-like objects (objects that have only translational energy), and extended, rigid objects, which may also have rotational energy. We have, however, implicitly assumed that all the objects we studied had some internal structure, or were to some extent deformable, whenever we allowed for the possibility of their storing other forms of energy, such as chemical or thermal.

This chapter deals with a very common type of organized (as opposed to incoherent) motion exhibited by extended elastic objects, namely, wave motion . (Often, the “object” in which the wave motion takes place is called a “medium.”) Waves can be “traveling” or “standing,” and we will start with the traveling kind, since they are the ones that most clearly exhibit the characteristics typically associated with wave motion.

A traveling wave in a medium is a disturbance of the medium that propagates through it, in a definite direction and with a definite velocity. By a “disturbance” we typically mean a displacement of the parts that make up the medium, away from their rest or equilibrium position. The idea here is to regard each part of an elastic medium as, potentially, an oscillator, which couples to the neighboring parts by pushing or pulling on them (for an example of how to model this mathematically, see Advanced Topic 12.6 at the end of this chapter). When the traveling wave reaches a particular location in the medium, it sets that part of the medium in motion, by giving it some energy and momentum, which it then passes on to a neighboring part, and so on down the line.

You can see an example of how this works in a slinky. Start by stretching the slinky somewhat, then grab a few coils, bunch them up at one end, and release them. You should see a “compression pulse” traveling down the slinky, with very little distortion; you may even be able to see it being reflected at the other end, and coming back, before all its energy is dissipated away.

Figure12-1-1.png

The compression pulse in the slinky in Figure \(\PageIndex{1}\) is an example of what is called a longitudinal wave , because the displacement of the parts that make up the medium (the rings, in this case) takes place along the same spatial dimension along which the wave travels (the horizontal direction, in the figure). The most important examples of longitudinal waves are sound waves , which work a bit like the longitudinal waves on the slinky: a region of air (or some other medium) is compressed, and as it expands it pushes on a neighboring region, causing it to compress, and passing the disturbance along. In the process, regions of rarefaction (where the density drops below its average value) are typically produced, alongside the regions of compression (increased density).

The opposite of a longitudinal wave is a transverse wave , in which the displacement of the medium’s parts takes place in a direction perpendicular to the wave’s direction of travel. It is actually also relatively easy to produce a transverse wave on a slinky: again, just stretch it somewhat and give one end a vigorous shake up and down. It is, however, a little hard to draw the resulting pulse on a long spring with all the coils, so in Figure \(\PageIndex{2}\) below I have instead drawn a transverse wave pulse on a string , which you can produce in the same way. (Strings have other advantages: they are also easier to describe mathematically, and they are very relevant, particularly to the production of musical sounds.)

Figure12-1-2.png

Perhaps the most important (and remarkable) property of wave motion is that it can carry energy and momentum over relatively long distances without an equivalent transport of matter . Again, think of the slinky: the “pulse” can travel through the slinky’s entire length, carrying momentum and energy with it, but each individual ring does not move very far away from its equilibrium position. Ideally, after the pulse has passed through a particular location in the medium, the corresponding part of the medium returns to its equilibrium position and does not move any more: all the energy and momentum it momentarily acquired is passed forward. The same is (ideally) true for the transverse wave on the string in Figure \(\PageIndex{2}\).

Since this is meant to be a very elementary introduction to waves, I will consider only this case of “ideal” (technically known as “linear and dispersion-free”) wave propagation, in which the speed of the wave does not depend on the shape or size of the disturbance. In that case, the disturbance retains its “shape” as it travels, as I have tried to illustrate in figures \(\PageIndex{1}\) and \(\PageIndex{2}\).

The "Wave Shape" Function- Displacement and Velocity of the Medium

In a slinky, what I have been calling the “parts” of the medium are very clearly seen (they are, naturally, the individual rings); in a “homogeneous” medium (one with no visible parts), the way to describe the wave is to break up the medium, in your mind, into infinitely many small parts or “particles” (as we have been doing for extended systems all semester), and write down equations that tell us how each part moves. Physically, you should think of each of these “particles” as being large enough to contain many molecules, but small enough that its position in the medium may be represented by a mathematical point.

The standard way to label each “particle” of the medium is by the position vector of its equilibrium position (the place where the particle sits at rest in the absence of a wave). In the presence of the wave, the particle that was initially at rest at the point \(\vec{r}\) will undergo a displacement that I am going to represent by the vector \(\vec{\xi}\) (where \(\xi\) is the Greek letter “xi”). This displacement will in general be a function of time, and it may also be different for different particles, so it will also be a function of \(\vec{r}\), the equilibrium position of the particle we are considering. The particle’s position under the influence of the wave becomes then

\[ \vec{r}+\vec{\xi}(\vec{r}, t) \label{eq:12.1} .\]

This is very general, and it can be given a simpler form for simple cases. For instance, for a transverse wave on a string, we can label each part of the string at rest by its \(x\) coordinate, and then take the displacement to lie along the \(y\) axis; the position vector, then, could be written in component form as \( (x, \xi(x, t), 0) \). Similarly, we can consider a “plane” sound wave as a longitudinal wave traveling in the \(x\) direction, where the density of the medium is independent of \(y\) and \(z\) (that is, it is constant on planes perpendicular to the direction of propagation). In that case, the equilibrium coordinate \(x\) can be used to refer to a whole “slice” of the medium, and the position of that slice, along the \(x\) axis, at the time \(t\) will be given by \(x+\xi(x, t)\). In both of these cases, the displacement vector \(\xi\) reduces to a single nonzero component (along the \(y\) or \(x\) axis, respectively), which can, of course, be positive or negative. I will restrict myself implicitly to these simple cases and treat \(\xi\) as a scalar from this point on.

Under these conditions, the function \(\xi(x, t)\) (which is often called the wave function ) gives us the shape of the “displacement wave,” that is to say, the displacement of every part of the medium, labeled by its equilibrium \(x\)-coordinate, at any instant in time. Accordingly, taking the derivative of \(\xi\) gives us the velocity of the corresponding part of the medium:

\[ v_{\operatorname{med}}=\frac{d \xi}{d t} \label{eq:12.2} .\]

This is also, in general, a vector (along the direction of motion of the wave, if the wave is longitudinal, or perpendicular to it if the wave is transverse). It is also a function of time, and in general will be different from the speed of the wave itself , which we have taken to be constant, and which I will denote by \(c\) instead.

Harmonic Waves

An important class of waves are those for which the wave function is sinusoidal. This means that the different parts of the medium execute simple harmonic motion, all with the same frequency, but each (in general) with a different phase. Specifically, for a sinusoidal wave we have

\[ \xi(x, t)=\xi_{0} \sin \left[\frac{2 \pi x}{\lambda}-2 \pi f t\right] \label{eq:12.3} .\]

In Equation (\ref{eq:12.3}), \(f\) stands for the frequency, and plays the same role it did in the previous chapter: it tells us how often (that is, how many times per second) the corresponding part of the medium oscillates around its equilibrium position. The constant \(\xi_0\) is just the amplitude of the oscillation (what we used to call \(A\) in the previous chapter). The constant \(\lambda\), on the other hand, is sometimes known as the “spatial period,” or, most often, the wavelength of the wave: it tells you how far you have to travel along the \(x\) axis, from a given point \(x\), to find another one that is performing the same oscillation with the same amplitude and phase.

A couple of snapshots of a harmonic wave are shown in Figure \(\PageIndex{3}\). The figure shows the displacement \(\xi\), at two different times, and as a function of the coordinate \(x\) used to label the parts into which we have broken up the medium (as explained in the previous subsection). As such, the wave it represents could equally well be longitudinal or transverse. If it is transverse, like a wave on a string, then you can think of \(\xi\) as being essentially just \(y\), and then the displacement curve (the blue line) just gives you the shape of the string. If the wave is longitudinal, however, then it is a bit harder to visualize what is going on just from the plot of \(\xi (x, t)\). This is what I have tried to do with the density plots at the bottom of the figure.

Figure12-1-3.png

Imagine the wave is longitudinal, and consider the \(x = \pi\) point on the \(t\) = 0 curve (the first zero, not counting the origin). A particle of the medium immediately to the left of that point has a positive displacement, that is, it is pushed towards \(x = \pi\), whereas a slice on the right has a negative displacement—which means it is also pushed towards \(x = \pi\). We therefore expect the density of the medium to be highest around that point, whereas around \(x = 2\pi\) the opposite occurs: particles to the left are pushed to the left and those to the right are pushed to the right, resulting in a low-density region. The density plot labeled \(t\) = 0 attempts to show this using a grayscale where darker and lighter correspond to regions of higher and smaller density, respectively. At the later time \(t = \Delta t\) the high and low density regions have moved a distance \(c\Delta t\) to the right, as shown in the second density plot.

Regardless of whether the wave is longitudinal or transverse, if it is harmonic, the spatial pattern will repeat itself every wavelength; you can think of the wavelength \(\lambda\) as the distance between two consecutive crests (or two consecutive troughs) of the displacement function, as shown in the figure. If the wave is traveling with a speed \(c\), an observer sitting at a fixed point \(x\) would see the disturbance pass through that point, the particles move up and down (or back and forth), and the motion repeat itself after the wave has traveled a distance \(\lambda\), that is, after a time \(\lambda/c\). This means the period of the oscillation at every point is \(T = \lambda/c\), and the corresponding frequency \(f = 1/T = c/ \lambda\):

\[ f=\frac{c}{\lambda} \label{eq:12.4} .\]

This is the most basic equation for harmonic waves. Making use of it, Equation (\ref{eq:12.3}) can be rewritten as

\[ \xi(x, t)=\xi_{0} \sin \left[\frac{2 \pi}{\lambda}(x-c t)\right] \label{eq:12.5} .\]

This suggests that if we want to have a wave moving to the left instead, all we have to do is change the sign of the term proportional to \(c\), which is indeed the case.

In contrast to the wave speed, which is a constant, the speed of any part of the medium, with equilibrium position \(x\), at the time \(t\), can be calculated from Eqs. (\ref{eq:12.2}) and (\ref{eq:12.3}) to be

\[ v_{m e d}(x, t)=2 \pi f \xi_{0} \cos \left[\frac{2 \pi x}{\lambda}-2 \pi f t\right]=\omega \xi_{0} \cos \left[\frac{2 \pi x}{\lambda}-2 \pi f t\right] \label{eq:12.6} \]

(where I have introduced the angular frequency \(\omega = 2\pi f\)). Again, this is a familiar result from the theory of simple harmonic motion: the velocity is “90 degrees out of phase” with the displacement, so it is maximum or minimum where the displacement is zero (that is, when the particle is passing through its equilibrium position in one direction or the other).

Note that the result (\ref{eq:12.6}) implies that, for a longitudinal wave, the “velocity wave” is in phase with the “density wave” : that is, the medium velocity is large and positive where the density is largest, and large and negative where the density is smallest (compare the density plots in Figure \(\PageIndex{3}\)). If we think of the momentum of a volume element in the medium as being proportional to the product of the instantaneous density and velocity, we see that for this wave, which is traveling in the positive \(x\) direction, there is more “positive momentum” than “negative momentum” in the medium at any given time (of course, if the wave had been traveling in the opposite direction, the sign of \(v_{med}\) in Equation (\ref{eq:12.6}) would have been negative, and we would have found the opposite result). This confirms our expectation that the wave carries a net amount of momentum in the direction of propagation. A detailed calculation (which is beyond the scope of this book) shows that the time-average of the “momentum density” (momentum per unit volume) can be written as

\[ \frac{p}{V}=\frac{1}{2 c} \rho_{0} \omega^{2} \xi_{0}^{2} \label{eq:12.7} \]

where \(\rho_{0}\) is the medium’s average mass density (mass per unit volume). Interestingly, this result applies also to a transverse wave!

As mentioned in the introduction, the wave also carries energy. Equation (\ref{eq:12.6}) could be used to calculate the kinetic energy of a small region of the medium (with volume \(V\) and density \(\rho_{0}\), and therefore \(m=\rho_{0} V\)), and its time average. This turns out to be equal to the time average of the elastic potential energy of the same part of the medium (recall that we had the same result for harmonic oscillators in the previous chapter). In the end, the total time-averaged energy density (energy per unit volume) in the region of the medium occupied by the wave is given by

\[ \frac{E}{V}=\frac{1}{2} \rho_{0} \omega^{2} \xi_{0}^{2} \label{eq:12.8} .\]

Comparing (\ref{eq:12.7}) and (\ref{eq:12.8}), you can see that

\[ \frac{E}{V}=\frac{c p}{V} \label{eq:12.9} .\]

This relationship between the energy and momentum densities (one is just \(c\) times the other) is an extremely general result that applies to all sorts of waves, including electromagnetic waves!

The Wave Velocity

You may ask, what determines the speed of a wave in a material medium? The answer, qualitatively speaking, is that \(c\) always ends up being something of the form

\[ c \sim \sqrt{\frac{\text { stiffness }}{\text { inertia }}} \label{eq:12.10} \]

where “stiffness” is some measure of how rigid the material is (how hard it is to compress it or, in the case of a transverse wave, shear it), whereas “inertia” means some sort of mass density.

For a transverse wave on a string, for instance, we find

\[ c=\sqrt{\frac{F t}{\mu}} \label{eq:12.11} \]

where \(F^t\) is the tension in the string and \(\mu\) is not the “reduced mass” of anything (sorry about the confusion!), but a common way to write the “mass per unit length” of the string. We could also just write \(\mu = M/L\), where \(M\) is the total mass of the string and \(L\) its length. Note that the tension is a measure of the stiffness of the string, so this is, indeed, of the general form (\ref{eq:12.10}). For two strings under the same tension, but with different densities, the wave will travel more slowly on the denser one.

For a sound wave in a fluid (liquid or gas), the speed of sound is usually written

\[ c=\sqrt{\frac{B}{\rho_{0}}} \label{eq:12.12} \]

where \(\rho_0\) is the regular density (mass per unit volume), and \(B\) is the so-called bulk modulus , which gives the fluid’s resistance to a change in volume when a pressure \(P\) is applied to it: \(B = P/(\Delta V /V )\). So, once again, we get something of the form (\ref{eq:12.10}). In this case, however, we find that for many fluids the density and the stiffness are linked, so they increase together, which means we cannot simply assume that the speed of sound will be automatically smaller in a denser medium. For gases, this does work well: the speed of sound in a lighter gas, like helium, is greater than in air, whereas in a denser gas like sulfur hexafluoride the speed of sound is less than in air 1 . However, if you compare the speed of sound in water to the speed of sound in air, you find it is much greater in water, since water is much harder to compress than air: in this case, the increase in stiffness more than makes up for the increase in density.

The same thing happens if you go from a liquid like water to a solid, where the speed of sound is given by

\[ c=\sqrt{\frac{Y}{\rho_{0}}} \label{eq:12.13} \]

where \(Y\) is, again, a measure of the stiffness of the material, called the Young modulus . Since a solid is typically even harder to compress than a liquid, the speed of sound in solids such as metals is much greater than in water, despite their being also denser. For reference, the speed of sound in steel would be about \(c\) = 5,000 m/s; in water, about 1,500 m/s; and in air, “only” about 340 m/s.

1 This effect can be used to produce “funny voices,” because of the relationship \(f = c/\lambda\) (Equation (\ref{eq:12.4})), which will be discussed in greater detail in the section on standing waves.

Reflection and Transmission of Waves at a Medium Boundary

Suppose that you have two different elastic media, joined in some way at a common boundary, and you have a wave in the first medium traveling towards the boundary. Examples of media connected this way could be two different strings tied together, or two springs with different spring constants joined at the ends; or, for sound waves, it could just be something like water with air above it: a compression wave in air traveling towards the water surface will push on the surface and set up a sound wave there, and vice-versa.

The first thing to notice is that, if the incident wave has a frequency \(f\), it will cause the medium boundary, when it arrives there, to oscillate at that frequency. As a result of that, the wave that is set up in the second medium—which we call the transmitted wave —will also have the same frequency \(f\). Again, think of the two strings tied together, so the first string “drives” the second one at the frequency \(f\); or the sound at the air-water boundary, driving (pushing) the water surface at the frequency \(f\).

So, the incident and transmitted waves will have the same frequency, but it is clear that, if the wave speeds in the two media are different, they cannot have the same wavelength: since the relation (\ref{eq:12.4}) has to hold, we will have \(\lambda_1 = c_1/f\), and \(\lambda_2 = c_2/f\). Thus, if a periodic wave goes from a slower to a faster medium, its wavelength will increase, and if it goes from a faster to a slower one, the wavelength will decrease.

It is easy to see physically why this happens, and how it has to be the case even for non-periodic waves, that is, wave pulses: a pulse going into a faster medium will widen in length (stretch), whereas a pulse going into a slower medium will become narrower (squeezed). Imagine, for example, several people walking in line, separated by the same distance \(d\), all at the same pace, until they reach a line beyond which they are supposed to start running. When the first person reaches the line, he starts running, but the second one is still walking, so by the time the second one reaches the line the first one has increased his distance from the second. The same thing will happen between the second and the third, and so on: the original “bunch” will become spread out. (If you watch car races, chances are you have seen this kind of thing happen already!)

Besides setting up a transmitted wave, with the properties I have just discussed, the incident wave will almost always cause a reflected wave to start traveling in the first medium, moving backwards from the boundary. The reflected wave also has the same frequency as the incident one, and since it is traveling in the same medium, it will also have the same wavelength. A non-periodic pulse, when reflected, will therefore not be stretched or squeezed, but it will be “turned around” back-to-front, since the first part to reach the boundary also has to be the first to leave. See Figure \(\PageIndex{4}\) (the top part) for an example.

Figure12-1-4.png

What is the physical reason for the reflected wave? Ultimately, it has to do with the energy carried by the incident wave, and whether it is possible for the transmitted wave alone to handle the incoming energy flux or not. As we saw earlier (Equation (\ref{eq:12.8})), the energy per unit volume in a harmonic wave of angular frequency \(\omega\) and amplitude \(\xi_0\) is \(E/V = \frac{1}{2} \rho_{0}\omega^{2}\xi^{2}_{0}\). If the wave is traveling at a speed \(c\), then the energy flux (energy transported per unit time per unit area) is equal to \((E/V )c\), which is to say

\[ I=\frac{1}{2} c \rho_{0} \omega^{2} \xi_{0}^{2} \label{eq:12.14} .\]

This is often called the intensity of the wave. It can be written as \(I=\frac{1}{2} Z \omega^{2} \xi_{0}^{2}\), where I have defined the medium’s mechanical impedance (or simply the impedance ) as

\[ Z = c \rho_{0} \label{eq:12.15} \]

(for a string, the mass per unit length \(\mu\) instead of the mass per unit volume \(\rho_0\) should be used). You can see that if the two media have the same impedance, then the energy flux in medium 2 will exactly match that in medium 1, provided the incident and transmitted waves have the same amplitudes. In that case, there will be no reflected wave: even if the two media have different densities and wave velocities, as long as they have the same impedance, the wave will be completely transmitted.

On the other hand, if the media have different impedances, then it will in general be impossible to match the energy flux with only a transmitted wave, and reflection will occur. This is not immediately obvious, since it looks like all you have to do, to compensate for the different impedances in Equation (\ref{eq:12.14}), is to give the transmitted wave an amplitude that is different from that of the incident wave. But the point is precisely that, mathematically, you cannot do that without introducing a reflected wave. This is because the actual amplitude of the oscillation at the boundary has to be the same on both sides, since the two media are connected there, and oscillating together; so, if \(\xi_{0, \text { inc }}\) is going to be different from \(\xi_{0, \text { trans }}\), you need to have another wave in medium 1, the reflected wave, to insure that \(\xi_{0, \text { inc }}+\xi_{0, \text { refl }}=\xi_{0, \text { trans }}\).

Another way to see this is to dig in a little deeper into the physical meaning of the impedance. This is a worthwhile detour, because impedance in various forms recurs in a number of physics and engineering problems. For a sound wave in a solid, for instance, we can see from Eqs. (\ref{eq:12.13}) and (\ref{eq:12.15}) that \(Z = c\rho_0 = \sqrt{Y \rho_0}\); so a medium can have a large impedance either by being very stiff (large \(Y\)) or very dense (large \(\rho_0\)) or both; either way, one would have to work harder to set up a wave in such a medium than in one with a smaller impedance. On the other hand, once the wave is set up, all that work gets stored as energy of the wave, so a wave in a medium with larger \(Z\) will also carry a larger amount of energy (as is also clear from Equation (\ref{eq:12.14})) 2 for a given displacement \(\xi_0\).

So, when a wave is trying to go from a low impedance to a large impedance medium, it will find it hard to set up a transmitted wave: the transmitted wave amplitude will be small (compared to that of the incident wave), and the only way to satisfy the condition \(\xi_{0, \text { inc }}+\xi_{0, \text { refl }}=\xi_{0, \text { trans }}\) will be to set up a reflected wave with a negative amplitude 3 —in effect, to flip the reflected wave upside down, in addition to left-to-right. This is the case illustrated in the bottom drawing in Figure \(\PageIndex{4}\).

Conversely, you might think that a wave trying to go from a high impedance to a low impedance medium would have no trouble setting up a transmitted wave there, and that is true—but because of its low impedance, the transmitted wave will still not be able to carry all the energy flux by itself. In this case, \(\xi_{0,trans}\) will be greater than \(\xi_{0,inc}\), and this will also call for a reflected wave in the first medium, only now it will be “upright,” that is, \(\xi_{0, \text { refl }}=\xi_{0, \text { trans }}-\xi_{0, \text { inc }}>0\).

To finish up the subject of impedance, note that the observation we just made, that impedance will typically go as the square root of the product of the medium’s “stiffness” times its density, is quite general. Hence, a medium’s density will typically be a good proxy for its impedance, at least as long as the “stiffness” factor is independent of the density (as for strings, where it is just equal to the tension) or, even better, increases with it (as is typically the case for sound waves in most materials). Thus, you will often hear that a reflected wave is inverted (flipped upside down) when it is reflected from a denser medium, without any reference to the impedance—it is just understood that “denser” also means “larger impedance” in this case. Also note, along these lines, that a “fixed end,” such as the end of a string that is tied down (or, for sound waves, the closed end of an organ pipe), is essentially equivalent to a medium with “infinite” impedance, in which case there is no transmitted wave at that end, and all the energy is reflected.

Finally, the expression \(\xi_{0,inc} + \xi_{0,refl}\) that I wrote earlier, for the amplitude of the wave in the first medium, implicitly assumes a very important property of waves, which is the phenomenon known as interference , or equivalently, the “linear superposition principle.” According to this principle, when two waves overlap in the same region of space, the total displacement is just equal to the algebraic sum of the displacements produced by each wave separately. Since the displacements are added with their signs, one may get destructive interference if the signs are different, or constructive interference if the signs are the same. This will play an important role in a moment, when we start the study of standing waves.

2 In this respect, it may help you to think of the impedance of an extended medium as being somewhat analog to the inertia (mass) of a single particle. The larger the mass, the harder it is to accelerate a particle, but once you have given it a speed v, the larger mass also carries more energy.

3 A better way to put this would be to say that the amplitude is positive as always, but the reflected wave is 180\(^{\circ}\) out of phase with the incident wave, so the amplitude of the total wave on the medium 1 side of the boundary is \(\xi_{0,inc} − \xi_{0,refl}\).

Accessibility Links

  • Skip to content
  • Skip to search IOPscience
  • Skip to Journals list
  • Accessibility help
  • Accessibility Help

Click here to close this panel.

Purpose-led Publishing is a coalition of three not-for-profit publishers in the field of physical sciences: AIP Publishing, the American Physical Society and IOP Publishing.

Together, as publishers that will always put purpose above profit, we have defined a set of industry standards that underpin high-quality, ethical scholarly communications.

We are proudly declaring that science is our only shareholder.

Recent theory of traveling-wave tubes: a tutorial-review

Patrick Wong 1 , Peng Zhang 1 and John Luginsland 2

Published 3 June 2020 • © 2020 IOP Publishing Ltd Plasma Research Express , Volume 2 , Number 2 Citation Patrick Wong et al 2020 Plasma Res. Express 2 023001 DOI 10.1088/2516-1067/ab9730

Article metrics

3873 Total downloads

Permissions

Get permission to re-use this article

Share this article

Author e-mails.

[email protected]

Author affiliations

1 Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, United States of America

2 Confluent Sciences, LLC, Albuquerque, NM 87111, United States of America

Patrick Wong https://orcid.org/0000-0002-8437-6990

Peng Zhang https://orcid.org/0000-0003-0606-6855

John Luginsland https://orcid.org/0000-0002-6130-6156

  • Received 3 March 2020
  • Revised 8 April 2020
  • Accepted 27 May 2020
  • Published 3 June 2020

Peer review information

Method : Single-anonymous Revisions: 1 Screened for originality? Yes

Buy this article in print

The traveling-wave tube (TWT), also known as the traveling-wave amplifier (TWA) or traveling-wave tube amplifier (TWTA), is a widely used amplifier in satellite communications and radar. An electromagnetic signal is inputted on one end of the device and is amplified over a distance until it is extracted downstream at the output. The physics behind this spatial amplification of an electromagnetic wave is predicated on the interaction of a linear DC electron beam with the surrounding circuit structure. Pierce, known as the 'father of communications satellites,' was the first to formulate the theory for this beam-circuit interaction, the basis of which has since been used to model other vacuum electronic devices such as free-electron lasers, gyrotrons, and Smith-Purcell radiators, just to name a few. In this paper, the traditional Pierce theory will first be briefly reviewed ; the classic Pierce theory will then be extended in several directions: harmonic generation and the effect of high beam current on both the beam mode and circuit mode as well as 'discrete effects', giving a brief tutorial of recent theories of TWTs.

Export citation and abstract BibTeX RIS

This article was updated on 22 April 2021 to add permission lines to the figures.

Introduction

Throughout the history of commercial and defense applications involving electromagnetics, there has always been a high demand for microwave and millimeter wave amplifiers that offer both high power output and wide bandwidth. One candidate that may fulfill these stringent requirements is the traveling-wave tube (TWT), also known as the traveling-wave amplifier (TWA) or traveling-wave tube amplifier (TWTA). This device is widely used in satellite communications and radar applications.

It turns out that a viable method of amplifying a given electromagnetic wave is by passing this signal through a periodic structure and co-propagating it with a linear DC electron beam. To reduce complexities, this beam-circuit interaction takes place in vacuum. In such a set-up, the traveling wave gains energy at the expense of the kinetic energy of the electron beam. This continuous interaction and subsequent amplification of the electromagnetic wave happens over a distance (the length of the interaction region of the TWT) until the amplified wave is extracted at the output.

Before the invention of the TWT, the klystron amplifier (1935) was one of the first microwave amplifiers used [ 1 – 4 ]. This device though had several limitations. The most notable one is that the klystron had narrow bandwidth, as amplification is restricted to the resonant frequencies of the klystron cavities. The idea then came about to couple the individual cavities of a multi-cavity klystron to a common transmission line so that there may be a continuous in-phase interaction between beam and circuit [ 3 , 4 ]. This eventually led to the development of the traveling-wave tube amplifier. Figure 1 shows the evolution of the klystron to the traveling-wave tube.

Figure 1.

Figure 1.  Schematic diagram showing the evolution of a multi-cavity klystron (a) to a traveling-wave tube amplifier (c). (b) shows the multi-cavity klystron with its cavities coupled to a common transmission line.

Download figure:

Rudolph Kompfner of England, interestingly an architect by profession, was the first to propose the idea of a traveling-wave tube. He and Nils Lindenblad of the United States used a metallic helix as the circuit structure for propagating the signal in phase with the centered pencil electron beam in vacuum and are credited with being the first to create the TWT as it is known today [ 5 , 6 ]. Prior to this, Haeff proposed a similar idea but had the electron beam on the outside of the circuit, leading to poorer efficiency [ 7 ]. Kompfner's invention of the TWT then aroused the intense interest of John R Pierce who then laid the foundation of the TWT theory and was later known as the 'father of communications satellites' [ 8 ]. A schematic diagram of a modern TWT is shown in figure 2 .

Figure 2.

Figure 2.  Schematic diagram of a traveling-wave tube amplifier. There are many components to a TWTA. In this paper, we will concentrate only on the middle section, the helix, of the above figure: the beam-circuit interaction region. The slow-wave circuit shown here is a helix (with support rods). Image from [ 9 ]. This diagram of helix TWT has been obtained by the author(s) from the Wikipedia website, where it is stated to have been released into the public domain. It is included within this article on that basis.

As can be seen in figure 2 , an actual TWT is a complex device, consisting of components (left: cathode, electron gun, etc) that create and form the electron beam, the beam-circuit interaction region (middle: helix, electron beam, etc), and components (right: collector) that collect the 'spent' beam. Each of these topics in and of themselves constitute a library of study. In this paper, the focus will be restricted to the physics in the beam-circuit interaction region within the helix (figure 2 ). In particular, the standard beam-circuit mode coupling theory of Pierce will be examined. Using the Pierce theory as a basis, we will then extend it to describe effects originally neglected by Pierce: harmonic generation and the effects of high beam current on the circuit as well as connecting the continuous wave picture of Pierce to a discrete circuit formulation.

To begin, it must be emphasized that the operation of a TWT, or virtually any beam-driven microwave source, is predicated on the interaction of an electron beam in vacuum with an in-phase, co-propagating electromagnetic wave on a circuit structure. As such, it is instructive to analyze each of these two components: beam and circuit separately and then describe their interaction, as was also done by Pierce [ 10 ]. This method of analysis and mode coupling is not restricted to TWTs, rather, the techniques presented here are quite general and applicable to all beam-circuit interactions between charged particles and electromagnetic radiation.

The electron beam

As can be seen in figure 2 , the electron beam is formed outside of the beam-circuit interaction region. Electrons are boiled off from a thermionic cathode (thermionic emission) or emitted from a material using strong electric fields (field emission). In either scenario, a voltage is externally applied to the cathode and this is essentially the kinetic energy of the beam entering the interaction region. We will denote the kinetic energy of this beam by voltage V b , corresponding to a DC beam velocity v 0 . Invoking conservation of energy in the non-relativistic regime,

For simplicity, it is assumed that there are no temperature effects on the beam so that the beam is 'cold' or mono-energetic (introducing temperature effects will be slightly more complicated but will essentially introduce a spread in electron velocities centered around a mean value). We further assume that the DC beam motion is one-dimensional, i.e., we implicitly assume that there is an infinite axial magnetic field confining the beam. In reality, this is provided by an external solenoid or periodic permanent magnets.

The two externally adjustable parameters of the beam: the DC beam voltage V b and the DC beam current I 0 control the unperturbed electron beam velocity v 0 and the electron charge density ρ 0 , respectively. In the absence of any perturbation, the electron beam travels through vacuum in the axial direction with these unperturbed properties.

The electromagnetic circuit

There are of course other SWS designs for different purposes. A listing and description of some of these SWS designs can be found in [ 11 ]. Traditionally speaking, all of the structures are metallic, but much effort has, in recent years, been placed in studying metamaterial SWS's [ 12 , 13 ]. These have very different dispersion characteristics than their metallic counterparts leading to different gain properties. However, their beam-circuit interaction may still be captured by the mode-coupling theory of Pierce (in the small-signal regime); the physics and mathematics of the beam and circuit coupling remains the same. Different beam configurations (e.g. pencil beam versus sheet beam versus annular beam) will also play a role in the gain characteristics of the tube (see [11a] for a comparison between pencil and annular beams and [11b] for a discussion comparing sheet beams, an 'unwrap' of annular beams, to pencil beams; [ 14 ] talks about the plasma frequency reduction factor for different beam configurations, a topic we will come back to soon). Still to this date though, the helix SWS coupled with a pencil electron beam remains the 'go-to' for high power microwave amplification because of its wide bandwidth and ease of construction.

The reason why a helix TWT may offer a wide bandwidth is qualitatively illustrated in figure 3 .

Figure 3.

Figure 3.  (a) Depiction of the TEM fields of a thin wire over a perfectly conducting metal plate. (b) The formation of a helix TWT from 'wrapping' the set-up in (a).

where E C is the circuit electric field, Γ 0 is the effective propagation constant of this circuit wave (= β ph with an imaginary part to account for attenuation in the circuit), and K is known as the interaction impedance (essentially a pure geometrical quantity that is a proportionality constant between the voltage and current of the 'cold' no beam system). Assuming time and spatial harmonic fields, equation ( 3 ) may be written as:

β 0 (or just β in the literature) is called the fundamental, and β n ( n ≠ 0) is called the n th space harmonic.

Figure 4.

What we have allured to but have neglected to discuss thus far is the space-charge electric field E SC coming from the beam, with good reason. Space-charge calculations in general are notoriously difficult, and there is no general model that can capture this internal self-force of the beam. It turns out, see the section 'Beyond Pierce', that one model of space-charge effects is intimately tied to the higher-order passbands of the circuit and the higher-order beam modes. This higher-order interaction was not initially captured by Pierce as his analysis was concerned with the interaction of the fundamental passband and beam mode only. It should be noted though that he did hypothesize the role of higher-order passbands of the circuit on space charge; for now, we will group space-charge effects under the term 'QC' or 'Q', following Pierce's original notation. Thus, accessing space-charge effects in TWTs boils down to finding an expression for 'Q', known as Pierce's AC space-charge parameter.

The Beam-Circuit Interaction and Pierce's theory of mode-coupling

Now that we have described the beam and circuit separately, how do they interact in such a way to produce amplification of the injected signal? When the beam enters the beam-circuit interaction region and feels the circuit wave, the electrons in the beam will respond to the sinusoidal nature of the electric field of the signal. Consequently, one can imagine that some electrons will be accelerated by the accelerating portion of the wave and some electrons will be decelerated by the decelerating portion of the wave. This leads to the formation of electron bunches in the electron beam. These bunches in turn cause the electric field in the circuit to increase by inducing more current in the circuit; electron motion inducing currents in external circuits can be attributed to the famous Shockley-Ramo Theorem [ 20 ]. The resulting amplitude increase in the electric field in the circuit causes more bunching in the beam. The physical mechanism is succinctly illustrated in Gilmour's book [ 1 , 2 ].

From a wave mechanics picture, a pair of space-charge waves on the electron beam are created from the beam interacting with the circuit wave. These space-charge waves on the beam are analogous to longitudinal pressure waves in air consisting of compressions and rarefactions. In this case, we have the fast and slow space-charge waves that co-propagate on the electron beam with the circuit wave. The energy difference from the slowing-down of the electrons in the beam to the slow space-charge wave phase velocity is given to the circuit, causing amplification of the circuit wave. This beam-circuit interaction has proven to be effective: the growth of the input signal, at least in the small-signal regime, is exponential with distance along the tube [ 1 , 2 ]. TWT power gain of 60 dB (1 million times) can be realized [ 21 ].

As stated before, the classical theory of beam-structure interactions in a TWT was developed by Pierce [ 10 ], whose treatment also provided the foundations for the understanding and design of a wide range of contemporary sources such as free-electron lasers [ 4 , 22 – 24 ], Smith-Purcell radiators [ 4 , 23 – 26 ], gyrotron amplifiers [ 4 , 26 – 30 ], metamaterials TWTs [ 12 , 13 , 31 ], and NonLinear Transmission Line (NLTL) based sources [ 32 ]. We will present the standard Pierce theory and further develop it to indicate other novel effects not previously considered by Pierce. Pierce described the energy transfer mechanism in terms of the interaction between the space-charge waves on the electron beam and the electromagnetic mode supported by the electromagnetic circuit. Amplification of a signal of frequency ω is described by the complex wavenumber β that is a solution of what is known as the Pierce dispersion relation, which, in its most basic form [ 1 , 3 , 4 , 10 , 13 , 22 – 24 , 26 – 30 ], is a third-degree polynomial for β(ω) . This dispersion relation describes the coupling between the beam mode and the circuit mode [ 33 , 34 ]. It has been used in the validation of non-linear, large-signal numerical codes in the small-signal regime.

An interesting side note is that unlike most topics, such as plasma physics and electromagnetic wave theory, here we consider the propagation constant as a function of the frequency. That is, one is generally given a frequency (the frequency of the input signal to be amplified) and is tasked with finding the corresponding propagation constant, which is in general complex. The imaginary part corresponds to either the growth (amplification) or decay (attenuation) of the wave in the system.

We are now in a position to self-consistently solve the governing equations for the beam (equations ( 1 ), ( 2 )) and circuit (equation ( 3 )) to yield solutions describing the evolution of both, as a coupled system. With some manipulation, the three equations can be combined to yield:

To simplify matters (this was back in the day when solving quartic polynomials was a chore!), a common approximation is to assume that C is small and to neglect the backward circuit wave, as it is primarily the forward waves that contribute to the gain of a TWT downstream. Doing so and using the notation of Pierce, the so-called 3-wave dispersion relation reads:

For comparison purposes (c.f. equation ( 5 ') below), Pierce's 3-wave dispersion relation for β reads:

It should be noted that reflection from the ends and waves near the band edges cannot be accounted for by Pierce's 3-wave theory. A classical look into Pierce's 4-wave theory and its reduction to the 3-wave theory is provided by Birdsall and Brewer [ 35 ]. The importance of the 4-wave description, and the effects of reflection on the TWT stability may be found in [ 36 – 38 ].

Beyond pierce

The above was a tour-de-force through electron beam dynamics and electromagnetic wave theory culminating in a simple yet powerful description of beam-circuit mode coupling and interaction in a TWT. One must remember that this type of analysis was done back in the early to mid-20th century. Many advancements in terms of modeling and mathematics and the advent of scientific supercomputing have taken analyses of such devices to new heights. Consequently, more complex and accurate descriptions on the inner workings of a TWT exist. Nevertheless, there is still a subtle charm to a simple theory. Not only can it give a 'back-of-the-envelope' assessment of a given problem but it can provide great physical insights into the nature of such interactions and processes. In this vein, we will now attempt to relax some of the assumptions of Pierce and extend his theory in a simple manner and draw further insights into the beam-circuit interaction. The topics covered here will be harmonic generation and space charge in TWTs as well as 'discrete cavity' effects, relating TWT analysis to klystron analysis. A brief synopsis of the first two topics are provided in the following paragraphs.

The classical Pierce theory was formulated for a single (fundamental) frequency, that of the input signal. However, in a TWT with an octave bandwidth or greater, in particular the widely used helix TWT, the second harmonic of the input signal may also be within the amplification band and thus may also be generated and amplified, with no input at this second harmonic frequency. An extension to the Pierce formulation that incorporates the generation of harmonics will be presented here. It is shown that the second harmonic arises mostly from a newly discovered dynamic synchronous interaction instead of by the kinematic orbital crowding mechanism that is the most dominant harmonic generation mechanism in other microwave devices. The methodology provided here, which is a natural extension of Pierce's original theory, may be applicable not only to TWTs but to other high-power microwave sources.

In beam-circuit interactions, the space-charge effect of the beam is important at high beam currents. In Pierce's TWT theory, as stated before, this space-charge effect is modelled by the parameter which he called Q in the beam mode. A reliable determination of Q remains elusive for a realistic TWT. Previously, Wong et al [ 39 ] constructed the first exact small-signal theory for the beam-circuit interaction for the tape helix TWT, from which Q may be unambiguously determined. In the process of doing so, it was discovered that the circuit mode in Pierce's theory must also be modified at high beam current, an aspect overlooked in Pierce's original analysis. This circuit mode modification is quantified by an entirely new parameter called q , introduced for the first time in TWT theory. For the example using a realistic tape helix TWT, we find that the effect of q is equivalent to a modification of the circuit phase velocity by as much as two percent, which is a very significant effect. A brief summary of q will be provided here.

Harmonic generation in TWTs

The subject of harmonic generation in a TWT has traditionally been studied in the non-linear, large-signal regime. We will not delve into large-signal theory here as it is beyond the scope of this text. The theory itself is more complete as it can not only capture non-linear phenomena such as saturation, wave-trapping of beam electrons, etc. but can also recover the linear, small-signal regime. However, it is much more complicated than the relatively simple linear theory presented here thus far and simple insights, which we stress here, cannot be easily drawn. We will note here that large-signal TWT theory and also the study of harmonics of the input signal can be traced back to the classical paper by Nordsieck [ 40 ], who also provided the first analysis of TWT efficiency. References may be made to Tien et al , Rowe, Giarola, and Dionne [ 41 ] whose subsequent works relied heavily on Nordsieck's initial theory.

Taking a step back, it is well-known in klystrons that the dominant cause of harmonics of the input signal to be generated is due to linearized orbital motion from an input signal leading to orbital crowding, which leads to harmonic generation kinematically [ 3 , 42 ] (see figure 5 below). In such scenarios, the extreme case of charge overtaking may also occur and contribute to harmonic generation. It was not until recently that such a theory on harmonic content in the beam current of a TWT was developed by Dong et al [ 43 ] in the small-signal regime. In that work, the linearized electron orbits might lead to a second harmonic AC current as high as 25% of the DC beam current, and this was favorably compared to the large-signal TWT code CHRISTINE [ 44 ].

Figure 5.

Figure 5.  An (exaggerated) illustration of harmonic generation due to orbital crowding. Crowding in the linear orbits may lead to harmonic current generation, as shown in klystron theory [ 3 ] and TWT theory [ 43 ].

A little after Dong et al 's paper, another source of harmonic content in a TWT was discovered: weak non-linearities in the electron orbits. It turns out, with respect to the RF power output, that this source of harmonic generation is much more important as it possesses the property of synchronism between the beam and circuit in both space and time , the underpinning concept behind the operation of TWTs. The effect of orbital crowding described in the preceding paragraph is negligible in comparison [ 45 ].

The main idea behind this alternative view on harmonic generation in TWTs stems from carrying out higher-order expansions of the governing equations for the beam [recall: Pierce's small-signal analysis is a linear theory; this is the answer to the above question posed in Footnote 1]. These minor corrections on the quantities of the beam corrects the linear quantities of Pierce [ 46 ] but also turn out to be the harmonics; basically, we regard the quantities of interest: the electric field (potential), beam density and velocity (current) to be represented by a Fourier series. This interpretation then is a quasi-linear theory.

The electromagnetic signal causes the electrons in the beam to bunch. To first order, these bunches are at the frequency of the signal itself, leading to Pierce's linear theory. The orbits of the beam electrons though are better represented at higher orders and thus contain more frequencies of the input signal. In response, the circuit picks up each of these frequencies individually (Maxwell's Equations are linear). The circuit then not only supports an amplifying signal at the fundamental (input) frequency but also a spectrum of harmonics of the original signal, all being spatially amplified. This of course can only happen if the spectrum of frequencies is within the bandwidth of the TWT: any frequency outside of the characteristic bandwidth will not be supported by the structure. The ubiquitous helix structure is one such circuit structure that can have octave bandwidth, as evidenced by figure 4 above.

Space-charge effects in TWTS

Let us now go back to the notorious problem of space-charge effects, or finding a closed form expression for Q in the Pierce picture. Many theories have been proposed that attempt to give a general formulation for calculating Q for a general beam and circuit structure. A treatise of the subtlety of this problem is given by Lau and Chernin [ 49 ], who ultimately advance the idea that Q is due to the interaction of the beam with the higher-order circuit modes (passbands higher than the fundamental). This interpretation of Q originating from residual interactions between beam and circuit, the 'remainder,' was actually also proposed by Pierce [ 10 ], who quickly abandoned such an interpretation [ 49 ], adhering instead to what the standard Pierce theory is now: the interaction between the fundamental circuit and beam modes.

Physically speaking, the calculation of the space-charge parameter boils down to calculating the 'reduced' plasma frequency ω q of the beam. Like a plasma, the beam, which can be considered to be a non-neutral plasma, has a natural frequency called the electron beam plasma frequency, which is the frequency of the space-charge waves in an unbounded medium. However, in reality, the beam is bounded in the sense that it has a definite shape and is enclosed by the finite circuit structure. Thus, the plasma frequency of the beam is 'reduced'. In this view, the task is to calculate the reduction factor to the plasma frequency that accounts for the circuit structure (including the SWS, taking into account, for example, the 'field leakage' in the opened sections of a helix). It turns out that this calculation is intimately tied to the higher-order circuit modes mentioned in the preceding paragraph. Before we delve deeper into the mystery of space charge, it is instructive to first understand the prior art.

A study using the pedagogical model of a dielectric TWT that consists of a planar dielectric slab and sheet electron beam was done by Simon et al [ 51 ]. In that model, where an exact 'hot tube (including the beam)' dispersion relation may be readily derived, the idea of the higher-order circuit modes yielding Q was established conclusively. In addition, that paper showed how to accurately evaluate the Pierce parameters once a closed analytic form was found. This model however is deficient in that there is no periodic slow-wave structure in this dielectric TWT (the dielectric slows down the wave), so higher harmonics in the beam mode are excluded.

If, however, we may find an exact dispersion relation for the circuit structure including all of the geometric complexities and the beam dynamics for a realistic TWT, then possibly we stand a chance at determining an expression for Q . As in [ 49 ] and [ 51 ], we may cast the derived analytic dispersion relation into the form of Pierce (equations ( 5 ) or ( 6 )) and extract an exact expression for the Pierce parameters. In [ 15 ], a formally exact treatment of the tape helix TWT without the electron beam (the 'cold-tube' dispersion relation) was presented by Chernin et al Later, in [ 52 ], this analysis was expanded on by the inclusion of the electron beam, giving a formally exact 'hot-tube' dispersion relation. However, because of the complexity of the analytical dispersion relation, it was unfeasible to rewrite it in a Pierce-like form. The alternative then was to numerically solve the dispersion relation and 'backtrack' to find what the Pierce parameters ought to be. In the process of doing so, it was found that it was necessary to introduce a new parameter, termed q , in order for the equations to be satisfied [ 39 ].

This new parameter q turns out to modify the circuit mode in much the same way that the original space-charge parameter Q that we were looking for modifies the beam mode, shown here for the 3-wave dispersion relation:

Because of this, the interpretation of q has been attributed to the effects of space charge of the beam on the circuit mode, i.e. beam loading of the circuit. This view is supported by the fact that q increases with beam current [ 39 ]. From a dispersion diagram perspective, much like how Q arises from the interaction of the fundamental beam mode with the 'remainder' of the circuit modes apart from the fundamental circuit mode, q arises from the interaction of the circuit mode with the beam modes of order higher than the fundamental, completing the symmetry. Because the dielectric TWT discussed above does NOT contain any periodic components that make up the slow-wave structure, the new parameter q  = 0 in this model as there are no spatial harmonics of the beam [ 39 ].

To demonstrate the effects of this q term, a plot of the roots of equation ( 6 a ) using the standard definition of C and ubiquitous models of Q (Branch and Mihran and sheath helix [ 14 , 44 ]) are given in figure 6 below along with the roots of the exact hot-tube dispersion relation for a tape helix. The three roots represent the resultant three waves from the beam-circuit interaction in the system: a neutral root, a decaying (complex with negative imaginary part) root, and an amplifying (complex with positive imaginary part) root. We, of course, care about the amplifying part as it represents the linear gain of the TWT. The other two waves however are also important as they constitute what is known as the launching loss of the TWT (where the remainder of the initial power of the signal is channeled to). Note that adding the fourth root (from solving the original 4-wave equation ( 5 )) representing the backward propagating wave will change the results. Qualitatively, the three (main) roots will remain the same, and the fourth root will be a neutral root with a negative propagation constant.

Figure 6.

Figure 6.  Plot of the three roots (propagation constant β ) as determined from: Pierce theory with Q modeled by Branch and Mihran (gray, dashed) [ 14 ], Pierce theory with Q modeled by the sheath helix model as used by CHRISTENE [ 44 ] (blue, dashed), and the exact dispersion relation (red, solid) as a function of input signal frequency. Image adapted from [ 39 ]. © [2018] IEEE. Reprinted, with permission, from [ 39 ].

As can be seen in figure 6 , there are differences in the solution and hence the predicted gain of a helix TWT depending on the models used. From the numerical solution to the exact hot-tube dispersion relation (red in figure 6 ), one can find what the Pierce parameters, namely C and Q here, ought to be. These are plotted as a function of frequency below in figure 7 along with the parameters' respective models/usual definitions. The new necessary parameter q is also plotted. We see that if q  = 0, the values of the traditional Pierce parameters take on unrealistic values, necessitating this new parameter to fit the prediction of the exact dispersion relation to Pierce theory.

Figure 7.

Discrete cavity analysis of TWTs

While the formulation thus far presented has centered around the groundwork laid by Pierce and therefore has concentrated on treating the beam-circuit interaction as a continuum, it is also important to consider 'discrete cavity' effects, relating back to the klystron-TWT picture presented in figure 1 . After all, the beam-circuit interaction is not always purely continuous in a real system and having discrete effects allows for more freedom on the part of the microwave engineer to account for various effects, as will be briefly discussed below, not amenable to standard Pierce theory. This also allows us to tie the TWT to other microwave structures such as the klystron [ 54 ] and the magnetron [ 28 ], where there is not a continuous interaction.

As we have shown, Pierce theory reduces the complex geometry of the slow wave circuit to individual traveling waves that must then interact with a driving beam. In high frequency tubes [ 55 ], as well as potential designs with relatively small numbers of cavities, the detailed adjustment of an individual cavity, either due to manufacturing errors, or for attempts to control oscillation and reflections, might be necessary. This can be challenging for Pierce theory, where we have implicitly used a wave analysis. One way to handle this is to simply allow the phase velocity mismatch, the gain parameter, and the cold-circuit loss to vary spatially in equations ( 6 )–( 9 ) [ 48 ]. Alternatively, another potential avenue of theoretical development comes from treating the individual cavities as discrete circuit elements with explicit coupling between the cavities. This allows individual cavities to be tuned, both in frequency and in shunt impedance, to engineer the dispersion between the beam and the electromagnetic structure, giving additional degrees of freedom to the tube designer. In terms of Pierce theory, however, the individual cavities and their coupling results in a set of normal modes. These normal modes have well-defined phase and group velocities due to the explicit nature of the coupling, just as in standard Pierce theory, but allow for non-uniform cavities to be used. Naturally, the transition from discrete normal modes to a wave formulation is well understood, as the number of cavities tends to large (infinite) values.

It should be noted that the discrete theory of TWTs has a long history. As a consequence, many developments have been made to account for various effects and ultimately advance the theory. Some more recent examples of this are listed in [ 56 ]. There is current, on-going theoretical, simulation, and experimental work in better understanding the transition from a discrete number of cavities to an infinite continuum in a tube, from the effects that a finite number of cavities introduces to the continuous wave interaction picture of Pierce. New tube designs with a low number of cavities have been set up for this purpose of determining the limits of the Pierce theory and the transition point to the discrete cavity regime. Here, we will lay out the basics of the theoretical formulation.

For illustration purposes, we can start by considering a traveling wave structure with just four cavities. After this pedagogical introduction, we will apply the method to a real TWT. We assume a cavity can be modeled as a circuit with bulk parameters that can be well described as a harmonic oscillator. This circuit then develops a voltage due to electron beam current, source signals, and coupling between the cavities. If we define the harmonic oscillator operator L as

we can then write the following set of equations for our four cavity system:

where ( A , B , C , D ) represent the cavity voltages, Z represents the cavity impedance, S is a general input source, c is the coupling between the cavities, and I ( x ) represents the electron beam current at a given cavities' location. This current naturally has the impact of the previous cavities, so needs to represent a history of the current interaction with the upstream cavities. Here, we only study the cold tube properties, so we set these currents to zero, and set the source term to zero. Furthermore, take the usual actions of normalizing the resonant frequency of the isolated cavities to one without loss of generality, assuming sinusoidal solutions, and setting the quality factor to infinity, equation ( 11 ) can then be recast into matrix form,

This equation has a number of nice features—first, it is tri-diagonal, which allows considerable linear algebra tools to be used. Second, all of the cavity details can be included in the matrix M . Thus, equation ( 12 ) gives us the homogenous solution to our slow wave structure. Sources can be added to the right hand side of equation ( 12 ) as we go forward.

Here, however, we will continue to focus on the eigenvalues of this system. These eigenvalues give the resonant frequencies of the system and the eigenvectors are the characteristic modes. Analytically, we can exploit the fact that tri-diagonal systems have a recursion relationship for the determinant. Furthermore, we use the fact that we have only four cavities and thus have a quartic equation, which is actual bi-quadratic. Alternatively, we can use Octave's symbolic package to solve for the roots of the system, or numerically solve for them. Because the resulting equations are polynomial in nature, the numerical solution with Octave is quite tedious, even for four cavities. Once a given root is found, the equations need to be deflated by hand for Octave's numerical solver to find the next root [ 57 ]. Despite this challenge, analytic, symbolic and numerical all gave the same resonant frequencies for our four cavity system:

where we note, for clarity, that ' c ' is the coupling between adjacent cavities.

We can then use equation ( 13 ) to solve for the eigenvectors of the system. The plot of these is shown in figure 8 . Initially, these eigenvectors were difficult to interpret, but noting that these are normal modes, the shape of these fields makes more sense. To do this, we need to fix the boundary conditions outside of the cavity structure—a typical choice is to set the fields to zero, which would be consistent with a TWT slow wave structure inside a cutoff waveguide. With these conditions, the normal modes become the typical phase advance per cavity if one assumes an additional 'virtual cavity' with amplitude of zero on either side of the slow wave structure. This is shown in figure 9 . Comparing figures 8 and 9 , we see that the orange points in the plot, representing the cavity field amplitude, are following a given sinusoidal pattern, albeit with a difference in phase for two of the eigenvectors between the Octave result and our normal model analysis. These modes can be directly tied to the phase advance per cavity, which is given by m π/4, where m is an integer. With this insight, it is possible to plot the dispersion relationship for our model with frequency versus phase advance (or mode number) as shown in figure 10 .

Figure 8.

Figure 8.  Eigenvectors for the four cavity system roots given in equation ( 13 ) from Octave. This plots mode amplitude versus cavity number. Thus, the horizontal axis gives the axial location of the field.

Figure 9.

Figure 9.  Analytic normal mode analysis of the 4 cavity eigenvectors.

Figure 10.

Figure 10.  Dispersion relation for our four cavity circuit model. Cavity coupling c  = 0.4.

It is worth noting that figure 10 has the typical characteristics of dispersion relationships that we expect from relatively short slow wave structures. The correct phase advance and mode shape have been found. Additionally, variation in the number of cavities and/or the cavity coupling allows the relative 'flatness' of the curve near the π mode to be observed, consistent with experimental and frequency domain analysis of TWT structures. Finally, it is a positive sign that we have tested the computational tools that are necessary to handle realistic numbers of cavities for a non-trivial if small, example, and verified these tools against analytic results.

Next, we extend the coupled-cavity circuit model to simulate a realistic TWT [e.g. using the parameters from [ 55 ]], as opposed to the four-cavity proof-of-principle. The effort proceeds by building a tridiagonal matrix with the harmonic oscillator operator on the diagonal and the coupling term on the bands. For our exercise here, we assume no ohmic loss and symmetric coupling between the cavities. In this case, we have a Toeplitz matrix, and there are recursion relations for the eigenvalues. This allows us to compute the dispersion relationship from the normal modes. Because the harmonic oscillator operation has two roots, there are two harmonic oscillator modes for the dispersion relation. The dispersion relation is shown in figure 11 . This result is in excellent agreement with HFSS results published in the literature [ 55 ].

Figure 11.

Figure 11.  Dispersion relation for a 20 cavity sine waveguide TWT constructed from a coupled cavity approach. Individual cavities have a resonant frequency of 8 GHz and the dimensionless coupling between the cavities is c  = 0.19.

The next passband is found in a similar manner, with the resonant frequency of an individual cavity changed to 12.5 GHz and the coupling reduced to c  = 0.03. The physical reason for the reduction in the cavity coupling for the second passband is the higher frequency allows the electromagnetic waves to be concentrated more tightly to the structure. It is also worth noting that both passbands have significantly more coupling than the klystron cases previously examined. This is simply the nature of the traveling wave tube structure. The second passband is shown in figure 12 .

Figure 12.

Figure 12.  Second passband for the sine waveguide TWT with individual cavity frequency of 12.5 GHz and cavity coupling of c  = 0.03.

Comparing the coupled circuit model and HFSS does unveil an interesting feature. Simply taking one harmonic oscillator mode (or 'branch cut' as it is referred to in figure 12 ) of the normal mode structure yields the orange curve in figure 12 . This is clearly different than that seen in the HFSS calculation. However, switching to the other harmonic oscillator mode allows us to find the correct dispersion relationship for both modes in the second passband. The jump between harmonic oscillator modes is not clear, and further work needs to be done to understand why this additional step is necessary to reproduce the second passband. That said, we believe this kink is physical, and structures like these are often found in band edges of TWTs. Additionally, the HFSS calculations were run on a unit cell, and it is not clear if that makes it easier to choose between the harmonic oscillator modes [ 58 ].

We have demonstrated that a coupled circuit model can reproduce the electromagnetic structure of a full TWT, and that this allows one to add spatial non-uniformity to the slow wave circuit. Additional work that we have done shows that as the number of cavities becomes large, the normal mode analysis naturally transitions to a wave description. This provides an explicit link between standard Pierce theory (wave) and a means to incorporate finite length effects, internal reflections, manufacturing errors, and individual cavity differences into a Pierce style analysis.

Concluding remarks

In conclusion, we have just glossed over the mode-coupling theory of Pierce in describing the beam-circuit interaction of a TWT. We have talked qualitatively and quantitatively about the linear DC electron beam and the electromagnetic signal on the surrounding structure constituting the circuit, separately, and then how both interact to produce the desired amplification of the signal. Even though Pierce's classical theory is a linear (so-called 'small-signal') theory, it is still powerful and widely used. The key to its endurance is its simplicity as well as tradition, as it was the first comprehensive theory to describe the inner workings of a TWT. Its simplicity allows for relatively fast 'back-of-the-envelope' calculations and insights into the underlying mechanisms of beam-circuit interaction. Here, we stress what we believe are the core ideas of Pierce's theory: synchronous interaction between and coupling of the beam and circuit producing amplification of the input signal.

In keeping with this, we attempt to provide several natural extensions to the Pierce theory to describe phenomena not previously considered in the small-signal regime. These include harmonic generation (from dynamical synchronous interaction between the beam and circuit in an octave bandwidth tube) and beam-loading on the circuit (stressing the symmetry between beam and circuit and their coupling). Additionally, we have discussed the links between normal mode analysis associated with coupling individual cavities and its relationship to Pierce theory.

It should be stressed that this paper is by no means a comprehensive overview of TWTs. This paper is much too immature to tackle such a subject. We wish to introduce the reader to the world of TWTs and beam-circuit interactions via the classical theory of Pierce and try to extend the core ideas behind the theory to tackle more modern topics: harmonic generation, space-charge effects, and discrete cavity effects. There are still many topics to explore. The reader is encouraged to look through the references (which is by no means extensive), especially [ 59 ] on the history of the TWT in telecommunication. Louisell's textbook [ 60 ] provides excellent, more in-depth discussions on many of the topics addressed here, especially space-charge waves (c.f. Chapters 2 and 3).

Acknowledgments

This work was supported by the Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Grant No. FA9550-18-1-0062, Air Force Office of Scientific Research (AFOSR) YIP Grant No. FA9550-18-1-0061, and the Office of Naval Research (ONR) YIP Grant No. N00014-20-1-2681.

Certain images in this publication have been obtained by the author(s) from the Wikipedia website, where they are stated to be in the public domain. Please see individual figure captions in this publication for details. To the extent that the law allows, IOP Publishing disclaim any liability that any person may suffer as a result of accessing, using or forwarding the image(s). Any reuse rights should be checked and permission should be sought if necessary from Wikipedia and/or the copyright owner (as appropriate) before using or forwarding the image(s).

The authors would also like to thank Professor Y. Y. Lau for valuable discussions and his advice and encouragement.

What happens if we keep more terms in the expansion? See the section, 'Beyond Pierce' below.

A travelling harmonic wave on a string is described by y ( x , t ) = 7.5 sin ( 0.0050 x + 12 t + π / 4 ) (a) What are the displacement and velocity of oscillation of a point at x = 1 cm and t = 1 s? Is this velocity equal to the velocity of wave propagation ? (b) Locate the points of the string which have the same transverse displacements and velocity as the x = 1 cm point at t = 2 s , 5 s and 11s

(a) the given harmonic wave is y ( x , t ) = 7.5 s i n ( 0.0050 x + 12 t + π 4 ) for x = 1 c m and t = 1 s , y ( 1 , 1 ) = 7.5 s i n ( 0.0050 x + 12 + π 4 ) = 7.5 s i n ( 12.0050 + π 4 ) = 7.5 s i n θ where, θ = 12.0050 + π 4 = 12.0050 + 3.14 4 = 12.79 r a d = 180 3.14 × 12.79 = 732.81 o ∴ y = ( 1 , 1 ) = 7.5 s i n ( 732.81 o ) = 7.5 s i n ( 90 × 8 + 12.81 o ) = 7.5 s i n 12.81 o = 7.5 × 0.2217 = 1.6629 ≈ 1.663 c m the velocity of the oscillation at a given point and times is given as: v = d d t y ( x , t ) = d d t [ 7.5 s i n ( 0.0050 x + 12 t + π 4 ) ] = 7.5 × 12 c o s ( 0.0050 x + 12 t + π 4 ) at x = 1 c m and t = 1 s v = y ( 1 , 1 ) = 90 c o s ( 12.005 + π 4 ) = 90 c o s s ( 732.81 o ) = 90 c o s ( 90 × 8 + 12.81 o ) = 90 c o s ( 12.81 o ) = 90 × 0.975 = 87.75 c m / s now, the equation of a propagating wave is given by: y ( x , t ) = a s i n ( k x + w t + ϕ ) where, k = 2 π / λ ∴ λ = 2 π / k and, ω = 2 π v ∴ v = ω / 2 π speed, v = v λ = ω / k where, ω = 12 r a d / s k = 0.0050 m − 1 ∴ v = 12 / 0.0050 = 2400 c m / s hence, the velocity of the wave oscillation at x = 1 c m and t = 1 s is not equal to the velocity of the wave propagation. (b) propagation constant is related to wavelength as: k = 2 π / λ ∴ λ = 2 π / k = 2 × 3.14 / 0.0050 = 1256 c m = 12.56 m therefore, all the points at distance n λ ( n = ± 1 ± 2 , . . . and so on ) , i.e. ± 12.56 m , ± 25.12 m , . . . and so on for x = 1 c m , will have the same displacement as the x = 1 c m points at t = 2 s , 5 s and 11 s ..

travelling harmonic wave

IMAGES

  1. PPT

    travelling harmonic wave

  2. 12.1: Traveling Waves

    travelling harmonic wave

  3. Explain what is a Plane progressive harmonic wave

    travelling harmonic wave

  4. A harmonic wave is travelling along +ve x-axis, on a stretched str

    travelling harmonic wave

  5. Derivation /Travelling Harmonic Waves/Wave motion/Physics.

    travelling harmonic wave

  6. A progressive wave travelling along the positive x

    travelling harmonic wave

VIDEO

  1. simple harmonic wave

  2. 2 The wave properties

  3. For a travelling haromonic wave , `y=2.0cos(10t-0.0080x+0.818)` where x and y are

  4. When a simple harmonic progressive wave is propogating the medium, all the particles

  5. wave length

  6. wave motion, mechanical waves

COMMENTS

  1. 12.1: Traveling Waves

    As we saw earlier (Equation ( 12.1.8 )), the energy per unit volume in a harmonic wave of angular frequency ω and amplitude ξ0 is E / V = 1 2ρ0ω2ξ2 0. If the wave is traveling at a speed c, then the energy flux (energy transported per unit time per unit area) is equal to (E / V)c, which is to say. I = 1 2cρ0ω2ξ2 0.

  2. 16.1 Traveling Waves

    Figure 16.3 An idealized surface water wave passes under a seagull that bobs up and down in simple harmonic motion. The wave has a wavelength λ λ, which is the distance between adjacent identical parts of the wave.The amplitude A of the wave is the maximum displacement of the wave from the equilibrium position, which is indicated by the dotted line. . In this example, the medium moves up and ...

  3. Traveling waves

    Consider a transverse harmonic wave traveling in the positive x-direction. Harmonic waves are sinusoidal waves. The displacement y of a particle in the medium is given as a function of x and t by. y(x,t) = A sin(kx - ωt + φ) Here k is the wave number, k = 2π/λ, and ω = 2π/T = 2πf is the angular frequency of the wave.

  4. 15.6: Wave Behavior and Interaction

    Mathematical Represenation of a Traveling Wave. The most general solution of the wave equation ∂ 2u ∂ t2 = c2 ∂ 2u ∂ x2 is given as u(x, t) = f(x + ct) + g(x − ct), where f and g are arbitrary functions. learning objectives. Formulate solution of the wave equation for a traveling wave. In general, one dimensional waves satisfy the 1D ...

  5. PDF Waves I

    Having discussed the general properties of traveling waves, we now focus specifically on harmonic traveling waves - i.e. waves that repeat themselves periodically in space and in time. Harmonic traveling waves are sinuosoids - and can be described either by sine or cosine functions - e.g. right-moving harmonic traveling waves can be mathematically

  6. Traveling Waves: Crash Course Physics #17

    Waves are cool. The more we learn about waves, the more we learn about a lot of things in physics. Everything from earthquakes to music! Ropes can tell us a ...

  7. PDF Traveling Waves

    Standing waves produced by the sum of waves traveling in opposite directions, shown as functions of the spatial coordinate at five different times. The sum is a spatial wave whose amplitude oscillates. 4.6 Anharmonic Traveling Waves, Dispersion Thus far the only traveling waves we have considered have been harmonic, i.e., consisting of a single

  8. 15.1 Simple Harmonic Motion

    In simple harmonic motion, the acceleration of the system, and therefore the net force, is proportional to the displacement and acts in the opposite direction of the displacement. A good example of SHM is an object with mass m attached to a spring on a frictionless surface, as shown in Figure 15.3.

  9. Traveling Waves

    Traveling Waves "The essence of science: ask an impertinent question, and you are on the way to a pertinent answer" ... Jacob Bronowski Imagine a sequence of particles undergoing identical Simple Harmonic Motion, such that each particle begins to move slightly after the one before it. The result is a traveling "Wave Motion". If all the ...

  10. 16.2 Mathematics of Waves

    The value 2 π λ is defined as the wave number. The symbol for the wave number is k and has units of inverse meters, m −1: k ≡ 2 π λ. 16.2. Recall from Oscillations that the angular frequency is defined as ω ≡ 2 π T. The second term of the wave function becomes. 2 π λ v t = 2 π λ ( λ T) t = 2 π T t = ω t.

  11. Travelling Waves

    A wave in which the positions of maximum and minimum amplitude travel through the medium is known as a travelling wave. To better understand a wave, let us think of the disturbance caused when we jump on a trampoline. ... An object in simple harmonic motion has an energy of \(\begin{array}{l}E=\frac{1}{2}kA^2\end{array} \). From this equation ...

  12. PDF Traveling Waves: Energy Transport

    Traveling Harmonic Waves: Harmonic waves have the form y = A sin(kx + φφφφ) at time t = 0, where k is the "wave number", k = 2 ππππ/λλλλ, λλλλis the "wave length". and A is the "amplitude". To construct a harmonic wave traveling to the right with speed v, replace x by x-vt as follows: y =Asin( k(x−vt )+φ) =Asin( kx −ωt ...

  13. Analyzing Waves on a String

    Harmonic Traveling Waves. Imagine that one end of a long taut string is attached to a simple harmonic oscillator, such as a tuning fork — this will send a harmonic wave down the string, f (x − v t) = A sin k (x − v t). The standard notation is. f (x − v t) = A sin (k x − ω t) where of course. ω = v k.

  14. 1.2: Wave Properties

    In the category of periodic waves, the easiest to work with mathematically are harmonic waves. The word "harmonic" is basically synonymous with "sinusoidal." For a one-dimensional wave, one might therefore assume that a harmonic wave function looks like: ... Find the speed at which this wave is traveling. Find the direction (\(\pm x\)) in which ...

  15. Standing waves review (article)

    Harmonic: A standing wave that is a positive integer multiple of the fundamental frequency. Standing wave harmonics. ... For a rope with two fixed ends, another wave travelling down the rope will interfere with the reflected wave. At certain frequencies, this produces standing waves where the nodes and antinodes stay at the same places over ...

  16. What are harmonic waves?

    A harmonic is a wave with a frequency that is a positive integer multiple of the fundamental frequency, the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the 1st harmonic, the other harmonics are known as higher harmonics. Quoting from Britannica. The properties of harmonic waves are ...

  17. 16.1 Traveling Waves

    Examples include gamma rays, X-rays, ultraviolet waves, visible light, infrared waves, microwaves, and radio waves. Electromagnetic waves can travel through a vacuum at the speed of light, v= c =2.99792458 × 108m/s. v = c = 2.99792458 × 10 8 m/s. For example, light from distant stars travels through the vacuum of space and reaches Earth.

  18. Waves

    The profile of the travelling wave is sinusoidal at any instant and each point in the track of the wave performs simple harmonic motion. The wave function y (z,t) is given by equation 1. (1) (2) where A is the amplitude, k is the propagation constant (or wave number), is the wavelength, is the angular frequency, f is the frequency, T the period ...

  19. 24.1: Traveling Waves

    Figure 24.1.3 24.1. 3: Top: two snapshots of a traveling harmonic wave at t t = 0 (solid) and at t = Δt t = Δ t (dashed). The quantity ξ ξ is the displacement of a typical particle of the medium at each point x x (the wave is traveling in the positive x x direction). Units for both x x and ξ ξ are arbitrary.

  20. Derivation /Travelling Harmonic Waves/Wave motion/Physics

    Explanation with Notes/Travelling Harmonic Waves/Wave motion/Physics.Transverse Waves on stringhttps://youtu.be/dJnYp5_uScU Superposition of two mutually per...

  21. For the travelling harmonic wave y(x,t)=2.0 cos 2pi (10t-0.0080 ...

    For the travelling harmonic wave y (x, t) = 2.0 c o s 2 π (10t-0.0080 x+0.35 ) where x and y are in cm and t in s. Calculate the phase difference between oscillatory motion of two points separated by a distance of x. x = 4 m, Δ ϕ = 6.4 π r a d; 0.5 m, Δ ϕ = 0.6 π r a d; λ / 2, Δ ϕ =.6 π r a d; 3 λ / 4, Δ ϕ = 2.5 π r a d.

  22. Recent theory of traveling-wave tubes: a tutorial-review

    This 'force" is a traveling wave at the second harmonic frequency. It has a phase velocity also synchronized with the electron beam because the condition for synchronism in a TWT. This "force" may then synchronously excite a second harmonic wave, both in time and space , which makes it a much more powerful contributor to second harmonic generation.

  23. A travelling harmonic wave on a string is described by

    A travelling harmonic wave on a string is described by y (x, t) = 7.5 sin (0.0050 x + 12 t + π / 4) (a) What are the displacement and velocity of oscillation of a point at x = 1 cm and t = 1 s? Is this velocity equal to the velocity of wave propagation ?