Chapter 15

The speed of sound in air at \(20^{\circ} \mathrm{C}\) is 344 \(\mathrm{m} / \mathrm{s} .\) (a) What is the wavelength of a sound wave with a frequency of 784 \(\mathrm{Hz}\) , corresponding to the note \(\mathrm{G}_{5}\) on a piano, and how many milliseconds does each vibration take? (b) What is the wavelength of a sound wave one octave higher than the note in part (a)?

Audible Sound. Provided the amplitude is sufficiently great, the human ear can respond to longitudinal waves over a range of frequencies from about 20.0 \(\mathrm{Hz}\) to about 20.0 \(\mathrm{kHz}\) . (a) If you were to mark the beginning of each complete wave pattern with a red dot for the long-wavelength sound and a blue dot for the short-wavelength sound, how far apart would the red dots be, and how far apart would the blue dots be? (b) In reality would adjacent dots in each set be far enough apart for you to easily measure their separation with a meter stick? (c) Suppose you repeated part (a) in water, where sound travels at 1480 \(\mathrm{m} / \mathrm{s}\) . How far apart would the dots be in each set? Could you readily measure their separation with a meter stick?

Tsunami! On December \(26,2004,\) a great earthquake occurred off the coast of Sumatra and triggered immense waves (tsunami) that killed some \(200,000\) people. Satellites observing these waves from space measured 800 \(\mathrm{km}\) from one wave crest to the next and a period between waves of 1.0 hour. What was the speed of these waves in \(\mathrm{m} / \mathrm{s}\) and in \(\mathrm{km} / \mathrm{h}\) ? Does your answer help you understand why the waves caused such devastation?

Ultrasound Imaging. Sound having frequencies above the range of human hearing (about \(20,000 \mathrm{Hz}\) ) is called ultrasound. Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of 1500 \(\mathrm{m} / \mathrm{s} .\) For a good, detailed image, the wavelength should be no more than 1.0 \(\mathrm{mm} .\) What frequency sound is required for a good scan?

A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.62 m. The fisherman sees that the wave crests are spaced 6.0 \(\mathrm{m}\) apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were 0.30 \(\mathrm{m}\) but the other data remained the same, how would the answers to parts (a) and (b) be affected?

Transverse waves on a string have wave speed 8.00 \(\mathrm{m} / \mathrm{s}\) , amplitude \(0.0700 \mathrm{m},\) and wavelength 0.320 \(\mathrm{m} .\) The waves travel in the \(-x\) -direction, and at \(t=0\) the \(x=0\) end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a par-ticle at \(x=0.360 \mathrm{m}\) at time \(t=0.150 \mathrm{s}\) . (d) How much time must elapse from the instant in part (c) until the particle at \(x=0.360 \mathrm{m}\) next has maximum upward displacement?

A certain transverse wave is described by $$y(x, t)=(6.50 \mathrm{mm}) \cos 2 \pi\left(\frac{x}{28.0 \mathrm{cm}}-\frac{t}{0.0360 \mathrm{s}}\right)$$ Determine the wave's (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation.

A water wave traveling in a straight line on a lake is described by the equation $$y(x, t)=(3.75 \mathrm{cm}) \cos \left(0.450 \mathrm{cm}^{-1} x+5.40 \mathrm{s}^{-1} t\right)$$ where \(y\) is the displacement perpendicular to the undisturbed surface of the lake. (a) How much time does it take for one complete wave pattem to go past a fisherman in a boat at anchor, and what horizontal distance does the wave crest travel in that time? (b) What are the wave number and the number of waves per second that pass the fisherman? (c) How fast does a wave crest travel past the fisherman, and what is the maximum speed of his cork floater as the wave causes it to bob up and down?

One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates the rope transversely at 120 \(\mathrm{Hz}\) . The other end passes over a pulley and supports a \(1.50-\mathrm{kg}\) mass. The linear mass density of the rope is 0.0550 \(\mathrm{kg} / \mathrm{m}\) . (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 \(\mathrm{kg}\) ?

With what tension must a rope with length 2.50 \(\mathrm{m}\) and mass 0.120 \(\mathrm{kg}\) be stretched for transverse waves of frequency 40.0 \(\mathrm{Hz}\) to have a wavelength of 0.750 \(\mathrm{m} ?\)

The upper end of a \(3.80-\) m-long steel wire is fastened to the ceiling, and a 54.0 -kg object is suspended from the lower end of the wire. You observe that it takes a transverse pulse 0.0492 s to travel from the bottom to the top of the wire. What is the mass of the wire?

A 1.50 -m string of weight 0.0125 \(\mathrm{N}\) is tied to the ceiling at its upper end, and the lower end supports a weight \(W\) . Neglect the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the string slightly, the waves traveling up the string obey the equation $$y(x, t)=(8.50 \mathrm{mm}) \cos \left(172 \mathrm{m}^{-1} x-4830 \mathrm{s}^{-1} t\right)$$ Assume that the tension of the string is constant and equal to \(W\) . (a) How much time does it take a pulse to travel the full length of the string? (b) What is the weight \(W ?\) (c) How many wavelengths are on the string at any instant of time? (d) What is the equation for wayes traveling down the string?

A thin, 75.0 -cm wire has a mass of 16.5 \(\mathrm{g} .\) One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of wavelength 3.33 \(\mathrm{cm}\) makes 875 vibrations per second? (b) How fast would this wave travel?

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 piano wire with mass 3.00 \(\mathrm{g}\) and length 80.0 \(\mathrm{cm}\) is stretched with a tension of 25.0 \(\mathrm{N}\) . A wave with frequency 120.0 \(\mathrm{Hz}\) and amplitude 1.6 \(\mathrm{mm}\) travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

A horizontal wire is stretched with a tension of 94.0 \(\mathrm{N}\) and the speed of transverse waves for the wire is 492 \(\mathrm{m} / \mathrm{s}\) . What must the amplitude of a traveling wave of frequency 69.0 \(\mathrm{Hz}\) be in order for the average power carried by the wave to be 0.365 \(\mathrm{w}\) ?

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?

\mathrm{A} jet plane at takeoff can produce sound of intensity 10.0 \(\mathrm{W} / \mathrm{m}^{2} at 30.0 \)\mathrm{m} away. But you prefer the tranquil 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 takeoff?

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are 7.5 \(\mathrm{m}\) from it, you measure its intensity to be 0.11 \(\mathrm{W} / \mathrm{m}^{2}\) . An intensity of 1.0 \(\mathrm{W} / \mathrm{m}^{2}\) is often used as the "threshold of pain." How much closer to the source can you move before the sound intensity reaches this threshold?

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is 0.026 \(\mathrm{W} / \mathrm{m}^{2}\) at a distance of 4.3 \(\mathrm{m}\) from the source. (a) What is the intensity at a distance of 3.1 m from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?

A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is \(y(x, t)=\) 2.30 \(\mathrm{mm} \cos [(6.98 \mathrm{rad} / \mathrm{m}) x+(742 \mathrm{rad} / \mathrm{s}) t] .\) Being more practical, you measure the rope to have a length of 1.35 \(\mathrm{m}\) and a mass of 0.00338 kg. You are then asked to determine the following: (a) amplitude; (b) frequency; (c) wavelength; (d) wave speed; (e) direction the wave is traveling; (f) tension in the rope; (g) average power transmitted by the wave.

At a distance of \(7.00 \times 10^{12} \mathrm{m}\) from a star, the intensity of the radiation from the star is 15.4 \(\mathrm{W} / \mathrm{m}^{2} .\) Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

15.36 .. CALC Adjacent antinodes of a standing wave on a string are 15.0 \(\mathrm{cm}\) apart. A particle at an antinode oscillates in simple harmonic motion with amplitude 0.850 \(\mathrm{cm}\) and period 0.0750 s. The string lies along the \(+x\) -axis and is fixed at \(x=0 .\) (a) How far apart are the adjacent nodes? (b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern?(c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?

15.40\(\cdot\) A 1.50 -m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 \(\mathrm{m} / \mathrm{s}\) . What are the wavelength and frequency of (a) the fundamental: (b) the second overtone; (c) the fourth harmonic?

A wire with mass 40.0 \(\mathrm{g}\) is stretched so that its ends are tied down at points 80.0 \(\mathrm{cm}\) apart. The wire vibrates in its fundamental mode with frequency 60.0 \(\mathrm{Hz}\) and with an amplitude at the antinodes of 0.300 \(\mathrm{cm} .\) (a) What is the speed of propagation oftransverse waves in the wire? (b) Compute the tension in the wire. (c) Find the maximum transverse velocity and acceleration of particles in the wire.

A piano tuner stretches a steel piano wire with a tension of 800 \(\mathrm{N}\) . The steel wire is 0.400 \(\mathrm{m}\) long and has a mass of 3.00 \(\mathrm{g}\) . (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to \(10,000 \mathrm{Hz} ?\)

CALC A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y(x, t)=\) \((5.60 \mathrm{cm}) \sin [(0.0340\) rad/ \(\mathrm{cm}) x] \sin [(50.0 \mathrm{rad} / \mathrm{s}) t],\) where the origin is at the left end of the string, the \(x\) -axis is along the string. and the \(y\) -axis is perpendicular to the string. (a) Draw a sketch that shows the standing-wave pattern. (b) Find the amplitude of the two traveling waves that make up this standing wave. (c) What is the length of the string? (d) Find the wavelength, frequency, period, and speed of the traveling waves. (e) Find the maximum transverse speed of a point on the string. (f) What would be the equation \(y(x, t)\) for this string if it were vibrating in its eighth harmonic?

15.44\(\cdot\) The wave function of a standing wave is \(y(x, t)\) 4.44 \(\mathrm{mm} \sin [(32.5 \mathrm{rad} / \mathrm{m}) x] \sin [(754 \mathrm{rad} / \mathrm{s}) t] .\) For the two traveing waves that make up this standing wave, find the (a) amplitude (b) wavelength; (c) frequency; (d) wave speed; (e) wave function (f) From the information given, can you determine which hamonic this is? Explain.

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 horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed \(v,\) frequency \(f,\) amplitude \(A,\) and wavelength \(\lambda\) . Calculate the maximum transverse velocity and maximum transverse accelerationof points located at (i) \(x=\lambda / 2,\) (ii) \(x=\lambda / 4,\) and (ii) \(x=\lambda / 8\) from the left-hand end of the string. (b) At each of the points in part (a) what is the amplitude of the motion? (c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

Guitar String. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) freguency 245 \(\mathrm{Hz})\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is \(344 \mathrm{m} / \mathrm{s},\) find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

Waves on a Stick. A flexible stick 2.0 \(\mathrm{m}\) long is not fixed in any way and is free to vibrate. Make clear drawings of this stick vibrating in its first three harmonics, and then use your drawings to find the wavelengths of each of these harmonics. (Hint: Should the ends be nodes or antinodes?

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 transverss velocity of a particle 1.35 \(\mathrm{m}\) to the right of the origin at time \(t=0.0625 \mathrm{s}\)

A transverse wave on a rope is given by $$y(x, t)=(0.750 \mathrm{cm}) \cos \pi\left[\left(0.400 \mathrm{cm}^{-1}\right) x+\left(250 \mathrm{s}^{-1}\right) t\right]$$ (a) Find the amplitude, period, frequency, wavelength, and speed of propagation. (b) Sketch the shape of the rope at these values of \(t : 0.0005\) s, 0.0010 s. (c) Is the wave wave traveling in the or -direction? (d) The mass per unit length of the rope is 0.0500 \(\mathrm{kg} / \mathrm{m} .\) Find the tension. (e) Find the average power of this wave.

Three pieces of string, each of length \(L,\) are joined together end to end, to make a combined string of length 3\(L .\) The first piece of string has mass per unit length \(\mu_{1},\) the second piecehas mass per unit length \(\mu_{2}=4 \mu_{1},\) and the third piece has mass per unit length \(\mu_{3}=\mu_{1} / 4 .(a)\) If the combined string is under tension \(F,\) how much time does it take a transverse wave to travel the entire length 3\(L ?\) Give your answer in terms of \(L, F,\) and \(\mu_{1}\) . (b) Does your answer to part (a) depend on the order in which the three pieces are joined together? Explain.

\(\mathrm{CP}\) A \(1750-\mathrm{N}\) irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires \((A\) and \(B),\) each 1.25 \(\mathrm{m}\) long and weighing 0.360 \(\mathrm{N}\) . The center of gravity of this beam is one-third of the way along the beam from the end where wire \(A\) is attached. If you pluck both strings at the same time at the beam, what is the time delay between the arrival of the two pulses at the ceiling? Which pulse arrives first? (Neglect the effect of the weight of the wires on the tension in the wires.)

CALC Ant Joy Ride. You place your pet ant Klyde (mass \(m )\) on top of a horizontal, stretched rope, where he holds on tightly. The rope has mass \(M\) and length \(L\) and is under tension \(F .\) You start a sinusoidal transverse wave of wavelength \(\lambda\) and amplitude \(A\) propagating along the rope. The motion of the rope is in a vertical plane. Klyde's mass is so small that his presence has no effect on the propagation of the wave. (a) What is Klyde's top speed as he oscillates up and down? (b) Klyde enjoys the ride and begs for more. You decide to double his top speed by changing the tension while keeping the wavelength and amplitude the same. Should the tension be increased or decreased, and by what factor?

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.

mass 0.0280 \(\mathrm{kg} .\) You measure that it takes 0.0600 s for a transverse pulse to travel from the lower end to the upper end of the string. On earth, for the same string and lead weight, it takes 0.0390 s for a transverse pulse to travel the length of the string. The weight of You are exploring a newly discovered planet. The radius of the planet is \(7.20 \times 10^{7} \mathrm{m} .\) You suspend a lead weight from the lower end of a light string that is 4.00 \(\mathrm{m}\) long and has mass 0.0280 kg. You measure that it takes 0.0600 s for a transverse pulse to travel from the lower end to the upper end of the string. On earth, for the same string and lead weight, it takes 0.0390 s for a transverse pulse to travel the length of the string. The weight of the string is small enough that its effect on the tension in the string can be neglected. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

For a string stretched between two supports, two succes- sive standing-wave frequencies are 525 Hz and 630 \(\mathrm{Hz}\) . There are other standing-wave frequencies lower than 525 \(\mathrm{Hz}\) and higher than 630 \(\mathrm{Hz}\) . If the speed of transverse waves on the string is \(384 \mathrm{m} / \mathrm{s},\) what is the length of the string? Assume that the mass of the wire is small enough for its effect on the tension in the wire to be neglected.

A \(5.00-\mathrm{m}, 0.732-\mathrm{kg}\) wire is used to support two uni-? form \(235-\mathrm{N}\) posts of equal length (Fig. P15.62). Assume that the wire is essentially horizontal and that the speed of sound is 344 \(\mathrm{m} / \mathrm{s} . \mathrm{A}\) strong wind is blowing, causing the wire to vibrate in its 5 th overtone. What are the frequency and wavelength of the sound this wire produces?

A continuous succession of sinusoidal wave pulses are produced at one end of a very long string and travel along the length of the string. The wave has frequency 70.0 \(\mathrm{Hz}\) , amplitude 5.00 \(\mathrm{mm}\) , and wavelength 0.600 \(\mathrm{m} .\) (a) How long does it take the wave to travel a distance of 8.00 \(\mathrm{m}\) along the length of the string? (b) How long does it take a point on the string to travel a distance of \(8.00 \mathrm{m},\) once the wave train has reached the point and set it into motion? (c) In parts (a) and (b), how does the time change if the amplitude is doubled?

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 \(u=x-\) vt. Then, to take partial derivatives of \(y(x, t),\) use the chain rule: $$\begin{aligned} \frac{\partial y(x, t)}{\partial t} &=\frac{d f(u)}{d u} \frac{\partial u}{\partial t}=\frac{d f(u)}{d u}(-v) \\ \frac{\partial y(x, t)}{\partial t} &=\frac{d f(u)}{d u} \frac{\partial u}{\partial x}=\frac{d f(u)}{d u} \end{aligned}$$ (c) A wave pulse is described by the function \(y(x, t)=\) \(D e^{-(B x-C t)^{2}},\) where \(B, C,\) and \(D\) are all positive constants. What is the speed of this wave?

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 \(\phi\) .

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 average power of \(50.0 \mathrm{W},\) what must be the amplitude of the wave? (b) For this same string, if the amplitude and wave-length are the same as in part (a), what is the average power for the wave if the tension is increased such the wave speed is doubled?

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 A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.386 \(\mathrm{m} .\) The maximum transverse accelera- tion of a point at the middle of the segment is \(8.40 \times 10^{3} \mathrm{m} / \mathrm{s}^{2}\) and the maximum transverse velocity is 3.80 \(\mathrm{m} / \mathrm{s}\) . (a) What is the amplitude of this standing wave? (b) What is the wave speed for the transverse traveling waves on this string?

A string with both ends held fixed is vibrating in its third harmonic. The waves have a speed of 192 \(\mathrm{m} / \mathrm{s}\) and a frequency of 240 \(\mathrm{Hz}\) . The amplitude of the standing wave at an antinode is 0.400 \(\mathrm{cm} .\) (a) Calculate the amplitude at points on the string a dis- tance of ( i ) \(40.0 \mathrm{cm} ;\) (ii) \(20.0 \mathrm{cm} ;\) and (iii) 10.0 \(\mathrm{cm}\) from the left end of the string. (b) At each point in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement? (c) Calculate the maximum transverse velocity and the maximum transverse acceleration of the string at each of the points in part (a).

Combining Standing Waves. A guitar string of length \(L\) is plucked in such a way that the total wave produced is the sum of the fundamental and the second harmonic. That is, the standing wave is given by $$y(x, t)=y_{1}(x, t)+y_{2}(x, t)$$ where $$\begin{aligned} y_{1}(x, t) &=C \sin \omega_{1} t \sin k_{1} x \\ y_{2}(x, t) &=C \sin \omega_{2} t \sin k_{2} x \end{aligned}$$ with \(\omega_{1}=v k_{1}\) and \(\omega_{2}=v k_{2}\) . (a) At what values of \(x\) are the nodes of \(y_{1} ?\left(\) b) At what values of \(x\) are the nodes of \(y_{2} ?(\mathrm{c})\) Graph \right the total wave at \(t=0, t=\frac{1}{8} f_{1}, t=\frac{1}{4} f_{1}, t=\frac{3}{8} f_{1},\) and \(t=\frac{1}{2} f_{1}\) . (d) Does the sum of the two standing waves \(y_{1}\) and \(y_{2}\) produce a standing wave? Explain.

A large rock that weighs 164.0 \(\mathrm{N}\) is suspended from the lower end of a thin wire that is 3.00 \(\mathrm{m}\) long. The density of the rock is 3200 \(\mathrm{kg} / \mathrm{m}^{3} .\) The mass of the wire is small enough that its effect on the tension in the wire can be neglected. The upper end of the wire is held fixed. When the rock is in air, the fundamental frequency for transverse standing waves on the wire is 42.0 Hz. When the rock is totally submerged in a liquid, with the top of the rock just below the surface, the fundamental frequency for the wire is 28.0 Hz. What is the density of the liquid?