Physics for Scientists and Engineers with Modern Physics

Douglas C. Giancoli

Chapter 15

Wave Motion - all with Video Answers

Educators

+ 7 more educators

Chapter Questions

Problem 1

(I) A fisherman notices that wave crests pass the bow of his anchored boat every 3.0 s. He measures the distance between two crests to be 8.0 $\mathrm{m}$ . How fast are the waves traveling?

Sachin Rao

Sachin Rao

Numerade Educator

Problem 2

(I) A sound wave in air has a frequency of 262 $\mathrm{Hz}$ and travels with a speed of 343 $\mathrm{m} / \mathrm{s} .$ How far apart are the wave crests (compressions)?

Kai Chen

Princeton University

Problem 3

(I) Calculate the speed of longitudinal waves in $(a)$ water, $(b)$ granite, and $(c)$ steel.

Sachin Rao

Numerade Educator

Problem 4

(I) AM radio signals have frequencies between 550 $\mathrm{kHz}$ and 1600 $\mathrm{kHz}$ (kilohertz) and travel with a speed of $3.0 \times 10^{8} \mathrm{m} / \mathrm{s}$ . What are the wavelengths of these signals? On FM the frequencies range from 88 $\mathrm{MHz}$ to 108 $\mathrm{MHz}$ (megahertz) and travel at the same speed. What are their wavelengths?

Kai Chen

Princeton University

Problem 5

(I) Determine the wavelength of a $5800-\mathrm{Hz}$ sound wave
traveling along an iron rod.

Sachin Rao

Numerade Educator

Problem 6

(II) $\mathrm{A}$ cord of mass 0.65 $\mathrm{kg}$ is stretched between two supports 8.0 $\mathrm{m}$ apart. If the tension in the cord is 140 $\mathrm{N}$ , how long will it take a pulse to travel from one support to the other?

Kai Chen

Princeton University

Problem 7

(II) $\mathrm{A} .0 .40$ -kg cord is stretched between two supports, 7.8 $\mathrm{m}$ apart. When one support is struck by a hammer, a transverse wave travels down the cord and reaches the other support in 0.85 $\mathrm{s}$ . What is the tension in the cord?

Sachin Rao

Numerade Educator

Problem 8

(II) A sailor strikes the side of his ship just below the
surface of the sea. He hears the echo of the wave reflected
from the ocean floor directly below 2.8 s later. How deep is the
ocean at this point?

Guilherme Barros

Guilherme Barros

Numerade Educator

Problem 9

(II) A ski gondola is connected to the top of a hill by a steel
cable of length 660 $\mathrm{m}$ and diameter 1.5 $\mathrm{cm} .$ As the gondola
comes to the end of its run, it bumps into the terminal and
sends a wave pulse along the cable. It is observed that it
took 17 s for the pulse to return. (a) What is the speed of the
pulse? (b) What is the tension in the cable?

Sachin Rao

Numerade Educator

Problem 10

(II) $\mathrm{P}$ and $\mathrm{S}$ waves from an earthquake travel at different speeds, and this difference helps locate the earthquake "epicenter" (where the disturbance took place). (a) Assuming
typical speeds of 8.5 $\mathrm{km} / \mathrm{s}$ and 5.5 $\mathrm{km} / \mathrm{s}$ for $\mathrm{P}$ and $\mathrm{S}$ waves, respectively, how far away did the earthquake occur if a particular seismic station detects the arrival of these two types of waves 1.7 min apart? (b) Is one seismic station sufficient to determine the position of the epicenter? Explain.

Kai Chen

Princeton University

Problem 11

(II) The wave on a string shown in Fig. 33 is moving to the right with a speed of 1.10 $\mathrm{m} / \mathrm{s}$ . (a) Draw the shape of the string 1.00 s later and indicate which parts of the string are moving up and which down at that instant. (b) Estimate the vertical speed of point $\mathrm{A}$ on the string at the instant shown in the Figure.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 12

(II) A 5.0 $\mathrm{kg}$ ball hangs from a steel wire 1.00 $\mathrm{mm}$ in
diameter and 5.00 $\mathrm{m}$ long. What would be the speed of a
wave in the steel wire?

Kai Chen

Princeton University

Problem 13

(II) Two children are sending signals along a cord of total mass 0.50 kg tied between tin cans with a tension of 35 $\mathrm{N}$ . It takes the vibrations in the string 0.50 $\mathrm{s}$ to go from one child
to the other. How far apart are the children?

Sachin Rao

Numerade Educator

Problem 14

(II) Dimensional analysis. Waves on the surface of the ocean do not depend significantly on the properties of water such as density or surface tension. The primary "return force" for water piled up in the wave crests is due to the gravitational attraction of the Earth. Thus the speed $v(\mathrm{m} / \mathrm{s})$ of ocean waves depends on the acceleration due to gravity $g .$ It is reasonable to expect that $v$ might also depend on water depth $h$ and the wave's wavelength $\lambda$ . Assume the wave speed is given by the functional form $v=C g^{\alpha} h^{\beta} \lambda^{\gamma},$ where $\alpha, \beta, \gamma,$ and $C$ are numbers without dimension. $(a)$ In deep water, the water deep below does not affect the motion of waves at the surface. Thus $v$ should be independent of depth $h$ (i.e., $\beta=0 ) .$ Using only dimensional analysis,
determine the formula for the speed of surface waves in deep water. $(b)$ In shallow water, the speed of surface waves is found experimentally to be independent of the wavelength (i.e., $\gamma=0 ) .$ Using only dimensional analysis, determine the formula for the speed of waves in shallow water.

Kai Chen

Princeton University

Problem 15

(I) Two earthquake waves of the same frequency travel through the same portion of the Earth, but one is carrying 3.0 times the energy. What is the ratio of the amplitudes of the two waves?

Evgeny Kozyrev

Numerade Educator

Problem 16

(II) What is the ratio of $(a)$ the intensities, and $(b)$ the ampli-
tudes, of an earthquake $P$ wave passing through the Earth
and detected at two points 15 $\mathrm{km}$ and 45 $\mathrm{km}$ from the source?

Kai Chen

Princeton University

Problem 17

(II) Show that if damping is ignored, the amplitude $A$ of circular water waves decreases as the square root of the distance $r$ from the source: $A \propto 1 / \sqrt{r} .$

Sachin Rao

Numerade Educator

Problem 18

(II) The intensity of an earthquake wave passing through the Earth is measured to be $3.0 \times 10^{6} \mathrm{J} / \mathrm{m}^{2} \cdot \mathrm{s}$ at a distance of 48 $\mathrm{km}$ from the source. (a) What was its intensity when it passed a point only 1.0 $\mathrm{km}$ from the source?
(b) At what rate did energy pass through an area of 2.0 $\mathrm{m}^{2}$ at 1.0 $\mathrm{km}$ ?

Kai Chen

Princeton University

Problem 19

(II) A small steel wire of diameter 1.0 $\mathrm{mm}$ is connected to an oscillator and is under a tension of 7.5 $\mathrm{N}$ . The frequency of the oscillator is 60.0 $\mathrm{Hz}$ and it is observed that the amplitude of the wave on the steel wire is 0.50 $\mathrm{cm} .$ (a) What is the power output of the oscillator, assuming that the wave is not reflected back? (b) If the power output stays constant but
the frequency is doubled, what is the amplitude of the wave?

Sachin Rao

Numerade Educator

Problem 20

(II) Show that the intensity of a wave is equal to the energy
density (energy per unit volume) in the wave times the
wave speed.

Kai Chen

Princeton University

Problem 21

(II) $(a)$ Show that the average rate with which energy is transported along a cord by a mechanical wave of frequency $f$ and amplitude $A$ is $$\overline{P}=2 \pi^{2} \mu v f^{2} A^{2},$$ where $v$ is the speed of the wave and $\mu$ is the mass per unit length of the cord. (b) If the cord is under a tension $F_{T}=135 \mathrm{N}$ and has mass per unit length $0.10 \mathrm{kg} / \mathrm{m},$ what power is required to transmit $120-\mathrm{Hz}$ transverse waves of amplitude 2.0 $\mathrm{cm} ?$

Sachin Rao

Numerade Educator

Problem 22

(I) A transverse wave on a wire is given by $D(x, t)=$ 0.015 $\sin (25 x-1200 t)$ where $D$ and $x$ are in meters and $t$ is in seconds. (a) Write an expression for a wave with the same amplitude, wavelength, and frequency but traveling in the opposite direction. (b) What is the speed of either wave?

Kai Chen

Princeton University

Problem 23

(1) Suppose at $t=0,$ a wave shape is represented by $D=A \sin (2 \pi x / \lambda+\phi) ;$ that is, it differs from Eq. 9 by a constant phase factor $\phi .$ What then will be the equation for a wave traveling to the left along the $x$ axis as a function of $x$ and $t ?$

Sachin Rao

Numerade Educator

Problem 24

(II) A transverse traveling wave on a cord is represented by $D=0.22 \sin (5.6 x+34 t)$ where $D$ and $x$ are in meters and $t$ is in seconds. For this wave determine $(a)$ the wavelength, (b) frequency, (c) velocity (magnitude and direction), (d) amplitude, and $(e)$ maximum and minimum speeds of particles of the cord.

Kai Chen

Princeton University

Problem 25

(II) Consider the point $x=1.00 \mathrm{m}$ on the cord of Example 5 of "Wave Motion." Determine $(a)$ the maximum velocity of this point, and $(b)$ its maximum acceleration. (c) What is its velocity and acceleration at $t=2.50 \mathrm{s} ?$

Sachin Rao

Numerade Educator

Problem 26

(II) A transverse wave on a cord is given by $D(x, t)=$ $0.12 \sin (3.0 x-15.0 t),$ where $D$ and $x$ are in $\mathrm{m}$ and $t$ is in s. At $t=0.20 \mathrm{s},$ what are the displacement and velocity of the
point on the cord where $x=0.60 \mathrm{m} ?$

Kai Chen

Princeton University

Problem 27

(II) A transverse wave pulse travels to the right along a string with a speed $v=2.0 \mathrm{m} / \mathrm{s} .$ At $t=0$ the shape of the pulse is given by the function $$D=0.45 \cos (2.6 x+1.2),$$
where $D$ and $x$ are in meters.
$(a)$ Plot $D$ vs. $x$ at $t=0$ .
(b) Determine a formula for the wave pulse at any time $t$ assuming there are no frictional losses. $(c)$ Plot $D(x, t)$ vs. $x$ at $t=1.0 \mathrm{s}$ .
(d) Repeat parts $(b)$ and $(c)$ assuming the pulse is traveling to the left. Plot all 3 graphs on the same axes for easy comparison.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 28

(II) A 524 -Hz longitudinal wave in air has a speed of 345 $\mathrm{m} / \mathrm{s}$ . (a) What is the wavelength? (b) How much time is required for the phase to change by $90^{\circ}$ at a given point in
space? (c) At a particular instant, what is the phase difference (in degrees) between two points 4.4 $\mathrm{cm}$ apart?

Kai Chen

Princeton University

Problem 29

(II) Write the equation for the wave in Problem 28 traveling to the right, if its amplitude is $0.020 \mathrm{cm},$ and $D=-0.020 \mathrm{cm},$ at $t=0$ and $x=0 .$

Lainey Roebuck

Numerade Educator

Problem 30

(II) A sinusoidal wave traveling on a string in the negative $x$ direction has amplitude $1.00 \mathrm{cm},$ wavelength $3.00 \mathrm{cm},$ and frequency 245 $\mathrm{Hz}$ . At $t=0,$ the particle of string at $x=0$ is displaced a distance $D=0.80 \mathrm{cm}$ above the origin and is moving upward. (a) Sketch the shape of the wave at $t=0$ and (b) determine the function of $x$ and $t$ that describes the wave.

Kai Chen

Princeton University

Problem 31

(II) Determine if the function $D=A \sin k x \cos \omega t$ is a solution of the wave equation.

Sachin Rao

Numerade Educator

Problem 32

(II) Show by direct substitution that the following functions satisfy the wave equation:
$(a) \quad D(x, t)=A \ln (x+v t)$
(b) $D(x, t)=(x-v t)^{4}$

Keshav Singh

Numerade Educator

Problem 33

(II) Show that the wave forms of Eqs. 13 and 15 satisfy the wave equation, Eq. $16 .$
$$D(x, t)=A \sin \left[\frac{2 \pi}{\lambda}(x+v t)\right] (13 a)$$
$$=A \sin \left(\frac{2 \pi x}{\lambda}+\frac{2 \pi t}{T}\right) (13 b)$$
$$=A \sin (k x+\omega t) (13 b) $$
$$D(x, t)=D(x+v t) (15)$$
$$\frac{\partial^{2} D}{\partial x^{2}}=\frac{1}{v^{2}} \frac{\partial^{2} D}{\partial t^{2}} (16)$$

Sachin Rao

Numerade Educator

Problem 34

(II) Let two linear waves be represented by $D_{1}=f_{1}(x, t)$ and $D_{2}=f_{2}(x, t) .$ If both these waves satisfy the wave equation $(\mathrm{Eq.} .16),$ show that any combination $D=C_{1} D_{1}+C_{2} D_{2}$ does as well, where $C_{1}$ and $C_{2}$ are constants.

Kai Chen

Princeton University

Problem 35

(II) Does the function $D(x, t)=e^{-(k x-\omega t)^{2}}$ satisfy the wave equation? Why or why not?

Sachin Rao

Numerade Educator

Problem 36

(II) In deriving Eq. $2, \quad v=\sqrt{F_{\mathrm{T}} / \mu,}$ for the speed of a transverse wave on a string, it was assumed that the wave's amplitude $A$ is much less than its wavelength $\lambda$ .
Assuming a sinusoidal wave shape $D=A \sin (k x-\omega t),$ show via the partial derivative $v^{\prime}=\partial D / \partial t \quad$ that the assumption $A \ll \lambda$ implies that the maximum transverse speed $v_{\text { max }}^{\prime}$ the string itself is much less than the wave
velocity. If $A=\lambda / 100$ determine the ratio $v_{\max }^{\prime} / v.$
$v=\sqrt{\frac{F_{\mathrm{T}}}{\mu}} \quad \quad \left[ \begin{array}{l}{\text { transverse }} \\ {\text { wave on a cord }}\end{array}\right]$ (2)

Kai Chen

Princeton University

Problem 37

(II) A cord has two sections with linear densities of 0.10 $\mathrm{kg} / \mathrm{m}$ and $0.20 \mathrm{kg} / \mathrm{m},$ Fig. $34 .$ An incident wave, given by $D=(0.050 \mathrm{m}) \sin (7.5 x-12.0 t),$ where $x$ is in meters and $t$ in seconds, travels along the lighter cord.
(a) What is the wavelength on the lighter section of the cord?
(b) What is the tension in the cord?
(c) What is the wavelength when the wave travels on the heavier section?

Penny Riley

Numerade Educator

Problem 38

(II) Consider a sine wave traveling down the stretched two- part cord of Fig. $19 .$ Determine a formula $(a)$ for the ratio of the speeds of the wave in the two sections, $v_{H} / v_{L},$ and $(b)$
for the ratio of the wavelengths in the two sections. (The frequency is the same in both sections. Why? (c) Is the wavelength larger in the heavier cord or the lighter?

Kai Chen

Princeton University

Problem 39

(II) Seismic reflection prospecting is commonly used to map deeply buried formations containing oil. In this technique, a seismic wave generated on the Earth's surface (for example, by an explosion or falling weight) reflects from the subsurface formation and is detected upon its return to ground level. By placing ground-level detectors at a variety of locations relative to the source, and observing the variation in the source-to- detector travel times, the depth of the subsurface formation can be determined. (a) Assume a ground-level detector is placed a distance $x$ away from a seismic-wave source and that a horizontal boundary between overlying rock and a subsurface formation exists at depth $D$ (Fig. 35a). Determine an expression for the time $t$ taken by the reflected wave to travel
from source to detector, assuming the seismic wave propagates at constant speed $v .(b)$ Suppose several detectors are placed along a line at different distances $x$ from the source as in Fig. 35 $\mathrm{b}$ . Then, when a seismic wave is generated, the different travel times $t$ for each detector are measured. Starting with your result from part $(a),$ explain how a graph of $t^{2}$ vs. $x^{2}$ can be used to determine $D .$

Lainey Roebuck

Numerade Educator

Problem 40

(III) $\mathrm{A}$ cord stretched to a tension $F_{\mathrm{T}}$ consists of two sections
(as in Fig. 19) whose linear densities are $\mu_{1}$ and $\mu_{2}$ . Take $x=0$ to be the point (a knot) where they are joined, with $\mu_{1}$ referring to that section of cord to the left and $\mu_{2}$ that to the right. A sinusoidal wave, $D=A \sin \left[k_{1}\left(x-v_{1} t\right)\right],$ starts at the left end of the cord. When it reaches the knot, part of it is reflected and part is transmitted. Let the equation of the reflected wave be $D_{\mathrm{R}}=A_{\mathrm{R}} \sin \left[k_{1}\left(x+v_{1} t\right)\right]$ and that for the transmitted wave be $D_{\mathrm{T}}=A_{\mathrm{T}} \sin \left[k_{2}\left(x-v_{2} t\right)\right] .$ since the frequency must be the same in both sections, we have $\omega_{1}=\omega_{2}$ or $k_{1} v_{1}=k_{2} v_{2} .(a)$ Because the cord is continuous, a point an infinitesimal distance to the left of the knot has the same displacement at any moment (due to incident plus reflected waves) as a point just to the right of the knot (due to the transmitted wave). Thus show that $A=A_{\mathrm{T}}+A_{\mathrm{R}}.$
(b) Assuming that the slope $(\partial D / \partial x)$ of the cord just to the left of the knot is the same as the slope just to the right of the knot, show that the amplitude of the reflected wave is given by
$$A_{\mathrm{R}}=\left(\frac{v_{1}-v_{2}}{v_{1}+v_{2}}\right) A=\left(\frac{k_{2}-k_{1}{k_{2}+k_{1}}\right) A.$$
(c) What is $A_{\mathrm{T}}$ in terms of $A ?$

Kai Chen

Princeton University

Problem 41

(I) The two pulses shown in Fig. 36 are moving toward each other. (a) Sketch the shape of the string at the moment they directly overlap. (b) Sketch the shape of the string a few moments later. $(c)$ In Fig. 22 $\mathrm{a}$ , at the moment the pulses pass each other, the string is straight. What has happened to the energy at this moment?

Keshav Singh

Numerade Educator

Problem 42

(II) Suppose two linear waves of equal amplitude and frequency have a phase difference $\phi$ as they travel in the same medium. They can be represented by
$$D_{1}=A \sin (k x-\omega t)$$
$$D_{2}=A \sin (k x-\omega t+\phi).$$
$\begin{array}{l}{\text { (a) Use the trigonometric identity } \sin \theta_{1}+\sin \theta_{2}=} \\ {2 \sin \frac{1}{2}\left(\theta_{1}+\theta_{2}\right) \cos \frac{1}{2}\left(\theta_{1}-\theta_{2}\right) \text { to show that the resultant }} \\ {\text { wave is given by }}\end{array}$
$$D=\left(2 A \cos \frac{\phi}{2}\right) \sin \left(k x-\omega t+\frac{\phi}{2}\right).$$
$\begin{array}{l}{\text { (b) What is the amplitude of this resultant wave? Is the wave }} \\ {\text { purely sinusoidal, or not? (c) Show that constructive interference}} \\ {\text { occurs if } \phi=0,2 \pi, 4 \pi, \text { and so on, and destructive }} \\ {\text { interference occurs if } \phi=\pi, 3 \pi, 5 \pi, \text { etc. (d) Describe the }} \\ {\text { resultant wave, by equation and in words, if } \phi=\pi / 2 .}\end{array}$

Kai Chen

Princeton University

Problem 43

(1) A violin string vibrates at 441 $\mathrm{Hz}$ when unfingered. At what frequency will it vibrate if it is fingered one-third of the way down from the end? (That is, only two-thirds of the string vibrates as a standing wave.)

Sachin Rao

Numerade Educator

Problem 44

(I) If a violin string vibrates at 294 $\mathrm{Hz}$ as its fundamental frequency, what are the frequencies of the first four harmonics?

Kai Chen

Princeton University

Problem 45

(1) In an earthquake, it is noted that a footbridge oscillated up and down in a one-loop (fundamental standing wave) pattern once every 1.5 $\mathrm{s}$ . What other possible resonant periods of motion are there for this bridge? What frequencies do they correspond to?

Thomas Quigley

Numerade Educator

Problem 46

(I) A particular string resonates in four loops at a frequency
of 280 $\mathrm{Hz}$ . Name at least three other frequencies at which it
will resonate.

Kai Chen

Princeton University

Problem 47

(II) A cord of length 1.0 $\mathrm{m}$ has two equal-length sections with linear densities of 0.50 $\mathrm{kg} / \mathrm{m}$ and 1.00 $\mathrm{kg} / \mathrm{m}$ . The tension in the entire cord is constant. The ends of the cord are oscillated so that a standing wave is set up in the cord with a single node where the two sections meet. What is the ratio of the oscillatory frequencies?

Sachin Rao

Numerade Educator

Problem 48

(II) The velocity of waves on a string is 96 $\mathrm{m} / \mathrm{s}$ . If the frequency of standing waves is 445 $\mathrm{Hz}$ , how far apart are the two adjacent nodes?

Kai Chen

Princeton University

Problem 49

(1I) If two successive harmonics of a vibrating string are 240 $\mathrm{Hz}$ and 320 $\mathrm{Hz}$ , what is the frequency of the fundamental?

PS

Phil Surks

Numerade Educator

Problem 50

(II) A guitar string is 90.0 $\mathrm{cm}$ long and has a mass of 3.16 $\mathrm{g}$ . From the bridge to the support post $(=\ell)$ is 60.0 $\mathrm{cm}$ and the string is under a tension of 520 $\mathrm{N}$ . What are the frequencies of the fundamental and first two overtones?

Kai Chen

Princeton University

Problem 51

(II) Show that the frequency of standing waves on a cord of length $\ell$ and linear density $\mu,$ which is stretched to a tension $F_{T, \text { is given by }}$
$$f=\frac{n}{2 \ell} \sqrt{\frac{F_{\mathrm{T}}}{\mu}}$$
where $n$ is an integer.

Sachin Rao

Numerade Educator

Problem 52

(II) One end of a horizontal string of linear density $6.6 \times 10^{-4} \mathrm{kg} / \mathrm{m}$ is attached to a small-amplitude mechanical $120-\mathrm{Hz}$ oscillator. The string passes over a pulley, a distance $\ell=1.50 \mathrm{m}$ away, and weights are hung from this end, Fig. $37 .$ What mass $\mathrm{m}$ must be hung from this end of the string to produce $(a)$ one loop, $(b)$ two loops, and $(c)$ five loops of a standing wave? Assume the string at the oscillator is a node, which is nearly true.

Kai Chen

Princeton University

Problem 53

(II) In Problem $52,$ Fig. $37,$ the length of the string may be adjusted by moving the pulley. If the hanging mass $m$ is fixed at 0.070 $\mathrm{kg}$ , how many different standing wave patterns may be achieved by varying $\ell$ between 10 $\mathrm{cm}$ and 1.5 $\mathrm{m}$ ?

Sachin Rao

Numerade Educator

Problem 54

(II) The displacement of a standing wave on a string is given by $D=2.4 \sin (0.60 x) \cos (42 t),$ where $x$ and $D$ are in centimeters and $t$ is in seconds. (a) What is the distance ( cm) between nodes? (b) Give the amplitude, frequency, and speed of each of the component waves. (c) Find the speed of a particle of the string at $x=3.20 \mathrm{cm}$ when $t=2.5 \mathrm{s}$

Kai Chen

Princeton University

Problem 55

(II) The displacement of a transverse wave traveling on a string is represented by $D_{1}=4.2 \sin (0.84 x-47 t+2.1)$ where $D_{1}$ and $x$ are in $\mathrm{cm}$ and $t$ in $s$ . $(a)$ Find an equation that represents a wave which, when traveling in the opposite direction, will produce a standing wave when added to this one. (b) What is the equation describing the standing wave?

Sachin Rao

Numerade Educator

Problem 56

(II) When you slosh the water back and forth in a tub at just the right frequency, the water alternately rises and falls at each end, remaining relatively calm at the center. Suppose the frequency to produce such a standing wave in a 45-cm-wide tub is 0.85 Hz. What is the speed of the water wave?

Kai Chen

Princeton University

Problem 57

(II) A particular violin string plays at a frequency of 294 $\mathrm{Hz}$. If the tension is increased $15 \%,$ what will the new frequency be?

Sachin Rao

Numerade Educator

Problem 58

(II) Two traveling waves are described by the functions
$$
\begin{aligned} D_{1} &=A \sin (k x-\omega t) \\ D_{2} &=A \sin (k x+\omega t) \end{aligned}
$$
where $A=0.15 \mathrm{m}, k=3.5 \mathrm{m}^{-1},$ and $\omega=1.8 \mathrm{s}^{-1} .$ (a) Plot these two waves, from $x=0$ to a point $x(>0)$ that includes one full wavelength. Choose $t=1.0 \mathrm{s}$ . (b) Plot the sum of the two waves and identify the nodes and antinodes in the plot, and compare to the analytic (mathematical) representation.

Kai Chen

Princeton University

Problem 59

(II) Plot the two waves given in Problem 58 and their sum, as a function of time from $t=0$ to $t=T$ (one period). Choose $(a) x=0$ and $(b) x=\lambda / 4 .$ Interpret your results.

Chai Santi

Numerade Educator

Problem 60

(II) A standing wave on a 1.64 -m-long horizontal string displays three loops when the string vibrates at 120 $\mathrm{Hz}$ . The maximum swing of the string (top to bottom) at the center of each loop is 8.00 $\mathrm{cm} .$ (a) What is the function describing the standing wave? (b) What are the functions describing the two equal-amplitude waves traveling in opposite directions that make up the standing wave?

Kai Chen

Princeton University

Problem 61

(II) On an electric guitar, a "pickup" under each string transforms the string's vibrations directly into an electrical signal. If a pickup is placed 16.25 $\mathrm{cm}$ from one of the fixed ends of a 65.00 -cm-long string, which of the harmonics from $n=1$ to $n=12$ will not be "picked up" by this pickup?

Sachin Rao

Numerade Educator

Problem 62

(II) $\mathrm{A} 65$ -cm guitar string is fixed at both ends. In the frequency range between 1.0 and 2.0 $\mathrm{kHz}$ , the string is found to resonate only at frequencies $1.2,1.5,$ and 1.8 $\mathrm{kHz}$ . What is the speed of traveling waves on this string?

Kai Chen

Princeton University

Problem 63

(II) Two oppositely directed traveling waves given by $D_{1}=(5.0 \mathrm{mm}) \cos \left[\left(2.0 \mathrm{m}^{-1}\right) x-(3.0 \mathrm{rad} / \mathrm{s}) t\right]$ and $D_{2}=$ $(5.0 \mathrm{mm}) \cos \left[\left(2.0 \mathrm{m}^{-1}\right) x+(3.0 \mathrm{rad} / \mathrm{s}) t\right]$ form a standing wave. Determine the position of nodes along the $x$ axis.

Lainey Roebuck

Numerade Educator

Problem 64

(II) A wire is composed of aluminum with length $\ell_{1}=0.600 \mathrm{m}$ and mass per unit length $\mu_{1}=2.70 \mathrm{g} / \mathrm{m}$ joined to a steel section with length $\ell_{2}=0.882 \mathrm{m}$ and mass per unit length $\mu_{2}=7.80 \mathrm{g} / \mathrm{m} .$ This composite wire is fixed at both ends and held at a uniform tension of 135 $\mathrm{N}$ . Find the lowest frequency standing wave that can exist on this wire, assuming there is a node at the joint between aluminum and steel. How many nodes (including the two at the ends) does this standing wave possess?

Kai Chen

Princeton University

Problem 65

(I) An earthquake $P$ wave traveling 8.0 $\mathrm{km} / \mathrm{s}$ strikes a boundary within the Earth between two kinds of material. If it approaches the boundary at an incident angle of $52^{\circ}$ and the angle of refraction is $31^{\circ},$ what is the speed in the second medium?

Sachin Rao

Numerade Educator

Problem 66

(I) Water waves approach an underwater "shelf" where the velocity changes from 2.8 $\mathrm{m} / \mathrm{s}$ to 2.5 $\mathrm{m} / \mathrm{s} .$ If the incident wave crests make a $35^{\circ}$ angle with the shelf, what will be the angle of refraction?

Kai Chen

Princeton University

Problem 67

(II) A sound wave is traveling in warm air $\left(25^{\circ} \mathrm{C}\right)$ when it hits a layer of cold $\left(-15^{\circ} \mathrm{C}\right)$ denser air. If the sound wave hits the cold air interface at an angle of $33^{\circ}$ , what is the angle of refraction? The speed of sound as a function of temperature can be approximated by $v=(331+0.60 T) \mathrm{m} / \mathrm{s},$ where $T$ is in $^{\circ} \mathrm{C} .$

Sachin Rao

Numerade Educator

Problem 68

(II) Any type of wave that reaches a boundary beyond which its speed is increased, there is a maximum incident angle if there is to be a transmitted refracted wave. This maximum incident angle $\theta_{\text { iM }}$ corresponds to an angle of refraction equal to $90^{\circ} .$ If $\theta_{1}>\theta_{\mathrm{iM}},$ all the wave is reflected at the boundary and none is refracted, because this would correspond to sin $\theta_{\mathrm{r}}>1$ (where $\theta_{\mathrm{r}}$ is the angle of refraction), which is impossible. This phenomenon is referred to as total internal reflection. (a) Find a formula for $\theta$ iM using the law of refraction, Eq. $19 .(b)$ How far from the bank should a trout fisherman stand (Fig. 38$)$ so trout won't be frightened by his voice $(1.8 \mathrm{m}$ above the ground)? The speed of sound is about 343 $\mathrm{m} / \mathrm{s}$ in air and 1440 $\mathrm{m} / \mathrm{s}$ in water.
$$
\frac{\sin \theta_{2}}{\sin \theta_{1}}=\frac{v_{2}}{v_{1}}
$$

Kai Chen

Princeton University

Problem 69

(II) A longitudinal earthquake wave strikes a boundary between two types of rock at a $38^{\circ}$ angle. As the wave crosses the boundary, the specific gravity of the rock changes from 3.6 to 2.8 . Assuming that the elastic modulus is the same for both types of rock, determine the angle of
refraction.

Sachin Rao

Numerade Educator

Problem 70

(1I) A satellite dish is about 0.5 $\mathrm{m}$ in diameter. According to the user's manual, the dish has to be pointed in the direction of the satellite, but an error of about $2^{\circ}$ to either side is allowed without loss of reception. Estimate the wavelength of the electromagnetic waves (speed = $3 \times 10^{8} \mathrm{m} / \mathrm{s} )$ received by the dish.

Kai Chen

Princeton University

Problem 71

A sinusoidal traveling wave has frequency 880 $\mathrm{Hz}$ and phase velocity 440 $\mathrm{m} / \mathrm{s}$ . (a) At a given time, find the distance between any two locations that correspond to a difference in phase of $\pi / 6$ rad. (b) At a fixed location, by how much does the phase change during a time interval of $1.0 \times 10^{-4} \mathrm{s} ?$

Vishal Gupta

Numerade Educator

Problem 72

When you walk with a cup of coffee (diameter 8 $\mathrm{cm}$ ) at just the right pace of about one step per second, the coffee sloshes higher and higher in your cup until eventually it starts to spill over the top, Fig $39 .$ Estimate the speed of the waves in the coffee.

Kai Chen

Princeton University

Problem 73

Two solid rods have the same bulk modulus but one is 2.5 times as dense as the other. In which rod will the speed of longitudinal waves be greater, and by what factor?

Sachin Rao

Numerade Educator

Problem 74

Two waves traveling along a stretched string have the same frequency, but one transports 2.5 times the power of the other. What is the ratio of the amplitudes of the two waves?

Kai Chen

Princeton University

Problem 75

A bug on the surface of a pond is observed to move up and down a total vertical distance of $0.10 \mathrm{m},$ lowest to highest point, as a wave passes. (a) What is the amplitude of the wave? (b) If the height increases to $0.15 \mathrm{m},$ by what factor does the bug's maximum kinetic energy change?

Noah Crow

Numerade Educator

Problem 76

A guitar string is supposed to vibrate at 247 $\mathrm{Hz}$ , but is measured to actually vibrate at 255 $\mathrm{Hz}$ . By what percentage should the tension in the string be changed to get the frequency to the correct value?

Kai Chen

Princeton University

Problem 77

An earthquake-produced surface wave can be approximated by a sinusoidal transverse wave. Assuming a frequency of 0.60 $\mathrm{Hz}$ (typical of earthquakes, which actually include a mixture of frequencies), what amplitude is needed so that objects begin to leave contact with the ground?[Hint: Set the acceleration $a>g . ]$

Averell Hause

Carnegie Mellon University

Problem 78

A uniform cord of length $\ell$ and mass $m$ is hung vertically from a support. $(a)$ Show that the speed of transverse waves in this cord is $\sqrt{g h}$ , where $h$ is the height above the lower
end. (b) How long does it take for a pulse to travel upward from one end to the other?

Kai Chen

Princeton University

Problem 79

A transverse wave pulse travels to the right along a string with a speed $v=2.4 \mathrm{m} / \mathrm{s} .$ At $t=0$ the shape of the pulse is given by the function
$$
D=\frac{4.0 \mathrm{m}^{3}}{x^{2}+2.0 \mathrm{m}^{2}}
$$
where $D$ and $x$ are in meters. (a) Plot $D$ vs. $x$ at $t=0$ from $x=-10 \mathrm{m}$ to $x=+10 \mathrm{m}$ . (b) Determine a formula for the wave pulse at any time $t$ assuming there are no
frictional losses. (c) Plot $D(x, t)$ vs. $x$ at $t=1.00 \mathrm{s}$ . (d) Repeat parts ( $b$ ) and (c) assuming the pulse is traveling to the left.

Lainey Roebuck

Numerade Educator

Problem 80

(a) Show that if the tension in a stretched string is changed by a small amount $\Delta F_{\mathrm{T}},$ the frequency of the fundamental is changed by an amount $\Delta f=\frac{1}{2}\left(\Delta F_{\mathrm{T}} / F_{\mathrm{T}}\right) f .$ (b) By what percent must the tension in a piano string be increased or decreased to raise the frequency from 436 $\mathrm{Hz}$ to 442 $\mathrm{Hz}$ . (c) Does the formula in part $(a)$ apply to the overtones as well?

Kai Chen

Princeton University

Problem 81

Two strings on a musical instrument are tuned to play at 392 $\mathrm{Hz}(\mathrm{G})$ and 494 $\mathrm{Hz}(\mathrm{B}) .(a)$ What are the frequencies of the first two overtones for each string? (b) If the two strings have the same length and are under the same tension, what must be the ratio of their masses $\left(m_{\mathrm{G}} / m_{\mathrm{B}}\right) ?(c)$ If the strings, instead, have the same mass per unit length and are under the same tension, what is the ratio of their lengths $\left(\ell_{G} / \ell_{B}\right) ?(d)$ If their masses and lengths are the same, what must be the ratio of the tensions in the two strings?

Prabhat Tyagi

Numerade Educator

Problem 82

The ripples in a certain groove 10.8 $\mathrm{cm}$ from the center of a 33 -rpm phonograph record have a wavelength of 1.55 $\mathrm{mm}$ . What will be the frequency of the sound emitted?

Kai Chen

Princeton University

Problem 83

A 10.0 -m-long wire of mass 152 $\mathrm{g}$ is stretched under a tension of 255 $\mathrm{N} .$ A pulse is generated at one end, and 20.0 $\mathrm{ms}$ later a second pulse is generated at the opposite end. Where will the two pulses first meet?

Sachin Rao

Numerade Educator

Problem 84

A wave with a frequency of 220 $\mathrm{Hz}$ and a wavelength of 10.0 $\mathrm{cm}$ is traveling along a cord. The maximum speed of particles on the cord is the same as the wave speed. What is the amplitude of the wave?

Kai Chen

Princeton University

Problem 85

A string can have a "free" end if that end is attached to a ring that can slide without friction on a vertical pole (Fig. 40). Determine the wavelengths of the resonant vibrations of such a string with one end fixed and the other free.

Sachin Rao

Numerade Educator

Problem 86

A highway overpass was observed to resonate as one full loop $\left(\frac{1}{2} \lambda\right)$ when a small earthquake shook the ground vertically at 3.0 $\mathrm{Hz}$ . The highway department put a support at the center of the overpass, anchoring it to the ground as shown in Fig. $41 .$ What resonant frequency would you now expect for the overpass? It is noted that earthquakes rarely
do significant shaking above 5 or 6 Hz. Did the modifications do any good? Explain.

Kai Chen

Princeton University

Problem 87

Figure 42 shows the wave shape at two instants of time for a sinusoidal wave traveling to the right. What is the mathematical representation of this wave?

Sachin Rao

Numerade Educator

Problem 88

Estimate the average power of a water wave when it hits the chest of an adult standing in the water at the seashore. Assume that the amplitude of the wave is 0.50 $\mathrm{m}$ , the wavelength is $2.5 \mathrm{m},$ and the period is 4.0 $\mathrm{s} .$

Kai Chen

Princeton University

Problem 89

A tsunami of wavelength 215 $\mathrm{km}$ and velocity 550 $\mathrm{km} / \mathrm{h}$ travels across the Pacific Ocean. As it approaches Hawaii, people observe an unusual decrease of sea level in the harbors. Approximately how much time do they have to run to safety? (In the absence of knowledge and warning, people have died during tsunamis, some of them attracted to the shore to see stranded fishes and boats.)

Sachin Rao

Numerade Educator

Problem 90

Two wave pulses are traveling in opposite directions with
the same speed of 7.0 $\mathrm{cm} / \mathrm{s}$ as shown in Fig. $43 . \mathrm{At}$
$t=0,$ the leading edges of the two pulses are 15 $\mathrm{cm}$ apart.
Sketch the wave pulses at $t=1.0,2.0$ and 3.0 $\mathrm{s} .$

Kai Chen

Princeton University

Problem 91

For a spherical wave traveling uniformly away from a point source, show that the displacement can be represented by $$D=\left(\frac{A}{r}\right) \sin (k r-\omega t)$$
where $r$ is the radial distance from the source and $A$ is a constant.

Sachin Rao

Numerade Educator

Problem 92

What frequency of sound would have a wavelength the
same size as a $1.0-\mathrm{m}$ -wide window? (The speed of sound is
344 $\mathrm{m} / \mathrm{s}$ at $20^{\circ} \mathrm{C} .$ What frequencies would diffract through
the window?

Kai Chen

Princeton University

Problem 93

(II) Consider a wave generated by the periodic vibration of a source and given by the expression $D(x, t)=A \sin ^{2} k(x-c t)$ where $x$ represents position (in meters), $t$ represents time (in seconds), and $c$ is a positive constant. We choose
$A=5.0 \mathrm{m}, k=1.0 \mathrm{m}^{-1},$ and $c=0.50 \mathrm{m} / \mathrm{s} .$ Use a spread-
sheet to make a graph with three curves of $D(x, t)$ from
$x=-5.0 \mathrm{m}$ to $+5.0 \mathrm{m}$ in steps of 0.050 $\mathrm{m}$ at times $t=0.0,$
$1.0,$ and 2.0 $\mathrm{s}$ . Determine the speed, direction of motion, period, and wavelength of the wave.

Lainey Roebuck

Numerade Educator

Problem 94

(II) The displacement of a bell-shaped wave pulse is described
by a relation that involves the exponential function:
$$D(x, t)=A e^{-\alpha(x-u l)^{2}}$$
where the constants $\quad A=10.0 \mathrm{m}, \quad \alpha=2.0 \mathrm{m}^{-2}, \quad$ and
$v=3.0 \mathrm{m} / \mathrm{s} .$ (a) Over the range $-10.0 \mathrm{m} \leq x \leq+10.0 \mathrm{m},$
use a graphing calculator or computer program to plot
$D(x, t)$ at each of the three times $t=0, \quad t=1.0,$ and
$t=2.0 \mathrm{s} .$ Do these three plots demonstrate the wave-pulse
shape shifting along the $x$ axis by the expected amount over
the span of each 1.0 -s interval? (b) Repeat part $(a)$ but
assume $D(x, t)=A e^{-\alpha(x+w)^{2}}.$

Kai Chen

Princeton University