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Chapter 15: Problem 24

A light wire is tightly stretched with tension \(F .\) Trans- verse traveling waves of amplitude \(A\) and wavelength \(\lambda_{1}\) carry average power \(P_{\mathrm{av}, 1}=0.400 \mathrm{W}\) . If the wavelength of the waves is doubled, so \(\lambda_{2}=2 \lambda_{1},\) while the tension \(F\) and amplitude \(A\) are not altered, what then is the average power \(P_{\mathrm{av}, 2}\) carried by the waves?

Short Answer

Expert verified
The average power of the wave with doubled wavelength is 0.100 W.

Step by step solution

01

Understand the Relationship of Power and Wavelength

The average power of a wave on a string is given by the equation \( P_{\text{av}} = \frac{1}{2} \mu \omega^2 A^2 v \), where \( \mu \) is the mass per unit length, \( \omega \) is the angular frequency, \( A \) is the amplitude, and \( v \) is the wave speed. For a wave on a string under tension, wave speed \( v \) is given by \( v = \sqrt{\frac{F}{\mu}} \).
02

Relate Wave Speed and Wavelength

Recognize that the wave speed \( v = f \lambda \), where \( f \) is the frequency and \( \lambda \) is the wavelength. Regardless of wavelength changes, \( v \) remains constant because \( F \) and \( \mu \) are unchanged. Thus, the product \( f \lambda \) must also be constant.
03

Determine Frequency Change

With \( \lambda_{2} = 2 \lambda_{1} \), and knowing \( v = f_1 \lambda_1 = f_2 \lambda_2 \), solve for \( f_2 \): \( f_2 = \frac{f_1 \lambda_1}{2 \lambda_1} = \frac{f_1}{2} \). The frequency halves as wavelength doubles.
04

Calculate New Angular Frequency

The angular frequency \( \omega \) is related to frequency by \( \omega = 2\pi f \). As frequency halves, \( \omega_2 = 2\pi (\frac{f_1}{2}) = \frac{\omega_1}{2} \).
05

Determine New Average Power

Substitute \( \omega_2 = \frac{\omega_1}{2} \) into the power equation: \( P_{\text{av}, 2} = \frac{1}{2} \mu (\frac{\omega_1}{2})^2 A^2 v = \frac{1}{4} \left( \frac{1}{2} \mu \omega_1^2 A^2 v \right) = \frac{1}{4} P_{\text{av}, 1} \).
06

Final Calculation

Since \( P_{\text{av}, 1} = 0.400 \) W, calculate \( P_{\text{av}, 2} = \frac{1}{4} \times 0.400 = 0.100 \) W.

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Key Concepts

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

Wave Speed
Wave speed on a string is an important concept in physics, particularly when studying waves under tension. Simply put, wave speed tells us how fast a wave travels along the string. The formula to calculate wave speed (\( v \) ) is: \[ v = \sqrt{\frac{F}{\mu}} \] where:
  • \( F \) is the tension in the string, and
  • \( \mu \) is the mass per unit length of the string.
IMPORTANT NOTE: Regardless of changes in wavelength, as long as the tension \( F \) and mass per unit length \( \mu \) remain unchanged, the wave speed \( v \) will also remain constant.
This understanding is crucial when analyzing how other wave properties affect each other, like frequency or power.
Wave Frequency
Wave frequency is essentially how often the wave oscillates in a given second. It's a vital property for determining the behavior of waves.The relationship between frequency (\( f \) ) and wave speed (\( v \) ) can be expressed as:\[ v = f \lambda \] where:
  • \( f \) is the frequency,
  • \( \lambda \) is the wavelength.
When the wavelength doubles, as it happens in the problem, the frequency changes correspondingly to keep the wave speed constant.
Using the relationship \( v = f_1 \lambda_1 = f_2 \lambda_2 \), we find that \( f_2 = \frac{f_1}{2} \), showing that the frequency halves when the wavelength is doubled.
This illustrates how changes in wavelength directly influence frequency when speed is constant. Understanding this relationship helps in comprehending how waves behave in different mediums or under varying conditions.
Angular Frequency
Angular frequency is another fundamental concept linked closely with wave frequency. It offers a way to express how fast an oscillation occurs in terms of radians.Angular frequency (\( \omega \) ) is calculated using:\[ \omega = 2\pi f \]where
  • \( f \) is the frequency.
If the wave frequency changes due to an alteration in wavelength鈥攁s shown when it halves鈥攁ngular frequency will also change.
Specifically, when \( f \) becomes \( \frac{f_1}{2} \), then \( \omega_2 = \frac{\omega_1}{2} \), meaning that angular frequency is directly proportional to the wave frequency. Understanding how angular frequency is related to frequency and how it impacts wave power calculations is crucial. It helps to determine the behavior and characteristics needed to solve more complex physics problems and understand wave dynamics deeply.

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Most popular questions from this chapter

One string of a certain musical instrument is 75.0 \(\mathrm{cm}\) long and has a mass of 8.75 g. It is being played in a room where the speed of sound is 344 \(\mathrm{m} / \mathrm{s}\) . (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength 0.765 \(\mathrm{m} ?\) (Assume that the breaking stress of the wire is very large and isn't exceeded.) (b) What frequency sound does this string produce in its fundamental mode of vibration?

A deep-sea diver is suspended beneath the surface of Loch Ness by a \(100-\) m-long cable that is attached to a boat on the surface (Fig. P15.84). The diver and his suit have a total mass of 120 \(\mathrm{kg}\) and a volume of 0.0880 \(\mathrm{m}^{3} .\) The cable has a diameter of 2.00 \(\mathrm{cm}\) and a linear mass density of \(\mu=\) 1.10 \(\mathrm{kg} / \mathrm{m} .\) The diver thinks he sees something moving in the murky depths and jerks the end of the cable back and forth to send transverse waves up the cable as a signal to his companions in the boat. (a) What is the tension in the cable at its lower end, where it is attached to the diver? Do not forget to include the buoyant force that the water (density 1000 \(\mathrm{kg} / \mathrm{m}^{3}\) ) exerts on him. (b)Calculate the tension in the cable a distance \(x\) above the diver. The buoyant force on the cable must be included in your calculation. (c) The speed of transverse waves on the cable is given by \(v=\sqrt{F / \mu}\) (Eq. 15.13). The speed therefore varies along the cable, since the tension is not constant. (This expression neglects the damping force that the water exerts on the moving cable.. Integrate to find the time required for the first signal to reach the surface.

chlc Energy in a Triangular Pulse. A triangular wave pulse on a taut string travels in the positive \(x\) -direction with speed \(v .\) The tension in the string is \(F,\) and the linear mass density of the string is \(\mu .\) At \(t=0,\) the shape of the pulse is given by $$y(x, 0)=\left\\{\begin{array}{ll}{0} & {\text { if } x < -L} \\ {h(L+x) / L} & {\text { for }-L< x <0} \\ {h(L-x) / L} & {\text { for } 0 < x < L} \\ {0} & {\text { for } x > L}\end{array}\right.$$ (a) Draw the pulse at \(t=0 .\) (b) Determine the wave function \(y(x, t)\) at all times \(t\) (c) Find the instantaneous power in the wave. Show that the power is zero except for \(-L<(x-v t)

CALC Equation \((15.7)\) for a sinusoidal wave can be made more general by including a phase angle \(\phi,\) where \(0 \leq \phi \leq 2 \pi(\) in radians). Then the wave function \(y(x, t)\) becomes $$y(x, t)=A \cos (k x-\omega t+\phi)$$ (a) Sketch the wave as a function of \(x\) at \(t=0\) for \(\phi=0,\) \(\phi=\pi / 4, \phi=\pi / 2, \phi=3 \pi / 4,\) and \(\phi=3 \pi / 2 .\) (b) Calculate the transverse velocity \(v_{y}=\partial y / \partial t .\) (c) At \(t=0,\) a particle on the string at \(x=0\) has displacement \(y=A / \sqrt{2} .\) Is this enough information to determine the value of \(\boldsymbol{\phi} ?\) In addition, if you are told that a particle at \(x=0\) is moving toward \(y=0\) at \(t=0,\) what is the value of \(\phi (\mathrm{d})\) Explain in general what you must know about the wave's behavior at a given instant to determine the value of \(\phi\) .

Weightless Ant. Ant. An ant with mass \(m\) is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length \(\mu\) and is under tension \(F .\) Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wave-length \(\lambda\) propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that \(m\) is so small that the presence of the ant has no effect on the propagation of the wave.
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