### Problems

### 16.1 Traveling Waves

Storms in the South Pacific can create waves that travel all the way to the California coast, 12,000 km away. How long does it take them to travel this distance if they travel at 15.0 m/s?

Waves on a swimming pool propagate at 0.75 m/s. You splash the water at one end of the pool and observe the wave go to the opposite end, reflect, and return in 30.00 s. How far away is the other end of the pool?

Wind gusts create ripples on the ocean that have a wavelength of 5.00 cm and propagate at 2.00 m/s. What is their frequency?

How many times a minute does a boat bob up and down on ocean waves that have a wavelength of 40.0 m and a propagation speed of 5.00 m/s?

Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 8.00 m apart. If they shake the bridge twice per second, what is the propagation speed of the waves?

What is the wavelength of the waves you create in a swimming pool if you splash your hand at a rate of 2.00 Hz and the waves propagate at a wave speed of 0.800 m/s?

What is the wavelength of an earthquake that shakes you with a frequency of 10.0 Hz and gets to another city 84.0 km away in 12.0 s?

Radio waves transmitted through empty space at the speed of light $\left(v=c=3.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)$ by the *Voyager* spacecraft have a wavelength of 0.120 m. What is their frequency?

Your ear is capable of differentiating sounds that arrive at each ear just 0.34 ms apart, which is useful in determining where low frequency sound is originating from. (a) Suppose a low-frequency sound source is placed to the right of a person, whose ears are approximately 18 cm apart, and the speed of sound generated is 340 m/s. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear? (b) Assume the same person was scuba diving and a low-frequency sound source was to the right of the scuba diver. How long is the interval between when the sound arrives at the right ear and the sound arrives at the left ear, if the speed of sound in water is 1500 m/s? (c) What is significant about the time interval of the two situations?

(a) Seismographs measure the arrival times of earthquakes with a precision of 0.100 s. To get the distance to the epicenter of the quake, geologists compare the arrival times of S- and P-waves, which travel at different speeds. If S- and P-waves travel at 4.00 and 7.20 km/s, respectively, in the region considered, how precisely can the distance to the source of the earthquake be determined? (b) Seismic waves from underground detonations of nuclear bombs can be used to locate the test site and detect violations of test bans. Discuss whether your answer to (a) implies a serious limit to such detection. (Note also that the uncertainty is greater if there is an uncertainty in the propagation speeds of the S- and P-waves.)

A Girl Scout is taking a 10.00-km hike to earn a merit badge. While on the hike, she sees a cliff some distance away. She wishes to estimate the time required to walk to the cliff. She knows that the speed of sound is approximately 343 meters per second. She yells and finds that the echo returns after approximately 2.00 seconds. If she can hike 1.00 km in 10 minutes, how long would it take her to reach the cliff?

A quality assurance engineer at a frying pan company is asked to qualify a new line of nonstick-coated frying pans. The coating needs to be 1.00 mm thick. One method to test the thickness is for the engineer to pick a percentage of the pans manufactured, strip off the coating, and measure the thickness using a micrometer. This method is a destructive testing method. Instead, the engineer decides that every frying pan will be tested using a nondestructive method. An ultrasonic transducer is used that produces sound waves with a frequency of $f=25\phantom{\rule{0.2em}{0ex}}\text{kHz}.$ The sound waves are sent through the coating and are reflected by the interface between the coating and the metal pan, and the time is recorded. The wavelength of the ultrasonic waves in the coating is 0.076 m. What should be the time recorded if the coating is the correct thickness (1.00 mm)?

### 16.2 Mathematics of Waves

A pulse can be described as a single wave disturbance that moves through a medium. Consider a pulse that is defined at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ by the equation $y\left(x\right)=\frac{6.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{3}}{{x}^{2}+2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}}$ centered around $x=0.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ The pulse moves with a velocity of $v=3.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ in the positive *x*-direction. (a) What is the amplitude of the pulse? (b) What is the equation of the pulse as a function of position and time? (c) Where is the pulse centered at time $t=5.00\phantom{\rule{0.2em}{0ex}}\text{s}$?

A transverse wave on a string is modeled with the wave function $y\left(x,t\right)=(0.20\phantom{\rule{0.2em}{0ex}}\text{cm})\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-3.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{16}\right).$ What is the height of the string with respect to the equilibrium position at a position $x=4.00\phantom{\rule{0.2em}{0ex}}\text{m}$ and a time $t=10.00\phantom{\rule{0.2em}{0ex}}\text{s}?$

Consider the wave function $y\left(x,t\right)=(3.00\phantom{\rule{0.2em}{0ex}}\text{cm})\text{sin}\left(0.4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+2.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{10}\right).$ What are the period, wavelength, speed, and initial phase shift of the wave modeled by the wave function?

A pulse is defined as $y\left(x,t\right)={e}^{\mathrm{-2.77}\phantom{\rule{0.2em}{0ex}}{\left(\frac{2.00\left(x-2.00\phantom{\rule{0.2em}{0ex}}\text{m/s}\left(t\right)\right)}{5.00\phantom{\rule{0.2em}{0ex}}\text{m}}\right)}^{2}}.$ Use a spreadsheet, or other computer program, to plot the pulse as the height of medium *y* as a function of position *x*. Plot the pulse at times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Where is the pulse centered at time $t=3.00\phantom{\rule{0.2em}{0ex}}\text{s}$? Use your spreadsheet to check your answer.

A wave is modeled at time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ with a wave function that depends on position. The equation is $y\left(x\right)=(0.30\phantom{\rule{0.2em}{0ex}}\text{m})\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x\right)$. The wave travels a distance of 4.00 meters in 0.50 s in the positive *x*-direction. Write an equation for the wave as a function of position and time.

A wave is modeled with the function $y\left(x,t\right)=(0.25\phantom{\rule{0.2em}{0ex}}\text{m})\text{cos}\left(0.30\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-0.90\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{3}\right).$ Find the (a) amplitude, (b) wave number, (c) angular frequency, (d) wave speed, (e) initial phase shift, (f) wavelength, and (g) period of the wave.

A surface ocean wave has an amplitude of 0.60 m and the distance from trough to trough is 8.00 m. It moves at a constant wave speed of 1.50 m/s propagating in the positive *x*-direction. At $t=0,$ the water displacement at $x=0$ is zero, and ${v}_{y}$ is positive. (a) Assuming the wave can be modeled as a sine wave, write a wave function to model the wave. (b) Use a spreadsheet to plot the wave function at times $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$ on the same graph. Verify that the wave moves 3.00 m in those 2.00 s.

A wave is modeled by the wave function $y\left(x,t\right)=(0.30\phantom{\rule{0.2em}{0ex}}\text{m})\text{sin}\left[\frac{2\pi}{4.50\phantom{\rule{0.2em}{0ex}}\text{m}}\left(x-18.00\frac{\text{m}}{\text{s}}t\right)\right].$ What are the amplitude, wavelength, wave speed, period, and frequency of the wave?

A transverse wave on a string is described with the wave function $y\left(x,t\right)=(0.50\phantom{\rule{0.2em}{0ex}}\text{cm})\text{sin}\left(1.57\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-6.28\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$. (a) What is the wave velocity of the wave? (b) What is the magnitude of the maximum velocity of the string perpendicular to the direction of the motion?

A swimmer in the ocean observes one day that the ocean surface waves are periodic and resemble a sine wave. The swimmer estimates that the vertical distance between the crest and the trough of each wave is approximately 0.45 m, and the distance between each crest is approximately 1.8 m. The swimmer counts that 12 waves pass every two minutes. Determine the simple harmonic wave function that would describes these waves.

Consider a wave described by the wave function $y\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(2.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-628.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ (a) How many crests pass by an observer at a fixed location in 2.00 minutes? (b) How far has the wave traveled in that time?

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi}{3.00\phantom{\rule{0.2em}{0ex}}\text{m}}x+\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.50\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi}{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ What are the similarities and differences between the two waves?

Consider two waves defined by the wave functions ${y}_{1}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{2\pi}{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right)$ and ${y}_{2}\left(x,t\right)=0.20\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{2\pi}{6.00\phantom{\rule{0.2em}{0ex}}\text{m}}x-\frac{2\pi}{4.00\phantom{\rule{0.2em}{0ex}}\text{s}}t\right).$ What are the similarities and differences between the two waves?

The speed of a transverse wave on a string is 300.00 m/s, its wavelength is 0.50 m, and the amplitude is 20.00 cm. How much time is required for a particle on the string to move through a distance of 5.00 km?

### 16.3 Wave Speed on a Stretched String

Transverse waves are sent along a 5.00-m-long string with a speed of 30.00 m/s. The string is under a tension of 10.00 N. What is the mass of the string?

A copper wire has a density of $\rho =8920\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3},$ a radius of 1.20 mm, and a length *L*. The wire is held under a tension of 10.00 N. Transverse waves are sent down the wire. (a) What is the linear mass density of the wire? (b) What is the speed of the waves through the wire?

A piano wire has a linear mass density of $\mu =4.95\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-3}}\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ Under what tension must the string be kept to produce waves with a wave speed of 500.00 m/s?

A string with a linear mass density of $\mu =0.0060\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ is tied to the ceiling. A 20-kg mass is tied to the free end of the string. The string is plucked, sending a pulse down the string. Estimate the speed of the pulse as it moves down the string.

A cord has a linear mass density of $\mu =0.0075\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ and a length of three meters. The cord is plucked and it takes 0.20 s for the pulse to reach the end of the string. What is the tension of the string?

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

Two strings are attached to poles, however the first string is twice as long as the second. If both strings have the same tension and mu, what is the ratio of the speed of the pulse of the wave from the first string to the second string?

Two strings are attached to poles, however the first string is twice the linear mass density mu of the second. If both strings have the same tension, what is the ratio of the speed of the pulse of the wave from the first string to the second string?

Transverse waves travel through a string where the tension equals 7.00 N with a speed of 20.00 m/s. What tension would be required for a wave speed of 25.00 m/s?

Two strings are attached between two poles separated by a distance of 2.00 m as shown below, both under the same tension of 600.00 N. String 1 has a linear density of ${\mu}_{1}=0.0025\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ and string 2 has a linear mass density of ${\mu}_{2}=0.0035\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?

Two strings are attached between two poles separated by a distance of 2.00 meters as shown in the preceding figure, both strings have a linear density of ${\mu}_{1}=0.0025\phantom{\rule{0.2em}{0ex}}\text{kg/m},$ the tension in string 1 is 600.00 N and the tension in string 2 is 700.00 N. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before the pulses pass one another?

The note ${E}_{4}$ is played on a piano and has a frequency of $f=393.88.$ If the linear mass density of this string of the piano is $\mu =0.012\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ and the string is under a tension of 1000.00 N, what is the speed of the wave on the string and the wavelength of the wave?

Two transverse waves travel through a taut string. The speed of each wave is $v=30.00\phantom{\rule{0.2em}{0ex}}\text{m/s}.$ A plot of the vertical position as a function of the horizontal position is shown below for the time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}.$ (a) What is the wavelength of each wave? (b) What is the frequency of each wave? (c) What is the maximum vertical speed of each string?

A sinusoidal wave travels down a taut, horizontal string with a linear mass density of $\mu =0.060\phantom{\rule{0.2em}{0ex}}\text{kg/m}$. The maximum vertical speed of the wave is ${v}_{y\phantom{\rule{0.2em}{0ex}}\text{max}}=0.30\phantom{\rule{0.2em}{0ex}}\text{cm/s}.$ The wave is modeled with the wave equation $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-24.00\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ (a) What is the amplitude of the wave? (b) What is the tension in the string?

The speed of a transverse wave on a string is $v=60.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$ and the tension in the string is ${F}_{T}=100.00\phantom{\rule{0.2em}{0ex}}\text{N}$. What must the tension be to increase the speed of the wave to $v=120.00\phantom{\rule{0.2em}{0ex}}\text{m/s?}$

### 16.4 Energy and Power of a Wave

A string of length 5 m and a mass of 90 g is held under a tension of 100 N. A wave travels down the string that is modeled as $y\left(x,t\right)=0.01\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(15.7\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-1170.12\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}\right).$ What is the power over one wavelength?

Ultrasound of intensity $1.50\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ is produced by the rectangular head of a medical imaging device measuring 3.00 cm by 5.00 cm. What is its power output?

The low-frequency speaker of a stereo set has a surface area of $A=0.05\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$ and produces 1 W of acoustical power. (a) What is the intensity at the speaker? (b) If the speaker projects sound uniformly in all directions, at what distance from the speaker is the intensity $0.1\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$?

To increase the intensity of a wave by a factor of 50, by what factor should the amplitude be increased?

A device called an insolation meter is used to measure the intensity of sunlight. It has an area of $100\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2}$ and registers 6.50 W. What is the intensity in ${\text{W/m}}^{2}$?

Energy from the Sun arrives at the top of Earth’s atmosphere with an intensity of $1400\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$. How long does it take for $1.80\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{9}\phantom{\rule{0.2em}{0ex}}\text{J}$ to arrive on an area of $1.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{2}$?

Suppose you have a device that extracts energy from ocean breakers in direct proportion to their intensity. If the device produces 10.0 kW of power on a day when the breakers are 1.20 m high, how much will it produce when they are 0.600 m high?

A photovoltaic array of (solar cells) is $10.0\text{\%}$ efficient in gathering solar energy and converting it to electricity. If the average intensity of sunlight on one day is $70.00\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$, what area should your array have to gather energy at the rate of 100 W? (b) What is the maximum cost of the array if it must pay for itself in two years of operation averaging 10.0 hours per day? Assume that it earns money at the rate of 9.00 cents per kilowatt-hour.

A microphone receiving a pure sound tone feeds an oscilloscope, producing a wave on its screen. If the sound intensity is originally $2.00\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-5}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$, but is turned up until the amplitude increases by $30.0\text{\%}$, what is the new intensity?

A string with a mass of 0.30 kg has a length of 4.00 m. If the tension in the string is 50.00 N, and a sinusoidal wave with an amplitude of 2.00 cm is induced on the string, what must the frequency be for an average power of 100.00 W?

The power versus time for a point on a string $\left(\mu =0.05\phantom{\rule{0.2em}{0ex}}\text{kg/m}\right)$ in which a sinusoidal traveling wave is induced is shown in the preceding figure. The wave is modeled with the wave equation $y\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{sin}\left(20.93\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-\omega t\right)$. What is the frequency and amplitude of the wave?

A string is under tension ${F}_{T1}$. Energy is transmitted by a wave on the string at rate ${P}_{1}$ by a wave of frequency ${f}_{1}$. What is the ratio of the new energy transmission rate ${P}_{2}$ to ${P}_{1}$ if the tension is doubled?

A 250-Hz tuning fork is struck and the intensity at the source is ${I}_{1}$ at a distance of one meter from the source. (a) What is the intensity at a distance of 4.00 m from the source? (b) How far from the tuning fork is the intensity a tenth of the intensity at the source?

A sound speaker is rated at a voltage of $P=120.00\phantom{\rule{0.2em}{0ex}}\text{V}$ and a current of $I=10.00\phantom{\rule{0.2em}{0ex}}\text{A}.$ Electrical power consumption is $P=IV$. To test the speaker, a signal of a sine wave is applied to the speaker. Assuming that the sound wave moves as a spherical wave and that all of the energy applied to the speaker is converted to sound energy, how far from the speaker is the intensity equal to $3.82\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}?$

The energy of a ripple on a pond is proportional to the amplitude squared. If the amplitude of the ripple is 0.1 cm at a distance from the source of 6.00 meters, what was the amplitude at a distance of 2.00 meters from the source?

### 16.5 Interference of Waves

Consider two sinusoidal waves traveling along a string, modeled as ${y}_{1}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ and ${y}_{2}\left(x,t\right)=0.6\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(8\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-6\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ What is the height of the resultant wave formed by the interference of the two waves at the position $x=0.5\phantom{\rule{0.2em}{0ex}}\text{m}$ at time $t=0.2\phantom{\rule{0.2em}{0ex}}\text{s?}$

Consider two sinusoidal sine waves traveling along a string, modeled as ${y}_{1}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{3}\right)$ and ${y}_{2}\left(x,t\right)=0.6\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(8\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-6\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ What is the height of the resultant wave formed by the interference of the two waves at the position $x=1.0\phantom{\rule{0.2em}{0ex}}\text{m}$ at time $t=3.0\phantom{\rule{0.2em}{0ex}}\text{s?}$

Consider two sinusoidal sine waves traveling along a string, modeled as ${y}_{1}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ and ${y}_{2}\left(x,t\right)=0.3\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(4\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+3\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right).$ What is the wave function of the resulting wave? [*Hint:* Use the trig identity $\text{sin}\left(u\pm v\right)=\text{sin}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}v\pm \phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}v$

Two sinusoidal waves are moving through a medium in the same direction, both having amplitudes of 3.00 cm, a wavelength of 5.20 m, and a period of 6.52 s, but one has a phase shift of an angle $\varphi $. What is the phase shift if the resultant wave has an amplitude of 5.00 cm? [*Hint:* Use the trig identity $\text{sin}\phantom{\rule{0.2em}{0ex}}u+\text{sin}\phantom{\rule{0.2em}{0ex}}v=2\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{u+v}{2}\right)\text{cos}\left(\frac{u-v}{2}\right)$

Two sinusoidal waves are moving through a medium in the positive *x*-direction, both having amplitudes of 6.00 cm, a wavelength of 4.3 m, and a period of 6.00 s, but one has a phase shift of an angle $\varphi =0.50\phantom{\rule{0.2em}{0ex}}\text{rad}.$ What is the height of the resultant wave at a time $t=3.15\phantom{\rule{0.2em}{0ex}}\text{s}$ and a position $x=0.45\phantom{\rule{0.2em}{0ex}}\text{m}$?

Two sinusoidal waves are moving through a medium in the positive *x*-direction, both having amplitudes of 7.00 cm, a wave number of $k=3.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}},$ an angular frequency of $\omega =2.50\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}},$ and a period of 6.00 s, but one has a phase shift of an angle $\varphi =\frac{\pi}{12}\phantom{\rule{0.2em}{0ex}}\text{rad}.$ What is the height of the resultant wave at a time $t=2.00\phantom{\rule{0.2em}{0ex}}\text{s}$ and a position $x=0.53\phantom{\rule{0.2em}{0ex}}\text{m?}$

Consider two waves ${y}_{1}\left(x,t\right)$ and ${y}_{2}\left(x,t\right)$ that are identical except for a phase shift propagating in the same medium. (a)What is the phase shift, in radians, if the amplitude of the resulting wave is 1.75 times the amplitude of the individual waves? (b) What is the phase shift in degrees? (c) What is the phase shift as a percentage of the individual wavelength?

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is ${y}_{R}\left(x,t\right)=0.70\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3.00\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-6.28\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\pi \text{/}16\phantom{\rule{0.2em}{0ex}}\text{rad}\right).$ What are the angular frequency, wave number, amplitude, and phase shift of the individual waves?

Two sinusoidal waves, which are identical except for a phase shift, travel along in the same direction. The wave equation of the resultant wave is ${y}_{R}\left(x,t\right)=0.35\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(6.28\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-1.57\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{4}\right).$ What are the period, wavelength, amplitude, and phase shift of the individual waves?

Consider two wave functions, ${y}_{1}\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\pi \phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-\pi \phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ and ${y}_{2}\left(x,t\right)=4.00\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\pi \phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-\pi \phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{3}\right).$ (a) Using a spreadsheet, plot the two wave functions and the wave that results from the superposition of the two wave functions as a function of position $\left(0.00\le x\le 6.00\phantom{\rule{0.2em}{0ex}}\text{m}\right)$ for the time $t=0.00\phantom{\rule{0.2em}{0ex}}\text{s}.$ (b) What are the wavelength and amplitude of the two original waves? (c) What are the wavelength and amplitude of the resulting wave?

Consider two wave functions, ${y}_{2}\left(x,t\right)=2.00\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{\pi}{2}{\text{m}}^{\mathrm{-1}}x-\frac{\pi}{3}{\text{s}}^{\mathrm{-1}}t\right)$ and ${y}_{2}\left(x,t\right)=2.00\phantom{\rule{0.2em}{0ex}}\text{m}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(\frac{\pi}{2}{\text{m}}^{\mathrm{-1}}x-\frac{\pi}{3}{\text{s}}^{\mathrm{-1}}t+\frac{\pi}{6}\right).$ (a) Verify that ${y}_{R}=2A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{\varphi}{2}\right)\text{sin}\left(kx-\omega t+\frac{\varphi}{2}\right)$ is the solution for the wave that results from a superposition of the two waves. Make a column for *x*, ${y}_{1}$, ${y}_{2}$, ${y}_{1}+{y}_{2}$, and ${y}_{R}=2A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{\varphi}{2}\right)\text{sin}\left(kx-\omega t+\frac{\varphi}{2}\right).$ Plot four waves as a function of position where the range of *x* is from 0 to 12 m.

Consider two wave functions that differ only by a phase shift, ${y}_{1}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx-\omega t\right)$ and ${y}_{2}\left(x,t\right)=A\phantom{\rule{0.2em}{0ex}}\text{cos}\left(kx-\omega t+\varphi \right).$ Use the trigonometric identities $\text{cos}\phantom{\rule{0.2em}{0ex}}u+\text{cos}\phantom{\rule{0.2em}{0ex}}v=2\phantom{\rule{0.2em}{0ex}}\text{cos}\left(\frac{u-v}{2}\right)\text{cos}\left(\frac{u+v}{2}\right)$ and $\text{cos}\left(\text{\u2212}\theta \right)=\text{cos}\left(\theta \right)$ to find a wave equation for the wave resulting from the superposition of the two waves. Does the resulting wave function come as a surprise to you?

### 16.6 Standing Waves and Resonance

A wave traveling on a Slinky® that is stretched to 4 m takes 2.4 s to travel the length of the Slinky and back again. (a) What is the speed of the wave? (b) Using the same Slinky stretched to the same length, a standing wave is created which consists of three antinodes and four nodes. At what frequency must the Slinky be oscillating?

A 2-m long string is stretched between two supports with a tension that produces a wave speed equal to ${v}_{w}=50.00\phantom{\rule{0.2em}{0ex}}\text{m/s}.$ What are the wavelength and frequency of the first three modes that resonate on the string?

Consider the experimental setup shown below. The length of the string between the string vibrator and the pulley is $L=1.00\phantom{\rule{0.2em}{0ex}}\text{m}.$ The linear density of the string is $\mu =0.006\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ The string vibrator can oscillate at any frequency. The hanging mass is 2.00 kg. (a)What are the wavelength and frequency of $n=6$ mode? (b) The string oscillates the air around the string. What is the wavelength of the sound if the speed of the sound is ${v}_{s}=343.00\phantom{\rule{0.2em}{0ex}}\text{m/s?}$

A cable with a linear density of $\mu =0.2\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ is hung from telephone poles. The tension in the cable is 500.00 N. The distance between poles is 20 meters. The wind blows across the line, causing the cable resonate. A standing waves pattern is produced that has 4.5 wavelengths between the two poles. The speed of sound at the current temperature $T=20\text{\xb0}\text{C}$ is $343.00\phantom{\rule{0.2em}{0ex}}\text{m/s}$. What are the frequency and wavelength of the hum?

Consider a rod of length *L*, mounted in the center to a support. A node must exist where the rod is mounted on a support, as shown below. Draw the first two normal modes of the rod as it is driven into resonance. Label the wavelength and the frequency required to drive the rod into resonance.

Consider two wave functions $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x-4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$ and $y\left(x,t\right)=0.30\phantom{\rule{0.2em}{0ex}}\text{cm}\phantom{\rule{0.2em}{0ex}}\text{sin}\left(3\phantom{\rule{0.2em}{0ex}}{\text{m}}^{\mathrm{-1}}x+4\phantom{\rule{0.2em}{0ex}}{\text{s}}^{\mathrm{-1}}t\right)$. Write a wave function for the resulting standing wave.

A 2.40-m wire has a mass of 7.50 g and is under a tension of 160 N. The wire is held rigidly at both ends and set into oscillation. (a) What is the speed of waves on the wire? The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 1.20 m. (b) What is the frequency used to drive the string into resonance?

A string with a linear mass density of 0.0062 kg/m and a length of 3.00 m is set into the $n=100$ mode of resonance. The tension in the string is 20.00 N. What is the wavelength and frequency of the wave?

A string with a linear mass density of 0.0075 kg/m and a length of 6.00 m is set into the $n=4$ mode of resonance by driving with a frequency of 100.00 Hz. What is the tension in the string?

Two sinusoidal waves with identical wavelengths and amplitudes travel in opposite directions along a string producing a standing wave. The linear mass density of the string is $\mu =0.075\phantom{\rule{0.2em}{0ex}}\text{kg/m}$ and the tension in the string is ${F}_{T}=5.00\phantom{\rule{0.2em}{0ex}}\text{N}.$ The time interval between instances of total destructive interference is $\text{\Delta}t=0.13\phantom{\rule{0.2em}{0ex}}\text{s}.$ What is the wavelength of the waves?

A string, fixed on both ends, is 5.00 m long and has a mass of 0.15 kg. The tension if the string is 90 N. The string is vibrating to produce a standing wave at the fundamental frequency of the string. (a) What is the speed of the waves on the string? (b) What is the wavelength of the standing wave produced? (c) What is the period of the standing wave?

A string is fixed at both end. The mass of the string is 0.0090 kg and the length is 3.00 m. The string is under a tension of 200.00 N. The string is driven by a variable frequency source to produce standing waves on the string. Find the wavelengths and frequency of the first four modes of standing waves.

The frequencies of two successive modes of standing waves on a string are 258.36 Hz and 301.42 Hz. What is the next frequency above 100.00 Hz that would produce a standing wave?

A string is fixed at both ends to supports 3.50 m apart and has a linear mass density of $\mu =0.005\phantom{\rule{0.2em}{0ex}}\text{kg/m}.$ The string is under a tension of 90.00 N. A standing wave is produced on the string with six nodes and five antinodes. What are the wave speed, wavelength, frequency, and period of the standing wave?

Sine waves are sent down a 1.5-m-long string fixed at both ends. The waves reflect back in the opposite direction. The amplitude of the wave is 4.00 cm. The propagation velocity of the waves is 175 m/s. The $n=6$ resonance mode of the string is produced. Write an equation for the resulting standing wave.