Chapter 15

Fans at a local football stadium are so excited that their team is winning that they start "the wave" in celebration. Which of the following four statements is (are) true? I. This wave is a traveling wave. II. This wave is a transverse wave. III. This wave is a longitudinal wave. IV. This wave is a combination of a longitudinal wave and a transverse wave. a) I and II c) III only e) I and III b) II only d) I and IV

Suppose that the tension is doubled for a string on which a standing wave is propagated. How will the velocity of the standing wave change? a) It will double. c) It will be multiplied by \(\sqrt{2}\). b) It will quadruple. d) It will be multiplied by \(\frac{1}{2}\).

Which of the following transverse waves has the greatest power? a) a wave with velocity \(v\), amplitude \(A\), and frequency \(f\) b) a wave of velocity \(v\), amplitude \(2 A\), and frequency \(f / 2\) c) a wave of velocity \(2 v\), amplitude \(A / 2\), and frequency \(f\) d) a wave of velocity \(2 v\), amplitude \(A\), and frequency \(f / 2\) e) a wave of velocity \(v\), amplitude \(A / 2\), and frequency \(2 f\)

The speed of light waves in air is greater than the speed of sound in air by about a factor of a million. Given a sound wave and a light wave of the same wavelength, both traveling through air, which statement about their frequencies is true? a) The frequency of the sound wave will be about a million times greater than that of the light wave. b) The frequency of the sound wave will be about a thousand times greater than that of the light wave. c) The frequency of the light wave will be about a thousand times greater than that of the sound wave. d) The frequency of the light wave will be about a million times greater than that of the sound wave. e) There is insufficient information to determine the relationship between the two frequencies.

A string is made to oscillate, and a standing wave with three antinodes is created. If the tension in the string is increased by a factor of 4 a) the number of antinodes increases. b) the number of antinodes remains the same. c) the number of antinodes decreases. d) the number of antinodes will equal the number of nodes.

The different colors of light we perceive are a result of the varying frequencies (and wavelengths) of the electromagnetic radiation. Infrared radiation has lower frequencies than does visible light, and ultraviolet radiation has higher frequencies than visible light does. The primary colors are red (R), yellow (Y), and blue (B). Order these colors by their wavelength, shortest to longest. a) \(\mathrm{B}, \mathrm{Y}, \mathrm{R}\) b) \(B, R, Y\) c) \(\mathrm{R}, \mathrm{Y}, \mathrm{B}\) d) \(R, B, Y\)

If transverse waves on a string travel with a velocity of \(50 \mathrm{~m} / \mathrm{s}\) when the string is under a tension of \(20 \mathrm{~N},\) what tension on the string is required for the waves to travel with a velocity of \(30 \mathrm{~m} / \mathrm{s} ?\) a) \(7.2 \mathrm{~N}\) c) \(33 \mathrm{~N}\) e) \(45 \mathrm{~N}\) b) \(12 \mathrm{~N}\) d) \(40 \mathrm{~N}\) f) \(56 \mathrm{~N}\)

You and a friend are holding the two ends of a Slinky stretched out between you. How would you move your end of the Slinky to create (a) transverse waves or (b) longitudinal waves?

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?

Noise results from the superposition of a very large number of sound waves of various frequencies (usually in a continuous spectrum), amplitudes, and phases. Can interference arise with noise produced by two sources?

The \(1 / R^{2}\) dependency for intensity can be thought of to be due to the fact that the same power is being spread out over the surface of a larger and larger sphere. What happens to the intensity of a sound wave inside an enclosed space, say a long hallway?

If two traveling waves have the same wavelength, frequency, and amplitude and are added appropriately, the result is a standing wave. Is it possible to combine two standing waves in some way to give a traveling wave?

A ping-pong ball is floating in the middle of a lake and waves begin to propagate on the surface. Can you think of a situation in which the ball remains stationary? Can you think of a situation involving a single wave on the lake in which the ball remains stationary?

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

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, \(v\)

One of the main things allowing humans to determine whether a sound is coming from the left or the right is the fact that the sound will reach one ear before the other. Given that the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\) and that human ears are typically \(20.0 \mathrm{~cm}\) apart, what is the maximum time resolution for human hearing that allows sounds coming from the left to be distinguished from sounds coming from the right? Why is it impossible for a diver to be able to tell from which direction the sound of a motor boat is coming? The speed of sound in water is \(1.50 \cdot 10^{3} \mathrm{~m} / \mathrm{s}\).

Hiking in the mountains, you shout "hey," wait \(2.00 \mathrm{~s}\) and shout again. What is the distance between the sound waves you cause? If you hear the first echo after \(5.00 \mathrm{~s}\), what is the distance between you and the point where your voice hit a mountain?

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

A traveling wave propagating on a string is described by the following equation: $$ y(x, t)=(5.00 \mathrm{~mm}) \sin \left(\left(157.08 \mathrm{~m}^{-1}\right) x-\left(314.16 \mathrm{~s}^{-1}\right) t+0.7854\right) $$ a) Determine the minimum separation, \(\Delta x_{\min }\), between two points on the string that oscillate in perfect opposition of phases (move in opposite directions at all times). b) Determine the separation, \(\Delta x_{A B}\), between two points \(A\) and \(B\) on the string, if point \(B\) oscillates with a phase difference of 0.7854 rad compared to point \(A\). c) Find the number of crests of the wave that pass through point \(A\) in a time interval \(\Delta t=10.0 \mathrm{~s}\) and the number of troughs that pass through point \(B\) in the same interval. d) At what point along its trajectory should a linear driver connected to one end of the string at \(x=0\) start its oscillation to generate this sinusoidal traveling wave on the string?

Consider a linear array of \(n\) masses, each equal to \(m,\) connected by \(n+1\) springs, all massless and having spring constant \(k\), with the outer ends of the first and last springs fixed. The masses can move without friction in the linear dimension of the array. a) Write the equations of motion for the masses. b) Configurations of motion for which all parts of a system oscillate with the same angular frequency are called normal modes of the system; the corresponding angular frequencies are the system's normal-mode angular frequencies. Find the normal-mode angular frequencies of this array.

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

A wave on a string has a wave function given by $$ y(x, t)=(0.0200 \mathrm{~m}) \sin \left[\left(6.35 \mathrm{~m}^{-1}\right) x+\left(2.63 \mathrm{~s}^{-1}\right) t\right] $$ a) What is the amplitude of the wave? b) What is the period of the wave? c) What is the wavelength of the wave? d) What is the speed of the wave? e) In which direction does the wave travel?

A sinusoidal wave traveling in the positive \(x\) -direction has a wavelength of \(12 \mathrm{~cm},\) a frequency of \(10.0 \mathrm{~Hz},\) and an amplitude of \(10.0 \mathrm{~cm}\). The part of the wave that is at the origin at \(t=0\) has a vertical displacement of \(5.00 \mathrm{~cm} .\) For this wave, determine the a) wave number, d) speed, b) period, e) phase angle, and c) angular frequency, f) equation of motion.

A particular steel guitar string has mass per unit length of \(1.93 \mathrm{~g} / \mathrm{m}\). a) If the tension on this string is \(62.2 \mathrm{~N},\) what is the wave speed on the string? b) For the wave speed to be increased by \(1.0 \%\), how much should the tension be changed?

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 \(f\)

A string with a mass of \(30.0 \mathrm{~g}\) and a length of \(2.00 \mathrm{~m}\) is stretched under a tension of \(70.0 \mathrm{~N}\). How much power must be supplied to the string to generate a traveling wave that has a frequency of \(50.0 \mathrm{~Hz}\) and an amplitude of \(4.00 \mathrm{~cm} ?\)

A sinusoidal wave on a string is described by the equation \(y=(0.100 \mathrm{~m}) \sin (0.75 x-40 t),\) where \(x\) and \(y\) are in meters and \(t\) is in seconds. If the linear mass density of the string is \(10 \mathrm{~g} / \mathrm{m}\), determine (a) the phase constant, (b) the phase of the wave at \(x=2.00 \mathrm{~cm}\) and \(t=0.100 \mathrm{~s}\) (c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

In an acoustics experiment, a piano string with a mass of \(5.00 \mathrm{~g}\) and a length of \(70.0 \mathrm{~cm}\) is held under tension by running the string over a frictionless pulley and hanging a \(250 .-\mathrm{kg}\) weight from it. The whole system is placed in an elevator. a) What is the fundamental frequency of oscillation for the string when the elevator is at rest? b) With what acceleration and in what direction (up or down) should the elevator move for the string to produce the proper frequency of \(440 .\) Hz, corresponding to middle A?

A string is \(35.0 \mathrm{~cm}\) long and has a mass per unit length of \(5.51 \cdot 10^{-4} \mathrm{~kg} / \mathrm{m}\). What tension must be applied to the string so that it vibrates at the fundamental frequency of \(660 \mathrm{Hz?}\)

A 2.00 -m-long string of mass \(10.0 \mathrm{~g}\) is clamped at both ends. The tension in the string is \(150 \mathrm{~N}\). a) What is the speed of a wave on this string? b) The string is plucked so that it oscillates. What is the wavelength and frequency of the resulting wave if it produces a standing wave with two antinodes?

A \(3.00-\mathrm{m}\) -long string, fixed at both ends, has a mass of \(6.00 \mathrm{~g}\). If you want to set up a standing wave in this string having a frequency of \(300 . \mathrm{Hz}\) and three antinodes, what tension should you put the string under?

A cowboy walks at a pace of about two steps per second, holding a glass of diameter \(10.0 \mathrm{~cm}\) that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk.

Students in a lab produce standing waves on stretched strings connected to vibration generators. One such wave is described by the wave function \(y(x, t)=(2.00 \mathrm{~cm}) \sin \left[\left(20.0 \mathrm{~m}^{-1}\right) x\right] \cos \left[\left(150 . \mathrm{s}^{-1}\right) t\right],\) where \(y\) is the transverse displacement of the string, \(x\) is the position along the string, and \(t\) is time. Rewrite this wave function in the form for a right- moving and a left-moving wave: \(y(x, t)=\) \(f(x-v t)+g(x+v t)\); that is, find the functions \(f\) and \(g\) and the speed, \(v\)

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 string with linear mass density \(\mu=0.0250 \mathrm{~kg} / \mathrm{m}\) under a tension of \(T=250 . \mathrm{N}\) is oriented in the \(x\) -direction. Two transverse waves of equal amplitude and with a phase angle of zero (at \(t=0\) ) but with different frequencies \((\omega=3000\). rad/s and \(\omega / 3=1000 . \mathrm{rad} / \mathrm{s}\) ) are created in the string by an oscillator located at \(x=0 .\) The resulting waves, which travel in the positive \(x\) -direction, are reflected at a distant point, so there is a similar pair of waves traveling in the negative \(x\) -direction. Find the values of \(x\) at which the first two nodes in the standing wave are produced by these four waves.

The equation for a standing wave on a string with mass density \(\mu\) is \(y(x, t)=2 A \cos (\omega t) \sin (\kappa x) .\) Show that the average kinetic energy and potential energy over time for this wave per unit length are given by \(K_{\text {ave }}(x)=\mu \omega^{2} A^{2} \sin ^{2} \kappa x\) and \(U_{\text {ave }}(x)=T(\kappa A)^{2}\left(\cos ^{2} \kappa x\right)\)

A sinusoidal wave traveling on a string is moving in the positive \(x\) -direction. The wave has a wavelength of \(4 \mathrm{~m}, \mathrm{a}\) frequency of \(50.0 \mathrm{~Hz},\) and an amplitude of \(3.00 \mathrm{~cm} .\) What is the wave function for this wave?

A guitar string with a mass of \(10.0 \mathrm{~g}\) is \(1.00 \mathrm{~m}\) long and attached to the guitar at two points separated by \(65.0 \mathrm{~cm} .\) a) What is the frequency of the first harmonic of this string when it is placed under a tension of \(81.0 \mathrm{~N} ?\) b) If the guitar string is replaced by a heavier one that has a mass of \(16.0 \mathrm{~g}\) and is \(1.00 \mathrm{~m}\) long, what is the frequency of the replacement string's first harmonic?

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 \(50.0-\mathrm{cm}\) -long wire with a mass of \(10.0 \mathrm{~g}\) is under a tension of \(50.0 \mathrm{~N}\). Both ends of the wire are held rigidly while it is plucked. a) What is the speed of the waves on the wire? b) What is the fundamental frequency of the standing wave? c) What is the frequency of the third harmonic?

What is the wave speed along a brass wire with a radius of \(0.500 \mathrm{~mm}\) stretched at a tension of \(125 \mathrm{~N}\) ? The density of brass is \(8.60 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).

Two steel wires are stretched under the same tension. The first wire has a diameter of \(0.500 \mathrm{~mm}\), and the second wire has a diameter of \(1.00 \mathrm{~mm}\). If the speed of waves traveling along the first wire is \(50.0 \mathrm{~m} / \mathrm{s}\), what is the speed of waves traveling along the second wire?

The middle-C key (key 52 ) on a piano corresponds to a fundamental frequency of about \(262 \mathrm{~Hz},\) and the sopranoC key (key 64) corresponds to a fundamental frequency of \(1046.5 \mathrm{~Hz}\). If the strings used for both keys are identical in density and length, determine the ratio of the tensions in the two strings.

The tension in a 2.7 -m-long, 1.0 -cm-diameter steel cable \(\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is \(840 \mathrm{~N}\). What is the fundamental frequency of vibration of the cable?

A wave traveling on a string has the equation of motion \(y(x, t)=0.02 \sin (5.00 x-8.00 t)\) a) Calculate the wavelength and the frequency of the wave. b) Calculate its velocity. c) If the linear mass density of the string is \(\mu=0.10 \mathrm{~kg} / \mathrm{m}\), what is the tension on the string?

Consider a guitar string stretching \(80.0 \mathrm{~cm}\) between its anchored ends. The string is tuned to play middle \(\mathrm{C},\) with a frequency of \(256 \mathrm{~Hz}\), when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced \(2.00 \mathrm{~mm}\) at its midpoint and released to produce this note, what are the wave speed, \(v\), and the maximum speed, \(V_{\text {max }}\), of the midpoint of the string?

The largest tension that can be sustained by a stretched string of linear mass density \(\mu\), even in principle, is given by \(\tau=\mu c^{2},\) where \(c\) is the speed of light in vacuum. (This is an enormous value. The breaking tensions of all ordinary materials are about 12 orders of magnitude less than this.) a) What is the speed of a traveling wave on a string under such tension? b) If a \(1.000-\mathrm{m}\) -long guitar string, stretched between anchored ends, were made of this hypothetical material, what frequency would its first harmonic have? c) If that guitar string were plucked at its midpoint and given a displacement of \(2.00 \mathrm{~mm}\) there to produce the fundamental frequency, what would be the maximum speed attained by the midpoint of the string?

A rubber band of mass \(0.21 \mathrm{~g}\) is stretched between two fingers, putting it under a tension of \(2.8 \mathrm{~N}\). The overall stretched length of the band is \(21.3 \mathrm{~cm} .\) One side of the band is plucked, setting up a vibration in \(8.7 \mathrm{~cm}\) of the band's stretched length. What is the lowest frequency of vibration that can be set up on this part of the rubber band? Assume that the band stretches uniformly.

Two waves traveling in opposite directions along a string fixed at both ends create a standing wave described by \(y(x, t)=1.00 \cdot 10^{-2} \sin (25 x) \cos (1200 t) .\) The string has a linear mass density of \(0.01 \mathrm{~kg} / \mathrm{m},\) and the tension in the string is supplied by a mass hanging from one end. If the string vibrates in its third harmonic, calculate (a) the length of the string, (b) the velocity of the waves, and (c) the mass of the hanging mass.