Chapter 16

If a wave \(y(x, t)=(6.0 \mathrm{~mm}) \sin (k x+(600 \mathrm{rad} / \mathrm{s}) t+\phi)\) travels along a string, how much time does any given point on the string take to move between displacements \(y=+2.0 \mathrm{~mm}\) and \(y=-2.0 \mathrm{~mm}\) ?

A human wave. During sporting events within large, densely packed stadiums, spectators will send a wave (or pulse) around the stadium (Fig. \(16-29)\). As the wave reaches a group of spectators, they stand with a cheer and then sit. At any instant, the width \(w\) of the wave is the distance from the leading edge (people are just about to stand) to the trailing edge (people have just sat down). Suppose a human wave travels a distance of 853 seats around a stadium in \(39 \mathrm{~s}\), with spectators requiring about \(1.8 \mathrm{~s}\) to respond to the wave's passage by standing and then sitting. What are (a) the wave speed \(v\) (in seats per second) and (b) width \(w\) (in number of seats)?

A wave has an angular frequency of \(110 \mathrm{rad} / \mathrm{s}\) and a wavelength of \(1.80 \mathrm{~m} .\) Calculate (a) the angular wave number and (b) the speed of the wave.

A sand scorpion can detect the motion of a nearby beetle (its prey) by the waves the motion sends along the sand surface (Fig. 16-30). The waves are of two types: transverse waves traveling at \(v_{t}=50 \mathrm{~m} / \mathrm{s}\) and longitudinal waves traveling at \(v_{l}=150 \mathrm{~m} / \mathrm{s}\). If a sudden motion sends out such waves, a scorpion can tell the distance of the beetle from the difference \(\Delta t\) in the arrival times of the waves at its leg nearest the beetle. If \(\Delta t=4.0 \mathrm{~ms}\) what is the beetle's distance?

A sinusoidal wave travels along} a string. The time for a particular point to move from maximum displacement to zero is \(0.170 \mathrm{~s}\). What are the (a) period and (b) frequency? (c) The wavelength is \(1.40 \mathrm{~m}\); what is the wave speed?

A transverse sinusoidal wave is moving along a string in the positive direction of an \(x\) axis with a speed of \(80 \mathrm{~m} / \mathrm{s}\). At \(t=0\), the string particle at \(x=0\) has a transverse displacement of \(4.0 \mathrm{~cm}\) from its equilibrium position and is not moving. The maximum transverse speed of the string particle at \(x=0\) is \(16 \mathrm{~m} / \mathrm{s}\). (a) What is the frequency of the wave? (b) What is the wavelength of the wave? If \(y(x, t)=y_{m} \sin (k x \pm \omega t+\phi)\) is the form of the wave equation, what are (c) \(y_{m},(\mathrm{~d}) k,(\mathrm{e}) \omega,(\mathrm{f}) \phi\), and \((\mathrm{g})\) the correct choice of sign in front of \(\omega\) ?

The equation of a transverse wave traveling along a very long string is \(y=6.0 \sin (0.020 \pi x+4.0 \pi t)\), where \(x\) and \(y\) are expressed in centimeters and \(t\) is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave, and (f) the maximum transverse speed of a particle in the string. (g) What is the transverse displacement at \(x=3.5 \mathrm{~cm}\) when \(t=\) \(0.26 \mathrm{~s} ?\)

The function \(y(x, t)=(15.0 \mathrm{~cm}) \cos (\pi x-15 \pi t)\), with \(x\) in meters and \(t\) in seconds, describes a wave on a taut string. What is the transverse speed for a point on the string at an instant when that point has the displacement \(y=+12.0 \mathrm{~cm}\) ?

A sinusoidal wave of frequency \(500 \mathrm{~Hz}\) has a speed of \(350 \mathrm{~m} / \mathrm{s}\). (a) How far apart are two points that differ in phase by \(\pi / 3\) rad? (b) What is the phase difference between two displacements at a certain point at times \(1.00 \mathrm{~ms}\) apart?

The equation of a transverse wave on a string is $$ y=(2.0 \mathrm{~mm}) \sin \left[\left(20 \mathrm{~m}^{-1}\right) x-\left(600 \mathrm{~s}^{-1}\right) t\right] $$ The tension in the string is \(15 \mathrm{~N}\). (a) What is the wave speed? (b) Find the linear density of this string in grams per meter.

A stretched string has a mass per unit length of \(5.00 \mathrm{~g} / \mathrm{cm}\) and a tension of \(10.0 \mathrm{~N}\). A sinusoidal wave on this string has an amplitude of \(0.12 \mathrm{~mm}\) and a frequency of \(100 \mathrm{~Hz}\) and is traveling in the negative direction of an \(x\) axis. If the wave equation is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (a) \(y_{m},(\mathrm{~b}) k,(\mathrm{c}) \omega\), and (d) the correct choice of sign in front of \(\omega\) ?

The speed of a transverse wave on a string is \(170 \mathrm{~m} / \mathrm{s}\) when the string tension is \(120 \mathrm{~N}\). To what value must the tension be changed to raise the wave speed to \(180 \mathrm{~m} / \mathrm{s} ?\)

The linear density of a string is \(1.6 \times 10^{-4} \mathrm{~kg} / \mathrm{m} .\) A transverse wave on the string is described by the equation $$ y=(0.021 \mathrm{~m}) \sin \left[\left(2.0 \mathrm{~m}^{-1}\right) x+\left(30 \mathrm{~s}^{-1}\right) t\right] $$ What are (a) the wave speed and (b) the tension in the string?

The heaviest and lightest strings on a certain violin have linear densities of \(3.0\) and \(0.29 \mathrm{~g} / \mathrm{m}\). What is the ratio of the diameter of the heaviest string to that of the lightest string, assuming that the strings are of the same material?

What is the speed of a transverse wave in a rope of length \(2.00 \mathrm{~m}\) and mass \(60.0 \mathrm{~g}\) under a tension of \(500 \mathrm{~N}\) ?

The tension in a wire clamped at both ends is doubled without appreciably changing the wire's length between the clamps. What is the ratio of the new to the old wave speed for transverse waves traveling along this wire?

A \(100 \mathrm{~g}\) wire is held under a tension of \(250 \mathrm{~N}\) with one end at \(x=0\) and the other at \(x=10.0 \mathrm{~m}\). At time \(t=0\), pulse 1 is sent along the wire from the end at \(x=10.0 \mathrm{~m}\). At time \(t=30.0\) ms, pulse 2 is sent along the wire from the end at \(x=0 .\) At what position \(x\) do the pulses begin to meet?

A sinusoidal wave is traveling on a string with speed \(40 \mathrm{~cm} / \mathrm{s}\). The displacement of the particles of the string at \(x=10 \mathrm{~cm}\) varies with time according to \(y=(5.0 \mathrm{~cm}) \sin \left[1.0-\left(4.0 \mathrm{~s}^{-1}\right) t\right] .\) The linear density of the string is \(4.0 \mathrm{~g} / \mathrm{cm}\). What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form \(y(x, t)=\) \(y_{m} \sin (k x \pm \omega t)\), what are (c) \(y_{m}\), (d) \(k\), (e) \(\omega\), and \((\mathrm{f})\) the correct choice of sign in front of \(\omega ?(\mathrm{~g})\) What is the tension in the string?

A uniform rope of mass \(m\) and length \(L\) hangs from a ceiling. (a) Show that the speed of a transverse wave on the rope is a function of \(y\), the distance from the lower end, and is given by \(v=\sqrt{g y}\). (b) Show that the time a transverse wave takes to travel the length of the rope is given by \(t=2 \sqrt{L / g} .\)

A string along which waves can travel is \(2.70 \mathrm{~m}\) long and has a mass of \(260 \mathrm{~g}\). The tension in the string is \(36.0 \mathrm{~N}\). What must be the frequency of traveling waves of amplitude \(7.70 \mathrm{~mm}\) for the average power to be \(85.0 \mathrm{~W} ?\)

Use the wave equation to find the speed of a wave given by $$ y(x, t)=(3.00 \mathrm{~mm}) \sin \left[\left(4.00 \mathrm{~m}^{-1}\right) x-\left(7.00 \mathrm{~s}^{-1}\right) t\right] . $$

Use the wave equation to find the speed of a wave given in terms of the general function \(h(x, t)\) : $$ y(x, t)=(4.00 \mathrm{~mm}) h\left[\left(30 \mathrm{~m}^{-1}\right) x+\left(6.0 \mathrm{~s}^{-1}\right) t\right] . $$

Two identical traveling waves, moving in the same direction, are out of phase by \(\pi / 2\) rad. What is the amplitude of the resultant wave in terms of the common amplitude \(y_{m}\) of the two combining waves?

What phase difference between two identical traveling waves, moving in the same direction along a stretched string, results in the combined wave having an amplitude \(1.50\) times that of the common amplitude of the two combining waves? Express your answer in (a) degrees, (b) radians, and (c) wavelengths.

A sinusoidal wave of angular frequency \(1200 \mathrm{rad} / \mathrm{s}\) and amplitude \(3.00 \mathrm{~mm}\) is sent along a cord with linear density \(2.00 \mathrm{~g} / \mathrm{m}\) and tension \(1200 \mathrm{~N}\). (a) What is the average rate at which energy is transported by the wave to the opposite end of the cord? (b) If, simultaneously, an identical wave travels along an adjacent, identical cord, what is the total average rate at which energy is transported to the opposite ends of the two cords by the waves? If, instead, those two waves are sent along the same cord simultaneously, what is the total average rate at which they transport energy when their phase difference is (c) \(0,(\mathrm{~d}) 0.4 \pi \mathrm{rad}\), and \((\mathrm{e}) \pi \mathrm{rad} ?\)

Two sinusoidal waves of the same frequency travel in the same direction along a string. If \(y_{m 1}=3.0 \mathrm{~cm}, y_{m 2}=4.0 \mathrm{~cm}\), \(\phi_{1}=0\), and \(\phi_{2}=\pi / 2 \mathrm{rad}\), what is the amplitude of the resultant wave?

Four waves are to be sent along the same string, in the same direction: $$ \begin{aligned} &y_{1}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t) \\ &y_{2}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.7 \pi) \\ &y_{3}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+\pi) \\ &y_{4}(x, t)=(4.00 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+1.7 \pi) . \end{aligned} $$ What is the amplitude of the resultant wave?

These two waves travel along the same string: $$ \begin{aligned} &y_{1}(x, t)=(4.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t) \\ &y_{2}(x, t)=(5.60 \mathrm{~mm}) \sin (2 \pi x-400 \pi t+0.80 \pi \mathrm{rad}) \end{aligned} $$ What are (a) the amplitude and (b) the phase angle (relative to wave 1 ) of the resultant wave? (c) If a third wave of amplitude \(5.00 \mathrm{~mm}\) is also to be sent along the string in the same direction as the first two waves, what should be its phase angle in order to maximize the amplitude of the new resultant wave?

Two sinusoidal waves of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of \(5.0 \mathrm{~mm}\), the other \(8.0 \mathrm{~mm}\). (a) What phase difference \(\phi_{1}\) between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference \(\phi_{2}\) results in the largest amplitude of the resultant wave? (d) What is that largest amplitude? (e) What is the resultant amplitude if the phase angle is \(\left(\phi_{1}-\phi_{2}\right) / 2 ?\)

Two sinusoidal waves of the same period, with amplitudes of \(5.0\) and \(7.0 \mathrm{~mm}\), travel in the same direction along a stretched string; they produce a resultant wave with an amplitude of \(9.0 \mathrm{~mm}\). The phase constant of the \(5.0 \mathrm{~mm}\) wave is \(0 .\) What is the phase constant of the \(7.0 \mathrm{~mm}\) wave?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string with a speed of \(10 \mathrm{~cm} / \mathrm{s}\). If the time interval between instants when the string is flat is \(0.50 \mathrm{~s}\), what is the wavelength of the waves?

A string fixed at both ends is \(8.40 \mathrm{~m}\) long and has a mass of \(0.120 \mathrm{~kg}\). It is subjected to a tension of \(96.0 \mathrm{~N}\) and set oscillating. (a) What is the speed of the waves on the string? (b) What is the longest possible wavelength for a standing wave? (c) Give the frequency of that wave.

A string under tension \(\tau_{i}\) oscillates in the third harmonic at frequency \(f_{3}\), and the waves on the string have wavelength \(\lambda_{3}\). If the tension is increased to \(\tau_{f}=4 \tau_{i}\) and the string is again made to oscillate in the third harmonic, what then are (a) the frequency of oscillation in terms of \(f_{3}\) and (b) the wavelength of the waves in terms of \(\lambda_{3}\) ?

What are (a) the lowest frequency, (b) the second lowest frequency, and (c) the third lowest frequency for standing waves on a wire that is \(10.0 \mathrm{~m}\) long, has a mass of \(100 \mathrm{~g}\), and is stretched under a tension of \(250 \mathrm{~N}\) ?

A \(125 \mathrm{~cm}\) length of string has mass \(2.00 \mathrm{~g}\) and tension \(7.00 \mathrm{~N}\). (a) What is the wave speed for this string? (b) What is the lowest resonant frequency of this string?

A string that is stretched between fixed supports separated by \(75.0 \mathrm{~cm}\) has resonant frequencies of 420 and \(315 \mathrm{~Hz}\), with no intermediate resonant frequencies. What are (a) the lowest resonant frequency and (b) the wave speed?

String \(A\) is stretched between two clamps separated by distance \(L\). String \(B\), with the same linear density and under the same tension as string \(A\), is stretched between two clamps separated by distance \(4 L\). Consider the first eight harmonics of string \(B\). For which of these eight harmonics of \(B\) (if any) does the frequency match the frequency of (a) \(A\) 's first harmonic, (b) \(A\) 's second harmonic, and (c) \(A\) 's third harmonic?

One of the harmonic frequencies for a particular string under tension is \(325 \mathrm{~Hz}\). The next higher harmonic frequency is \(390 \mathrm{~Hz}\). What harmonic frequency is next higher after the harmonic frequency \(195 \mathrm{~Hz}\) ?

If a transmission line in a cold climate collects ice, the increased diameter tends to cause vortex formation in a passing wind. The air pressure variations in the vortexes tend to cause the line to oscillate (gallop), especially if the frequency of the variations matches a resonant frequency of the line. In long lines, the resonant frequencies are so close that almost any wind speed can set up a resonant mode vigorous enough to pull down support towers or cause the line to short out with an adjacent line. If a transmission line has a length of \(347 \mathrm{~m}\), a linear density of \(3.35 \mathrm{~kg} / \mathrm{m}\), and a tension of \(65.2 \mathrm{MN}\), what are (a) the frequency of the fundamental mode and (b) the frequency difference between successive modes?

A nylon guitar string has a linear density of \(7.20 \mathrm{~g} / \mathrm{m}\) and is under a tension of \(150 \mathrm{~N}\). The fixed supports are distance \(D=90.0 \mathrm{~cm}\) apart. The string is oscillating in the standing wave pattern shown in Fig. 16-39. Calculate the (a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.

Two waves are generated on a string of length \(3.0 \mathrm{~m}\) to produce a three-loop standing wave with an amplitude of \(1.0 \mathrm{~cm}\). The wave speed is \(100 \mathrm{~m} / \mathrm{s}\). Let the equation for one of the waves be of the form \(y(x, t)=y_{m} \sin (k x+\omega t)\). In the equation for the other wave, what are (a) \(y_{m}\), (b) \(k\), (c) \(\omega\), and (d) the sign in front of \(\omega\) ?

A rope, under a tension of \(200 \mathrm{~N}\) and fixed at both ends, oscillates in a second-harmonic standing wave pattern. The displacement of the rope is given by $$ y=(0.10 \mathrm{~m})(\sin \pi x / 2) \sin 12 \pi t $$ where \(x=0\) at one end of the rope, \(x\) is in meters, and \(t\) is in seconds. What are (a) the length of the rope, (b) the speed of the waves on the rope, and (c) the mass of the rope? (d) If the rope oscillates in a third-harmonic standing wave pattern, what will be the period of oscillation?

A string oscillates according to the equation $$ y^{\prime}=(0.50 \mathrm{~cm}) \sin \left[\left(\frac{\pi}{3} \mathrm{~cm}^{-1}\right) x\right] \cos \left[\left(40 \pi \mathrm{s}^{-1}\right) t\right] $$ What are the (a) amplitude and (b) speed of the two waves (identical except for direction of travel) whose superposition gives this oscillation? (c) What is the distance between nodes? (d) What is the transverse speed of a particle of the string at the position \(x=1.5 \mathrm{~cm}\) when \(t=\frac{9}{8} \mathrm{~s} ?\)

The following two waves are sent in opposite directions on a horizontal string so as to create a standing wave in a vertical plane: $$ \begin{aligned} &y_{1}(x, t)=(6.00 \mathrm{~mm}) \sin (4.00 \pi x-400 \pi t) \\ &y_{2}(x, t)=(6.00 \mathrm{~mm}) \sin (4.00 \pi x+400 \pi t) \end{aligned} $$ with \(x\) in meters and \(t\) in seconds. An antinode is located at point \(A\). In the time interval that point takes to move from maximum upward displacement to maximum downward displacement, how far does each wave move along the string?

A standing wave pattern on a string is described by $$ y(x, t)=0.040(\sin 5 \pi x)(\cos 40 \pi t) $$ where \(x\) and \(y\) are in meters and \(t\) is in seconds. For \(x \geq 0\), what is the location of the node with the (a) smallest, (b) second smallest, and (c) third smallest value of \(x\) ? (d) What is the period of the oscillatory motion of any (nonnode) point? What are the (e) speed and (f) amplitude of the two traveling waves that interfere to produce this wave? For \(t \geq 0\), what are the \((\mathrm{g})\) first, \((\mathrm{h})\) second, and (i) third time that all points on the string have zero transverse velocity?

A generator at one end of a very long string creates a wave given by $$ y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x+\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ and a generator at the other end creates the wave $$ y=(6.0 \mathrm{~cm}) \cos \frac{\pi}{2}\left[\left(2.00 \mathrm{~m}^{-1}\right) x-\left(8.00 \mathrm{~s}^{-1}\right) t\right] $$ Calculate the (a) frequency, (b) wavelength, and (c) speed of each wave. For \(x \geq 0\), what is the location of the node having the (d) smallest, (e) second smallest, and (f) third smallest value of \(x\) ? For \(x \geq 0\), what is the location of the antinode having the (g) smallest, (h) second smallest, and (i) third smallest value of \(x\) ?

In Fig. \(16-42\), a string, tied to a sinusoidal oscillator at \(P\) and running over a support at \(Q\), is stretched by a block of mass \(m\). The separation \(L\) between \(P\) and \(Q\) is \(1.20 \mathrm{~m}\), and the frequency \(f\) of the oscillator is fixed at \(120 \mathrm{~Hz}\). The amplitude of the motion at \(P\) is small enough for that point to be considered a node. A node also exists at \(Q .\) A standing wave appears when the mass of the hanging block is \(286.1 \mathrm{~g}\) or \(447.0 \mathrm{~g} .\) but not for any intermediate mass. What is the linear density of the string?

A sinusoidal transverse wave traveling in the positive direction of an \(x\) axis has an amplitude of \(2.0 \mathrm{~cm}\), a wavelength of \(10 \mathrm{~cm}\), and a frequency of \(400 \mathrm{~Hz}\). If the wave equation is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (a) \(y_{m}\), (b) \(k\), (c) \(\omega\), and (d) the correct choice of sign in front of \(\omega ?\) What are (e) the maximum transverse speed of a point on the cord and (f) the speed of the wave?

A wave has a speed of \(240 \mathrm{~m} / \mathrm{s}\) and a wavelength of \(3.2 \mathrm{~m}\). What are the (a) frequency and (b) period of the wave?

The equation of a transverse wave traveling along a string is $$ y=0.15 \sin (0.79 x-13 t) $$ in which \(x\) and \(y\) are in meters and \(t\) is in seconds. (a) What is the displacement \(y\) at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s} ?\) A second wave is to be added to the first wave to produce standing waves on the string. If the second wave is of the form \(y(x, t)=y_{m} \sin (k x \pm \omega t)\), what are (b) \(y_{m},(\mathrm{c})\) \(k\), (d) \(\omega\), and (e) the correct choice of sign in front of \(\omega\) for this second wave? (f) What is the displacement of the resultant standing wave at \(x=2.3 \mathrm{~m}, t=0.16 \mathrm{~s}\) ?