College Physics With an Integrated Approach to Forces and Kinematics

Alan Giambattista, Betty McCarthy Richardson , Robert C. Richardson

Chapter 11

Waves - all with Video Answers

Educators

Chapter Questions

Problem 1

The intensity of sunlight that reaches Earth's atmosphere is $1400 \mathrm{~W} / \mathrm{m}^{2}$. What is the intensity of the sunlight that reaches Jupiter? Jupiter is $5.2$ times as far from the Sun as Earth. [Hint: Treat the Sun as an isotropic source of light waves.

Narayan Hari

Narayan Hari

Numerade Educator

Problem 2

Michelle is enjoying a picnic across the valley from a cliff. She is playing music on her radio (assume it to be an isotropic source) and notices an echo from the cliff. She claps her hands and the echo takes $1.5 \mathrm{~s}$ to return.
(a) Given that the speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$ on that day, how far away is the cliff? (b) If the intensity of the music $1.0 \mathrm{~m}$ from the radio is $1.0 \times 10^{-5} \mathrm{~W} / \mathrm{m}^{2}$, what is the intensity of the music arriving at the cliff?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 3

The intensity of the sound wave from a jet airplane as it is taking off is $1.0 \times 10^{2} \mathrm{~W} / \mathrm{m}^{2}$ at a distance of $5.0 \mathrm{~m}$. What is the intensity of the sound wave that reaches the ears of a person standing at a distance of $120 \mathrm{~m}$ from the runway? Assume that the sound wave radiates from the airplane equally in all directions.

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 4

At what rate in watts does the jet airplane in Problem 3 radiate energy in the form of sound waves?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 5

The Sun emits electromagnetic waves (including light) equally in all directions. The intensity of the waves at Earth's upper atmosphere is $1.4 \mathrm{~kW} / \mathrm{m}^{2}$. At what rate does the Sun emit electromagnetic waves? (In other words, what is the power output?)

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 6

(a) What is the speed of propagation of the pulse shown in the figure? (b) At what average speed does the point at $x=2.0 \mathrm{~m}$ move during this time interval?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 7

(a) What is the position of the peak of the pulse shown in the figure with Problem 6 at $t=3.00 \mathrm{~s} ?$ (b) When does the peak of the pulse arrive at $x=4.00 \mathrm{~m}$ ?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 8

When the tension in a cord is $75 \mathrm{~N}$, the wave speed is $140 \mathrm{~m} / \mathrm{s}$. What is the linear mass density of the cord?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 9

A metal guitar string has a linear mass density of $\mu=3.20 \mathrm{~g} / \mathrm{m} .$ What is the speed of transverse waves on this string when its tension is $90.0 \mathrm{~N}$ ?

Eduardo Muntaner

Eduardo Muntaner

Numerade Educator

Problem 10

Two strings, each $15.0 \mathrm{~m}$ long, are stretched side by side. One string has a mass of $78.0 \mathrm{~g}$ and a tension of $180.0 \mathrm{~N}$. The second string has a mass of $58.0 \mathrm{~g}$ and a tension of $160.0 \mathrm{~N}$. A pulse is generated at one end of each string simultaneously. On which string will the pulse move faster? Once the faster pulse reaches the far end of its string, how much additional time will the slower pulse require to reach the end of its string?

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 11

A uniform string of length $10.0 \mathrm{~m}$ and weight $0.25 \mathrm{~N}$ is attached to the ceiling. A weight of $1.00 \mathrm{kN}$ hangs from its lower end. The lower end of the string is suddenly displaced horizontally. How long does it take the resulting wave pulse to travel to the upper end? [Hint: Is the weight of the string negligible in comparison with that of the hanging mass?]

Anthony Jansen-Yochim

Anthony Jansen-Yochim

Numerade Educator

Problem 12

What is the speed of a wave whose frequency and wavelength are $500.0 \mathrm{~Hz}$ and $0.500 \mathrm{~m}$, respectively?

Narayan Hari

Numerade Educator

Problem 13

What is the wavelength of a wave whose speed and period are $75.0 \mathrm{~m} / \mathrm{s}$ and $5.00 \mathrm{~ms}$, respectively?

Eduardo Muntaner

Eduardo Muntaner

Numerade Educator

Problem 14

What is the frequency of a wave whose speed and wavelength are $120 \mathrm{~m} / \mathrm{s}$ and $30.0 \mathrm{~cm}$, respectively?

Alick Cushing

Numerade Educator

Problem 15

The speed of sound in air at room temperature is $340 \mathrm{~m} / \mathrm{s}$.
(a) What is the frequency of a sound wave in air with wavelength $1.0 \mathrm{~m}$ ? (b) What is the frequency of a radio wave with the same wavelength? (Radio waves are electromagnetic waves that travel at $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$ in air or in vacuum.)

Alick Cushing

Numerade Educator

Problem 16

Light visible to humans consists of electromagnetic waves with wavelengths (in air) in the range $400-700 \mathrm{~nm}$ $\left(4.0 \times 10^{-7} \mathrm{~m}\right.$ to $\left.7.0 \times 10^{-7} \mathrm{~m}\right)$. The speed of light in air is $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}$. What are the frequencies of electromagnetic waves that are visible?

Alick Cushing

Numerade Educator

Problem 17

A fisherman notices a buoy bobbing up and down in the water in ripples produced by waves from a passing speedboat. These waves travel at $2.5 \mathrm{~m} / \mathrm{s}$ and have a wavelength of $7.5 \mathrm{~m}$. At what frequency does the buoy bob up and down?

Alick Cushing

Numerade Educator

Problem 18

You are swimming in the ocean as water waves with wavelength $9.6 \mathrm{~m}$ pass by. What is the closest distance that another swimmer could be so that his motion is exactly opposite yours (he goes up when you go down)?

Narayan Hari

Numerade Educator

Problem 19

What is the speed of the wave represented by $y(x, t)=$ $A \sin (k x-\omega t)$, where $k=6.0 \mathrm{rad} / \mathrm{cm}$ and $\omega=5.0 \mathrm{rad} / \mathrm{s} ?$

Alick Cushing

Numerade Educator

Problem 20

The equation of a wave is
$$
y(x, t)=(3.5 \mathrm{~cm}) \sin \left\{\frac{\pi}{3.0 \mathrm{~cm}}[x-(66 \mathrm{~cm} / \mathrm{s}) t]\right\}
$$
Find (a) the amplitude and (b) the wavelength of this wave.

Alick Cushing

Numerade Educator

Problem 21

A wave on a string has equation
$$
y(x, t)=(4.0 \mathrm{~mm}) \sin (\omega t-k x)
$$
where $\omega=6.0 \times 10^{2} \mathrm{rad} / \mathrm{s}$ and $k=6.0 \mathrm{rad} / \mathrm{m} .$ (a) What is
the amplitude of the wave? (b) What is the wavelength?
(c) What is the period? (d) What is the wave speed?
(e) In which direction does the wave travel?

Alick Cushing

Numerade Educator

Problem 22

A transverse wave on a string is described by the equation $y(x, t)=(2.20 \mathrm{~cm}) \sin [(130 \mathrm{rad} / \mathrm{s}) t+(15 \mathrm{rad} / \mathrm{m}) x]$.
(a) What is the maximum transverse speed of a point on the string? (b) What is the maximum transverse acceleration of a point on the string? (c) How fast does the wave move along the string? (d) Why is your answer to
(c) different from the answer to (a)?

Alick Cushing

Numerade Educator

Problem 23

Write an equation for a sine wave with amplitude $0.120 \mathrm{~m}$, wavelength $0.300 \mathrm{~m}$, and wave speed $6.40 \mathrm{~m} / \mathrm{s}$ traveling in the $-x$ -direction.

Alick Cushing

Numerade Educator

Problem 24

Write the equation for a transverse sinusoidal wave with a maximum amplitude of $2.50 \mathrm{~cm}$ and an angular frequency of $2.90 \mathrm{rad} / \mathrm{s}$ that is moving along the positive $x$ -direction with a wave speed that is $5.00$ times as fast as the maximum speed of a point on the string. Assume that at time $t=0$, the point $x=0$ is at $y=0$ and then moves in the $-y$ -direction in the next instant of time.

Alick Cushing

Numerade Educator

Problem 25

A sine wave is traveling to the right on a cord. The lighter line in the figure represents the shape of the cord at time $t=0 ;$ the darker line represents the shape of the cord at time $t=0.10 \mathrm{~s}$. (Note that the horizontal and vertical scales are different.) What are (a) the amplitude and (b) the wavelength of the wave? (c) What is the speed of the wave? What are (d) the frequency and
(e) the period of the wave?

Narayan Hari

Numerade Educator

Problem 26

(a) Plot a graph for
$$
y(x, t)=(4.0 \mathrm{~cm}) \sin [(378 \mathrm{rad} / \mathrm{s}) t-(314 \mathrm{rad} / \mathrm{cm}) x]
$$
versus $x$ at $t=0$ and at $t=\frac{1}{480} \mathrm{~s}$. From the plots determine the amplitude, wavelength, and speed of the wave.
(b) For the same function, plot a graph of $y(x, t)$ versus $t$ at $x=0$ and find the period of the vibration. Show that $\lambda=v T .$

Manish Jain

Numerade Educator

Problem 27

For a transverse wave on a string described by
$y(x, t)=(0.0050 \mathrm{~m}) \cos [(4.0 \pi \mathrm{rad} / \mathrm{s}) t-(1.0 \pi \mathrm{rad} / \mathrm{m}) x]$
find the maximum speed and the maximum acceleration of a point on the string. Plot graphs for one cycle of displacement $y$ versus $t$, velocity $v_{r}$ versus $t$, and acceleration $a_{y}$ versus $t$ at the point $x=0$.

Manish Jain

Numerade Educator

Problem 28

A transverse wave on a string is described by
$y(x, t)=(1.2 \mathrm{~mm}) \sin [(2.0 \pi \mathrm{rad} / \mathrm{s}) t-(0.50 \pi \mathrm{rad} / \mathrm{m}) x]$
Plot the displacement $y$ and the velocity $v_{y}$ versus $t$ for one complete cycle of the point $x=0$ on the string.

Manish Jain

Numerade Educator

Problem 29

(a) Sketch graphs of $y$ versus $x$ for the function
$$
y(x, t)=(0.80 \mathrm{~mm}) \sin (k x-\omega t)
$$
for the times $t=0,0.96 \mathrm{~s}$, and $1.92 \mathrm{~s}$. Make all three graphs of the same axes, using a solid line for the first, a dashed line for the second, and a dotted line for the third. Use the values $k=\pi /(5.0 \mathrm{~cm})$ and $\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s} .$
(b) Repeat part (a) for the function
$$
y(x, t)=(0.50 \mathrm{~mm}) \sin (k x+\omega t)
$$
(c) Which function represents a wave traveling in the $-x$ direction and which represents a wave traveling in the $+x$ -direction?

Manish Jain

Numerade Educator

Problem 30

The drawing shows a snapshot of a transverse wave traveling along a string at $10.0 \mathrm{~m} / \mathrm{s}$. The equation for the wave is $y(x, t)=A \cos (\omega t+k x)$. (a) Is the wave moving to the right or to the left? (b) What are the numerical values of $A$, $\omega$ and $k ?$ (c) At what times could this snapshot have been taken? (Give the three smallest nonnegative possibilities.)

Narayan Hari

Numerade Educator

Problem 31

Two pulses on a cord at time $t=0$ are moving toward each other; the speed of each pulse is $40 \mathrm{~cm} / \mathrm{s}$. Sketch the shape of the cord at $0.15,0.25$, and $0.30 \mathrm{~s}$.

Manish Jain

Numerade Educator

Problem 32

Two pulses on a cord at time $t=0$ are moving toward one another; the speed of each pulse is $2.5 \mathrm{~m} / \mathrm{s}$. Sketch the shape of the cord at $0.60,0.80$, and $0.90 \mathrm{~s}$.

Manish Jain

Numerade Educator

Problem 33

Using graph paper, sketch two identical sine waves of amplitude $4.0 \mathrm{~cm}$ that differ in phase by (a) $\pi / 3 \mathrm{rad}$ $\left(60^{\circ}\right)$ and (b) $\pi / 2$ rad $\left(90^{\circ}\right)$. Find the amplitude of the superposition of the two waves in each case.

Brandy Heflin

Numerade Educator

Problem 34

Two traveling sine waves, identical except for a phase difference $\phi$, add so that their superposition produces another traveling wave with the same amplitude as the two component waves. What is the phase difference hetween the two wayes?

Alick Cushing

Numerade Educator

Problem 35

A traveling sine wave is the result of the superposition of two other sine waves with equal amplitudes, wavelengths, and frequencies. The two component waves each have amplitude $5.00 \mathrm{~cm}$. If the superposition wave has amplitude $6.69 \mathrm{~cm}$, what is the phase difference $\phi$ between the component waves? [Hint: Let $y_{1}=A \sin (\omega t+k x)$ and $y_{2}=A \sin (\omega t+k x-\phi) .$ Make
use of the trigonometric identity (Appendix A.7) for $\sin \alpha+\sin \beta$ when finding $y=y_{1}+y_{2}$ and identify the
new amplitude in terms of the original amplitude.]

Alick Cushing

Numerade Educator

Problem 36

Light of wavelength $0.500 \mu \mathrm{m}$ (in air) enters the water in a swimming pool. The speed of light in water is $0.750$ times the speed in air. What is the wavelength of the light in water?

Narayan Hari

Numerade Educator

Problem 37

When does the string first look completely flat for $t>0 ?$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

When is the first time for $t>0$ that the string looks exactly as it does at $t=0 ?$

Manik Pulyani

Numerade Educator

Problem 39

Two waves with identical frequency but different amplitudes $A_{1}=5.0 \mathrm{~cm}$ and $A_{2}=3.0 \mathrm{~cm}$, occupy the same region of space (are superimposed). (a) At what phase difference does the resulting wave have the largest amplitude? What is the amplitude of the resulting wave in that case? (b) At what phase difference does the resulting wave have the smallest amplitude and what is its amplitude? (c) What is the ratio of the largest and smallest amplitudes?

Alick Cushing

Numerade Educator

Problem 40

Two waves with identical frequency but different amplitudes $A_{1}=6.0 \mathrm{~cm}$ and $A_{2}=3.0 \mathrm{~cm}$, occupy the same region of space (i.e., are superimposed). (a) At what phase difference will the resulting wave have the highest intensity? What is the amplitude of the resulting wave in that case? (b) At what phase difference will the resulting wave have the lowest intensity and what will its amplitude be? (c) What is the ratio of the two intensities?

Alick Cushing

Numerade Educator

Problem 41

A sound wave with intensity $25 \mathrm{~mW} / \mathrm{m}^{2}$ interferes constructively with a sound wave that has an intensity of $15 \mathrm{~mW} / \mathrm{m}^{2}$. What is the intensity of the superposition of the two? ( Wh tutorial: superposition)

Alick Cushing

Numerade Educator

Problem 42

A sound wave with intensity $25 \mathrm{~mW} / \mathrm{m}^{2}$ interferes destructively with a sound wave that has an intensity of $28 \mathrm{~mW} / \mathrm{m}^{2} .$ What is the intensity of the superposition of the two?

Alick Cushing

Numerade Educator

Problem 43

Two coherent sound waves have intensities of $0.040 \mathrm{~W} / \mathrm{m}^{2}$ and $0.090 \mathrm{~W} / \mathrm{m}^{2}$ where you are listening. (a) If the waves interfere constructively, what is the intensity that you hear? (b) What if they interfere destructively? (c) If they were incoherent, what would be the intensity? [Hint: If your answers are correct, then (c) is the average of (a) and (b).]

Alick Cushing

Numerade Educator

Problem 44

While testing speakers for a concert, Tomás sets up two speakers to produce sound waves at the same frequency, which is between $100 \mathrm{~Hz}$ and $150 \mathrm{~Hz}$. The two speakers vibrate in phase with one another. He notices that when he listens at certain locations, the sound is very soft (a minimum intensity compared to nearby points). One such point is $25.8 \mathrm{~m}$ from one speaker and $37.1 \mathrm{~m}$ from the other. What are the possible frequencies of the sound waves coming from the speakers? (The speed of sound in air is $343 \mathrm{~m} / \mathrm{s}$.)

Manish Jain

Numerade Educator

Problem 45

In order to decrease the fundamental frequency of a guitar string by $4.0 \%$, by what percentage should you reduce the tension?

Narayan Hari

Numerade Educator

Problem 46

The tension in a guitar string is increased by $15 \%$. What happens to the fundamental frequency of the string?

Alick Cushing

Numerade Educator

Problem 47

A standing wave has wavenumber $2.0 \times 10^{2} \mathrm{rad} / \mathrm{m} .$ What is the distance between two adjacent nodes?

Alick Cushing

Numerade Educator

Problem 48

A harpsichord string of length $1.50 \mathrm{~m}$ and linear mass density $25.0 \mathrm{mg} / \mathrm{m}$ vibrates at a (fundamental) frequency of $450.0 \mathrm{~Hz}$. (a) What is the speed of the transverse string waves? (b) What is the tension? (c) What are the wavelength and frequency of the sound wave in air produced by vibration of the string? (The speed of sound in air at room temperature is $340 \mathrm{~m} / \mathrm{s}$.)

Alick Cushing

Numerade Educator

Problem 49

A cord of length $1.5 \mathrm{~m}$ is fixed at both ends. Its mass per unit length is $1.2 \mathrm{~g} / \mathrm{m}$ and the tension is $12 \mathrm{~N}$. (a) What is the frequency of the fundamental oscillation? (b) What tension is required if the $n=3$ mode has a frequency of $0.50 \mathrm{kHz} ?$

Alick Cushing

Numerade Educator

Problem 50

Tension is maintained in a string by attaching one. end to a wall and by hanging a $2.20-\mathrm{kg}$ object from the other end of the string after it passes over a pulley that is $2.00 \mathrm{~m}$ from the wall. The string has a mass per unit length of $3.55 \mathrm{mg} / \mathrm{m}$. What is the fundamental frequency of this string?

Alick Cushing

Numerade Educator

Problem 51

A guitar's E-string has length $65 \mathrm{~cm}$ and is stretched to a tension of $82 \mathrm{~N}$. It vibrates at a fundamental frequency of $329.63 \mathrm{~Hz}$. Determine the mass per unit length of the string.

Alick Cushing

Numerade Educator

Problem 52

A string $2.0 \mathrm{~m}$ long is held fixed at both ends. If a sharp blow is applied to the string at its center, it takes $0.050 \mathrm{~s}$ for the pulse to travel to the ends of the string and return to the middle. What is the fundamental frequency of oscillation for this string?

Alick Cushing

Numerade Educator

Problem 53

A $1.6$ -m-long string fixed at both ends vibrates at resonan frequencies of $780 \mathrm{~Hz}$ and $1040 \mathrm{~Hz}$, with no other reso nant frequency between these values. (a) What is the fun. damental frequency of this string? (b) When the tension in the string is $1200 \mathrm{~N}$, what is the total mass of the string?

Alick Cushing

Numerade Educator

Problem 54

A certain string has a mass per unit length of $0.120 \mathrm{~g} / \mathrm{m}$. It is attached to a vibrating device and weight similar to that shown in Figure $11.22 .$ The vibrator oscillates at a constant frequency of $110 \mathrm{~Hz}$. How heavy should the weight be in order to produce standing waves in a string of length $42 \mathrm{~cm}$ ?

Manish Jain

Numerade Educator

Problem 55

The longest "string" (a thick metal wire) on a particular piano is $2.0 \mathrm{~m}$ long and has a tension of $300.0 \mathrm{~N}$. It vibrates with a fundamental frequency of $27.5 \mathrm{~Hz}$. What is the total mass of the wire?

Alick Cushing

Numerade Educator

Problem 56

Suppose that a string of length $L$ and mass $\bar{m}$ is under tension $F$. (a) Show that $\sqrt{F L / m}$ has units of speed.
(b) Show that there is no other combination of $L, m$, and $F$ with units of speed. [Hint: Of the dimensions of the three quantities $L, m$, and $F$, only $F$ includes time. $]$ Thus, the speed of transverse waves on the string can only be some dimensionless constant times $\sqrt{F L / m}$.

Alick Cushing

Numerade Educator

Problem 57

The speed of waves on a lake depends on frequency. For waves of frequency $1.0 \mathrm{~Hz}$, the wave speed is $1.56 \mathrm{~m} / \mathrm{s} ;$ for $2.0-\mathrm{Hz}$ waves, the speed is $0.78 \mathrm{~m} / \mathrm{s}$. The 2.0-Hz waves from a speedboat's wake reach you $120 \mathrm{~s}$ after the $1.0-\mathrm{Hz}$ waves generated by the same boat. How far away is the boat?

Narayan Hari

Numerade Educator

Problem 58

A transverse wave on a string is described by
$y(x, t)=(1.2 \mathrm{~cm}) \sin [(0.50 \pi \mathrm{rad} / \mathrm{s}) t-(1.00 \pi \mathrm{rad} / \mathrm{m}) x]$
Find the maximum velocity and the maximum acceleration of a point on the string. Plot graphs for displacement $y$ versus $t$, velocity $v_{y}$ versus $t$, and acceleration $a_{y}$ versus $t$ at $x=0$.

Manish Jain

Numerade Educator

Problem 59

What is the wavelength of the radio waves transmitted by an FM station at $90 \mathrm{MHz}$ ? (Radio waves travel at $3.0 \times 10^{8} \mathrm{~m} / \mathrm{s}_{.}$

Alick Cushing

Numerade Educator

Problem 60

A longitudinal wave has a wavelength of $10 \mathrm{~cm}$ and an amplitude of $5.0 \mathrm{~cm}$ and travels in the $y$ -direction. The wave speed in this medium is $80 \mathrm{~cm} / \mathrm{s}$. (a) Describe the motion of a particle in the medium as the wave travels through the medium. (b) How would your answer differ if the wave were transverse instead?

Keshav Singh

Numerade Educator

Problem 61

An underground explosion sends out both transverse (S waves) and longitudinal (P waves) mechanical wave pulses (seismic waves) through the crust of the Earth. Suppose the speed of transverse waves is $8.0 \mathrm{~km} / \mathrm{s}$ and that of longitudinal waves is $10.0 \mathrm{~km} / \mathrm{s}$. On one occasion, both waves follow the same path from a source to a detector (a seismograph); the longitudinal pulse arrives $2.0 \mathrm{~s}$ before the transverse pulse. What is the distance between the source and the detector?

Alick Cushing

Numerade Educator

Problem 62

The graph shows ground vibrations recorded by a seismograph $180 \mathrm{~km}$ from the focus of a small earthquake. It took the waves $30.0 \mathrm{~s}$ to travel from their source to the seismograph. Estimate the wavelength.

Narayan Hari

Numerade Educator

Problem 63

When the string of a guitar is pressed against a fret, the shortened string vibrates at a frequency $5.95 \%$ higher than when the previous fret is pressed. If the length of the part of the string that is free to vibrate is $64.8 \mathrm{~cm}$, how far from one end of the string are the first three frets located?

Manish Jain

Numerade Educator

Problem 64

A guitar string has a fundamental frequency of $300.0 \mathrm{~Hz}$.
(a) What are the next three lowest standing wave frequencies? (b) If you press a finger lightly against the string at its midpoint so that both sides of the string can still vibrate, you create a node at the midpoint. What are the lowest four standing wave frequencies now? (c) If you press hard at the same point, only one side of the string can vibrate. What are the lowest four standing wave frequencies?

Manish Jain

Numerade Educator

Problem 65

A sign is hanging from a single metal wire, as shown in part (a) of the drawing. The shop owner notices that the wire vibrates at a fundamental resonance frequency of $660 \mathrm{~Hz}$, which irritates his customers.
(a)
(b)
In an attempt to fix the problem, the shop owner cuts the wire in half and hangs the sign from the two halves, as shown in part (b). Assuming the tension in the two wires to be the same, what is the new fundamental frequency of each wire?

Manish Jain

Numerade Educator

Problem 66

(a) Write an equation for a surface seismic wave moving along the $-x$ -axis with amplitude $2.0 \mathrm{~cm}$, period $4.0 \mathrm{~s}$, and wavelength $4.0 \mathrm{~km}$. Assume the wave is harmonic, $x$ is measured in $\mathrm{m}$, and $t$ is measured in s. (b) What is the maximum speed of the ground as the wave moves by?
(c) What is the wave speed?

Alick Cushing

Numerade Educator

Problem 67

The formula for the speed of transverse waves on a spring is the same as for a string. (a) A spring is stretched to a length much greater than its relaxed length. Explain why the tension in the spring is approximately proportional to the length. (b) A wave takes $4.00 \mathrm{~s}$ to travel from one end of such a spring to the other. Then the length is increased $10.0 \%$. Now how long does a wave take to travel the length of the spring? [Hint: Is the mass per unit length constant?]

Manish Jain

Numerade Educator

Problem 68

Deep-water waves are dispersive (their wave speed depends on the wavelength). The restoring force is provided by gravity. Using dimensional analysis, find out how the speed of deep-water waves depends on wavelength $\lambda$, assuming that $\lambda$ and $g$ are the only relevant quantities. (Mass density does not enter into the expression because the restoring force, arising from the weight of the water, is itself proportional to the mass density.)

Narayan Hari

Numerade Educator

Problem 69

In contrast to deep-water waves, shallow ripples on the surface of a pond are due to surface tension. The surface tension $\gamma$ of water characterizes the restoring force; the mass density $\rho$ of water characterizes the water's inertia. Use dimensional analysis to determine whether the surface waves are dispersive (the wave speed depends on the wavelength) or nondispersive (their wave speed is independent of wavelength). [Hint: Start by assuming that the wave speed is determined by $\gamma, \rho$, and the wavelength $\lambda$.]

Narayan Hari

Numerade Educator

Problem 70

A seismic wave is described by the equation
$y(x, t)=(7.00 \mathrm{~cm}) \cos [(6.00 \pi \mathrm{rad} / \mathrm{cm}) x+(20.0 \pi \mathrm{rad} / \mathrm{s}) t]$
The wave travels through a uniform medium in the $x$ -direction. (a) Is this wave moving right $(+x$ -direction) or left (-x-direction)? (b) How far from their equilibrium positions do the particles in the medium move?
(c) What is the frequency of this wave? (d) What is the wavelength of this wave? (e) What is the wave speed?
(f) Describe the motion of a particle that is at $y=7.00 \mathrm{~cm}$ and $x=0$ when $t=0 .(\mathrm{g})$ Is this wave transverse or longitudinal?

Narayan Hari

Numerade Educator

Problem 71

The drawing shows a snapshot of a transverse wave moving to the left on a string. The wave speed is $10.0 \mathrm{~m} / \mathrm{s}$. At the instant the snapshot is taken, (a) in what direction is point $A$ moving? (b) In what direction is point $B$ moving? (c) At which of these points is the speed of the string segment (not the wave speed) larger? Explain.

Manish Jain

Numerade Educator

Problem 72

Consider a point just to the left of point $A$ in the drawing with Problem 71 . Plot the position of that point and the velocity of that point as a function of time as the wave passes the point.

Narayan Hari

Numerade Educator

Problem 73

Two speakers spaced a distance $1.5 \mathrm{~m}$ apart emit coherent sound waves at a frequency of $680 \mathrm{~Hz}$ in all directions. The waves start out in phase with each other. A listener walks in a circle of radius greater than one meter centered on the midpoint of the two speakers. At how many points does the listener observe destructive interference? The listener and the speakers are all in the same horizontal plane and the speed of sound is $340 \mathrm{~m} / \mathrm{s}$. [Hint: Start with a diagram; then determine the maximum path difference between the two waves at points on the circle.] Experiments like this must be done in a special room so that reflections are negligible.

Manish Jain

Numerade Educator

Problem 74

(a) Use a graphing calculator or computer graphing program to plot $y$ versus $x$ for the function
$$
y(x, t)=(5.0 \mathrm{~cm})[\sin (k x-\omega t)+\sin (k x+\omega t)]
$$
for the times $t=0,1.0 \mathrm{~s}$, and $2.0 \mathrm{~s}$. Use the values $k=\pi /(5.0 \mathrm{~cm})$ and $\omega=(\pi / 6.0) \mathrm{rad} / \mathrm{s} .$ (b) Is this a trav-
eling wave? If not. what kind of wave is it?

Alick Cushing

Numerade Educator

Problem 75

Show that the amplitudes of the graphs you made in Problem 74 satisfy the equation $A^{\prime}=2 A \cos (\omega t)$, where $A^{\prime}$ is the amplitude of the wave you plotted and $A$ is $5.0 \mathrm{~cm}$, the amplitude of the waves that were added together.

Narayan Hari

Numerade Educator

Problem 76

The pulse travels on a string whose ends at $x=0$ and $x=4.0 \mathrm{~m}$ are both fixed in place. Sketch the shape of the string at $t=2.2 \mathrm{~s}$.

Manish Jain

Numerade Educator

Problem 77

The pulse travels on a string whose ends at $x=0$ and $x=4.0 \mathrm{~m}$ are both fixed in place. Sketch the shape of the string at $t=1.6 \mathrm{~s}$.

Manish Jain

Numerade Educator