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Chapter 15: Problem 16

Consider a monochromatic wave on a string, with amplitude \(A\) and wavelength \(\lambda\), traveling in one direction. Find the relationship between the maximum speed of any portion of string, \(v_{\max },\) and the wave speed, \(v\)

Short Answer

Expert verified
Answer: The maximum speed of any portion of the string, \(v_{\max}\), is related to the wave speed, \(v\), through the equation: $$ v_{\max} = 2\pi A f $$ where \(A\) is the amplitude of the wave, and \(f\) is the frequency. The frequency can be expressed in terms of the wave speed and wavelength through the equation \(v = f\lambda\).

Step by step solution

01

Understanding the properties of a monochromatic wave on a string

A monochromatic wave on a string is a wave with a single frequency. The wave is characterized by its amplitude, \(A\), and wavelength, \(\lambda\). The amplitude is the maximum displacement of the string from its equilibrium position, and the wavelength is the distance between two consecutive points with the same phase (e.g. two consecutive peaks or troughs) in the wave.
02

Writing down the wave equation

The wave equation that describes a monochromatic wave on a string traveling in one direction can be written as: $$ y(x,t)=A\sin\left(\frac{2\pi}{\lambda}(x-vt)\right) $$ where \(y(x,t)\) represents the displacement of the string as a function of position, \(x\), and time, \(t\), and \(v\) is the wave speed.
03

Calculating the velocity of a point on the wave

To find the velocity of a point on the wave, we need to determine the rate of change of the displacement with respect to time. We can do this by taking the partial derivative of the wave equation with respect to time: $$ v_y(x,t) = \frac{\partial y(x,t)}{\partial t} $$ Applying the chain rule and differentiating the wave equation with respect to time: $$ v_y(x,t) = A \frac{2\pi}{\lambda} (-v) \cos\left(\frac{2\pi}{\lambda}(x-vt)\right) $$
04

Finding the maximum value of the velocity

To find the maximum speed of any portion of the string, we need to find the maximum value of \(v_y(x,t)\). Since the maximum value of the cosine function is \(1\), the maximum value of \(v_y(x,t)\) is obtained when \(\cos\left(\frac{2\pi}{\lambda}(x-vt)\right)=1\). Hence, $$ v_{\max} = A \frac{2\pi}{\lambda}(-v) $$
05

Expressing the maximum speed in terms of the wave speed

To find the relationship between the maximum speed, \(v_{\max}\), and the wave speed, \(v\), we can rearrange the equation we derived in the previous step: $$ v_{\max} = 2\pi A \frac{-v}{\lambda} $$ Since the frequency, \(f\), is related to the wave speed and wavelength by \(v=f\lambda\), we can rewrite the above equation in terms of frequency: $$ v_{\max} = 2\pi A f $$ This equation shows the relationship between the maximum speed of any portion of the string, \(v_{\max}\), and the wave speed, \(v\).

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Most popular questions from this chapter

a) Starting from the general wave equation (equation 15.9 ), prove through direct derivation that the Gaussian wave packet described by the equation \(y(x, t)=(5.00 m) e^{-0.1(x-5 t)^{2}}\) is indeed a traveling wave (that it satisfies the differential wave equation). b) If \(x\) is specified in meters and \(t\) in seconds, determine the speed of this wave. On a single graph, plot this wave as a function of \(x\) at \(t=0, t=1.00 \mathrm{~s}, t=2.00 \mathrm{~s},\) and \(t=3.00 \mathrm{~s}\) c) More generally, prove that any function \(f(x, t)\) that depends on \(x\) and \(t\) through a combined variable \(x \pm v t\) is a solution of the wave equation, irrespective of the specific form of the function \(f\)

Write the equation for a sinusoidal wave propagating in the negative \(x\) -direction with a speed of \(120 . \mathrm{m} / \mathrm{s}\), if a particle in the medium in which the wave is moving is observed to swing back and forth through a \(6.00-\mathrm{cm}\) range in \(4.00 \mathrm{~s}\). Assume that \(t=0\) is taken to be the instant when the particle is at \(y=0\) and that the particle moves in the positive \(y\) -direction immediately after \(t=0\).

A small ball floats in the center of a circular pool that has a radius of \(5.00 \mathrm{~m}\). Three wave generators are placed at the edge of the pool, separated by \(120 .\). The first wave generator operates at a frequency of \(2.00 \mathrm{~Hz}\). The second wave generator operates at a frequency of \(3.00 \mathrm{~Hz}\). The third wave generator operates at a frequency of \(4.00 \mathrm{~Hz}\). If the speed of each water wave is \(5.00 \mathrm{~m} / \mathrm{s}\), and the amplitude of the waves is the same, sketch the height of the ball as a function of time from \(t=0\) to \(t=2.00 \mathrm{~s}\), assuming that the water surface is at zero height. Assume that all the wave generators impart a phase shift of zero. How would your answer change if one of the wave generators was moved to a different location at the edge of the pool?

A steel cable consists of two sections with different cross-sectional areas, \(A_{1}\) and \(A_{2}\). A sinusoidal traveling wave is sent down this cable from the thin end of the cable. What happens to the wave on encountering the \(A_{1} / A_{2}\) boundary? How do the speed, frequency and wavelength of the wave change?

A wave travels along a string in the positive \(x\) -direction at \(30.0 \mathrm{~m} / \mathrm{s}\). The frequency of the wave is \(50.0 \mathrm{~Hz}\). At \(x=0\) and \(t=0,\) the wave velocity is \(2.50 \mathrm{~m} / \mathrm{s}\) and the vertical displacement is \(y=4.00 \mathrm{~mm} .\) Write the function \(y(x, t)\) for the wave
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