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Chapter 15: Problem 35

Wave Equation and Standing Waves. (a) Prove by direct substitution that \(y(x, t)=\left(A_{\text { sw }} \sin k x\right)\) sin \(\omega t\) is a solution of the wave equation, Eq. \((15.12),\) for \(v=\omega / k .\) (b) Explain why the relationship \(v=\omega / k\) for traveling waves also applies to standing waves.

Short Answer

Expert verified
(a) The function satisfies the wave equation for \( v = \frac{\omega}{k} \). (b) The relationship holds as standing waves are formed from traveling waves.

Step by step solution

01

Understand the Wave Equation

The wave equation for a function \( y(x,t) \) is given by: \( \frac{\partial^2 y}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 y}{\partial t^2} \), where \( v \) is the wave speed. Our goal is to prove that the given function \( y(x, t)=(A_{\text{sw}} \sin k x) \sin \omega t \) is a solution to this equation.
02

Compute the Partial Derivatives with Respect to x

First, we compute the first partial derivative of \( y(x, t) \) with respect to \( x \): \[ \frac{\partial y}{\partial x} = A_{\text{sw}} k \cos(kx) \sin(\omega t) \]Next, compute the second partial derivative with respect to \( x \): \[ \frac{\partial^2 y}{\partial x^2} = -A_{\text{sw}} k^2 \sin(kx) \sin(\omega t) \]
03

Compute the Partial Derivatives with Respect to t

Now, compute the first partial derivative of \( y(x, t) \) with respect to \( t \): \[ \frac{\partial y}{\partial t} = A_{\text{sw}} \sin(kx) \omega \cos(\omega t) \]Then, compute the second partial derivative with respect to \( t \): \[ \frac{\partial^2 y}{\partial t^2} = -A_{\text{sw}} \sin(kx) \omega^2 \sin(\omega t) \]
04

Substitute into the Wave Equation

Substitute the second derivatives into the wave equation: \[ -A_{\text{sw}} k^2 \sin(kx) \sin(\omega t) = \frac{1}{v^2}(-A_{\text{sw}} \sin(kx) \omega^2 \sin(\omega t)) \]This simplifies to: \[ k^2 = \frac{\omega^2}{v^2} \]
05

Verify the Relationship v = 蠅/k

From the expression \( k^2 = \frac{\omega^2}{v^2} \), taking square root on both sides gives \( k = \frac{\omega}{v} \) or \( v = \frac{\omega}{k} \). This confirms that the relationship holds, showing that \( y(x, t) \) satisfies the wave equation.
06

Explain the Applicability to Standing Waves

The relationship \( v = \frac{\omega}{k} \) applies to both traveling and standing waves because it arises from the dispersion relation, which dictates how frequency and wave number are related to wave speed. For standing waves, the individual components (sines or cosines of \( kx \) and \( \omega t \)) form via superposition of traveling waves, preserving the same relation between \( \omega \) and \( k \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standing Waves
Standing waves represent a fascinating concept in wave physics. They are formed by the interference of two traveling waves moving in opposite directions. When these waves have the same amplitude and frequency, and their combination sets up nodes and antinodes (areas of no movement and maximum movement, respectively), we get a standing wave.

In these waves, certain points, called nodes, remain stationary as they experience no net displacement over time. Between these nodes are antinodes, which are points with maximum amplitude oscillation. Unlike traveling waves that transfer energy from one place to another, standing waves store energy by confining it to a fixed space.

Standing waves are not just a theoretical concept鈥攖hey have practical applications in musical instruments, where the waves confined to the length of the strings produce harmonic tones. Another application is in microwave ovens, where standing waves help heat food evenly. By understanding standing waves, students can better appreciate the behavior of waves in fixed-boundary conditions.
Partial Derivatives
Partial derivatives are a crucial tool in understanding functions of multiple variables, such as those involved in wave equations. They measure how a function changes as one of the input variables changes, while the others are held constant.

Consider a wave function, such as \( y(x, t) = (A_{\text{sw}} \sin(kx)) \sin(\omega t) \). When we compute the partial derivatives with respect to \( x \) and \( t \), it reveals how the wave's position and oscillation change over space and time, respectively.
  • By differentiating with respect to \( x \), we determine the rate of change of the wave's shape as it moves through space.
  • By differentiating with respect to \( t \), we observe how the wave's oscillation changes over time.
These derivatives play a pivotal role in verifying if a function satisfies the wave equation. For example, in proving that a standing wave function satisfies the wave equation, one must show that the spatial and temporal partial derivatives adequately fulfill the equation's requirements. Mastery of partial derivatives is thus essential for anyone working with multi-variable equations.
Dispersion Relation
The dispersion relation is a fundamental concept in wave physics. It describes the relationship between the wave frequency \( \omega \) and wave number \( k \), linking these to the wave's speed \( v \). For many wave types, including standing and traveling waves, this relationship is expressed as \( v = \frac{\omega}{k} \).

Understanding the dispersion relation allows us to see how wave characteristics are interrelated. For instance, knowing the speed of a wave and its frequency helps us determine the wave's wavelength (or the inverse of wave number), which is important in various practical settings from oceanography to telecommunications.
  • For electromagnetic waves in a vacuum, the dispersion relation is simple and direct. It implies a linear relationship between frequency and wave number because the speed is constant.
  • For other media, such as dispersive media where wave speed varies with frequency, the dispersion relation becomes more complex.
The fact that the dispersion relation applies to standing waves, though derived from traveling waves, shows its fundamental role in wave theory. This universality explains phenomena like resonance and wave superposition, which are core aspects of wave dynamics.

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Most popular questions from this chapter

Waves of Arbitrary Shape. (a) Explain why any wave described by a function of the form \(y(x, t)=f(x-v t)\) moves in the \(+x\) -direction with speed \(v\) . (b) Show that \(y(x, t)=f(x-v t)\) satisfies the wave equation, no matter what the functional form of \(f\) . To do this, write \(y(x, t)=f(u),\) where \(u=x-v t\) . Then, to take partial derivatives of \(y(x, t),\) use the chain rule: $$ \begin{array}{l}{\frac{\partial y(x, t)}{\partial t}=\frac{d f(u)}{d u} \frac{\partial u}{\partial t}=\frac{d f(u)}{d u}(-v)} \\ {\frac{\partial y(x, t)}{\partial x}=\frac{d f(u)}{d u} \frac{\partial u}{\partial x}=\frac{d f(u)}{d u}}\end{array} $$ (c) A wave pulse is described by the function \(y(x, t)=D e^{-(B x-C)^{2}}\) where \(B, C,\) and \(D\) are all positive constants. What is the speed of this wave?

A simple harmonic oscillator at the point \(x=0\) generates a wave on a rope. The oscillator operates at a frequency of 40.0 \(\mathrm{Hz}\) and with an amplitude of 3.00 \(\mathrm{cm}\) . The rope has a linear mass density of 50.0 \(\mathrm{g} / \mathrm{m}\) and is stretched with a tension of 5.00 \(\mathrm{N}\) . (a) Determine the speed of the wave. (b) Find the wavelength. (c) Write the wave function \(y(x, t)\) for the wave. Assume that the oscillator has its maximum upward displacement at time \(t=0\) . (d) Find the maximum transverse acceleration of points on the rope. (e) In the discussion of transverse waves in this chapter, the force of gravity was ignored. Is that a reasonable approximation for this wave? Explain.

A sinusoidal transverse wave travels on a string. The string has length 8.00 \(\mathrm{m}\) and mass 6.00 \(\mathrm{g}\) . The wave speed is 30.0 \(\mathrm{m} / \mathrm{s}\) , and the wavelength is 0.200 \(\mathrm{m}\) (a) If the wave is to have an aver- age power of 50.0 \(\mathrm{W}\) , what must be the amplitude of the wave? (b) For this same string, if the amplitude and wavelength are the same as in part (a), what is the average power for the wave if the tension is increased such that the wave speed is doubled?

A jet plane at take-off can produce sound of intensity 10.0 \(\mathrm{W} / \mathrm{m}^{2}\) at 30.0 \(\mathrm{m}\) away. But you prefer the tranguil sound of normal conversation, which is 1.0\(\mu \mathrm{W} / \mathrm{m}^{2}\) . Assume that the plane behaves like a point source of sound. (a) What is the closest distance you should live from the airport runway to preserve your peace of mind? (b) What intensity from the jet does your friend experience if she lives twice as far from the runway as you do? (c) What power of sound does the jet produce at take-off?

A transverse sine wave with an amplitude of 2.50 \(\mathrm{mm}\) and a wavelength of 1.80 \(\mathrm{m}\) travels from left to right along a long, horizontal, stretched string with a speed of 36.0 \(\mathrm{m} / \mathrm{s}\) . Take the origin at the left end of the undisturbed string. At time \(t=0\) the left end of the string has its maximum upward displacement, (a) What are the frequency, angular frequency, and wave number of the wave? (b) What is the function \(y(x, t)\) that describes the wave? (c) What is \(y(t)\) for a particle at the left end of the string? (d) What is \(y(t)\)for a particle 1.35 \(\mathrm{m}\) to the right of the origin? (e) What is the maximum magnitude of transverse velocity of any particle of the string? (f) Find the transverse displacement and the transverse velocity of a particle 1.35 \(\mathrm{m}\) to the right of the origin at time \(t=0.0625 \mathrm{s}\) .
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