|| FIGURE P16.45 is a snapshot graph at $t=0\text{s}$ of a 5.0 Hz wave traveling to the left.

1. What is the wave speed?

2. What is the phase constant of the wave?

3. Write the displacement equation for this wave.

FIGURE P16.45

|| FIGURE P16.46 is a history graph at $x=0\text{m}$ of a wave traveling in the positive x-direction at $4.0\text{m}\text{/}\text{s}$.

1. What is the wavelength?

2. What is the phase constant of the wave?

3. Write the displacement equation for this wave.

FIGURE P16.46

| String 1 in FIGURE P16.47 has linear density $2.0\text{g}\text{/}\text{m}$ and string 2 has linear density $4.0\text{g}\text{/}\text{m}$. A student sends pulses in both directions by quickly pulling up on the knot, then releasing it. What should the string lengths $L_1$ and $L_2$ be if the pulses are to reach the ends of the strings simultaneously?

FIGURE P16.47

|| Oil explorers set off explosives to make loud sounds, then listen for the echoes from underground oil deposits. Geologists suspect that there is oil under 500-m-deep Lake Physics. It’s known that Lake Physics is carved out of a granite basin. Explorers detect a weak echo 0.94 s after exploding dynamite at the lake surface. If it’s really oil, how deep will they have to drill into the granite to reach it?

| A helium-neon laser beam has a wavelength in air of 633 nm. It takes 1.38 ns for the light to travel through 30 cm of an unknown liquid. What is the wavelength of the laser beam in the liquid?

| Earthquakes are essentially sound waves—called seismic waves—traveling through the earth. Because the earth is solid, it can support both longitudinal and transverse seismic waves. The speed of longitudinal waves, called P waves, is $8000\text{m}\text{/}\text{s}$. Transverse waves, called S waves, travel at a slower $4500\text{m}\text{/}\text{s}$. A seismograph records the two waves from a distant earthquake. If the S wave arrives 2.0 min after the P wave, how far away was the earthquake? You can assume that the waves travel in straight lines, although actual seismic waves follow more complex routes.

|| BIO One cue your hearing system uses to localize a sound (i.e., to tell where a sound is coming from) is the slight difference in the arrival times of the sound at your ears. Your ears are spaced approximately 20 cm apart. Consider a sound source 5.0 m from the center of your head along a line $45\text{°}$ to your right. What is the difference in arrival times? Give your answer in microseconds.

Hint: You are looking for the difference between two numbers that are nearly the same. What does this near equality imply about the necessary precision during intermediate stages of the calculation?

|| A $10\text{cm}\times 10\text{cm}\times 10\text{cm}$ cube contains 970 g of liquid. A transducer on one wall of the cube emits a short ultrasound pulse that is detected $80\mu \text{s}$ later by a receiver on the opposite wall. What is the liquid’s bulk modulus?

||| INT A 20.0-cm-long, 10.0-cm-diameter cylinder with a piston at one end contains 1.34 kg of an unknown liquid. Using the piston to compress the length of the liquid by 1.00 mm increases the pressure by 41.0 atm. What is the speed of sound in the liquid?

| A sound wave is described by $D(y\text{,}t)=(0.0200\text{mm})\times \text{sin}[(8.96\text{rad/m})y+(3140\text{rad/s})t+\pi \text{/}4\text{rad}]$, where y is in m and t is in s.

1. In what direction is this wave traveling?

2. Along which axis is the air oscillating?

3. What are the wavelength, the wave speed, and the period of oscillation?

|| A wave on a string is described by $D(x\text{,\hspace{0.17em}}t)=(3.0\text{cm})\times \text{sin}[2\pi (x\text{/}(2.4\text{m})+t\text{/}(0.20\text{s})+1)]$, where x is in m and t is in s.

1. In what direction is this wave traveling?

2. What are the wave speed, the frequency, and the wave number?

3. At $t=0.50\text{s,}$ what is the displacement of the string at $x=0.20\text{m}?$

| A wave on a string is described by $D(x\text{,\hspace{0.17em}}t)=(2.00\text{cm})\times \text{sin}[(12.57\text{rad/m})x-(638\text{rad/s})t]$, where x is in m and t in s. The linear density of the string is $5.00\text{g}\text{/}\text{m}$. What are

1. The string tension?

2. The maximum displacement of a point on the string?

3. The maximum speed of a point on the string?

| FIGURE P16.57 shows a snapshot graph of a wave traveling to the right along a string at $45\text{m}\text{/}\text{s}$. At this instant, what is the velocity of points 1, 2, and 3 on the string?

FIGURE P16.57

|| INT FIGURE P16.58 shows two masses hanging from a steel wire. The mass of the wire is 60.0 g. A wave pulse travels along the wire from point 1 to point 2 in 24.0 ms. What is mass m?

FIGURE P16.58

| The string in FIGURE P16.59 has linear density $\mu$. Find an expression in terms of M, $\mu$, and $\theta$ for the speed of waves on the string.

FIGURE P16.59

|| A wire is made by welding together two metals having different densities. FIGURE P16.60 shows a 2.00-m-long section of wire centered on the junction, but the wire extends much farther in both directions. The wire is placed under 2250 N tension, then a 1500 Hz wave with an amplitude of 3.00 mm is sent down the wire. How many wavelengths (complete cycles) of the wave are in this 2.00-m-long section of the wire?

FIGURE P16.60

|| One way to monitor global warming is to measure the average temperature of the ocean. Researchers are doing this by measuring the time it takes sound pulses to travel underwater over large distances. At a depth of 1000 m, where ocean temperatures hold steady near $4\text{°C}$, the average sound speed is $1480\text{m}\text{/}\text{s}$. It’s known from laboratory measurements that the sound speed increases $4.0\text{m}\text{/}\text{s}$ for every 1.$0\text{°C}$ increase in temperature. In one experiment, where sounds generated near California are detected in the South Pacific, the sound waves travel 8000 km. If the smallest time change that can be reliably detected is 1.0 s, what is the smallest change in average temperature that can be measured?

|| A string that is under 50.0 N of tension has linear density $5.0\text{g}\text{/}m$. A sinusoidal wave with amplitude 3.0 cm and wavelength 2.0 m travels along the string. What is the maximum speed of a particle on the string?

|| INT The G string on a guitar is a 0.46-mm-diameter steel string with a linear density of 1.3 g/m. When the string is properly tuned to 196 Hz, the wave speed on the string is 250 m/s. Tuning is done by turning the tuning screw, which slowly tightens—and stretches—the string. By how many mm does a 75-cm-long G string stretch when it’s first tuned?

| A sinusoidal wave travels along a stretched string. A particle on the string has a maximum speed of $2.0\text{m}\text{/}\text{s}$ and a maximum acceleration of $200\text{m}\text{/}{\text{s}}^2$. What are the frequency and amplitude of the wave?

||| A 1000 Hz sound wave traveling through 20°C air causes the pressure to oscillate around atmospheric pressure by $\pm 0.050\%$. What is the maximum speed of an oscillating air molecule? Give your answer in mm/s.

|| A long, vertical, 2.0-mm-diameter metal wire is pulled down by a 15 kg weight. A 1500 Hz horizontal vibration of the top of the wire sends waves with a wavelength of 5.0 cm down the wire. What is the density of the metal?

|| An AM radio station broadcasts with a power of 25 kW at a frequency of 920 kHz. Estimate the intensity of the radio wave at a point 10 km from the broadcast antenna.

|| BIO LASIK eye surgery uses pulses of laser light to shave off tissue from the cornea, reshaping it. A typical LASIK laser emits a 1.0-mm-diameter laser beam with a wavelength of 193 nm. Each laser pulse lasts 15 ns and contains 1.0 mJ of light energy.

1. What is the power of one laser pulse?

2. During the very brief time of the pulse, what is the intensity of the light wave?

|| The sound intensity 50 m from a wailing tornado siren is $0.10\text{W}\text{/}{\text{m}}^2$.

1. What is the intensity at 1000 m?

2. The weakest intensity likely to be heard over background noise is $\approx 1\mu \text{W}\text{/}{\text{m}}^2$. Estimate the maximum distance at which the siren can be heard.

|| INT A battery-powered siren is attached to the rim of a 20.0-cm-diameter rotating disk. As the disk spins, you hear the sound varying between 724 Hz and 778 Hz. What are (a) the siren’s frequency and (b) the disk’s angular velocity in rpm? The air temperature is 20°C.

|| INT A battery-powered siren emits 0.50 W of sound power at 1000 Hz. It is dropped from 100 m directly over your head on a 20°C day. 4.0 s after it is released, what are (a) the frequency and (b) the sound intensity level you hear?

|| CALC Some modern optical devices are made with glass whose index of refraction changes with distance from the front surface. FIGURE P16.72 shows the index of refraction as a function of the distance into a slab of glass of thickness L. The index of refraction increases linearly from $n_1$ at the front surface to $n_2$ at the rear surface.

1. Find an expression for the time light takes to travel through this piece of glass.

2. Evaluate your expression for a 1.0-cm-thick piece of glass for which $n_1=1.50$ and $n_2=1.60$.

FIGURE P16.72

|| A $20.0\text{m}\times 20.0\text{m}$ square is marked off around a loudspeaker that is in the center of the square. Carla stands at the center of one side of the square to listen to the sound, and Diego stands at one of the corners. Measurements find that the sound waves they hear have a phase difference of 180° when the temperature is 20°C. What is the sound frequency?

||| The intensity of a sound source is described by an inverse-square law only if the source is very small (a point source) and only if the waves can travel unimpeded in all directions. For an extended source or in a situation where obstacles absorb or reflect the waves, the intensity at distance r can often be expressed as $I=c{P}_{\text{source}}\text{/}r{}^x$, where c is a constant and the exponent x—which would be 2 for an ideal spherical wave—depends on the situation. In one such situation, you use a sound meter to measure the sound intensity level at different distances from a source, acquiring the data in the table. Use the best-fit line of an appropriate graph to determine the exponent x that characterizes this sound source.

|| A physics professor demonstrates the Doppler effect by tying a 600 Hz sound generator to a 1.0-m-long rope and whirling it around her head in a horizontal circle at 100 rpm. What are the highest and lowest frequencies heard by a student in the classroom?

|| CALC A loudspeaker on a pole is radiating 100 W of sound energy in all directions. You are walking directly toward the speaker at 0.80 m/s. When you are 20 m away, what are (a) the sound intensity level and (b) the rate (dB/s) at which the sound intensity level is increasing?

Hint: Use the chain rule and the relationship ${\log }_{10}x=\text{ln }x/\text{ln }10$.

|| Show that the Doppler frequency $f_{-}$ of a receding source is $f_{-}=f_0\text{/}(1+v_{\text{s}}\text{/}v)$.

| A starship approaches its home planet at a speed of 0.10c. When it is $54\times {10}^6\text{km}$ away, it uses its green laser beam $(\text{λ}=540\text{nm})$ to signal its approach.

1. How long does the signal take to travel to the home planet?

2. At what wavelength is the signal detected on the home planet?

|| Wavelengths of light from a distant galaxy are found to be 0.50% longer than the corresponding wavelengths measured in a terrestrial laboratory. Is the galaxy approaching or receding from the earth? At what speed?

| You have just been pulled over for running a red light, and the police officer has informed you that the fine will be $250. In desperation, you suddenly recall an idea that your physics professor recently discussed in class. In your calmest voice, you tell the officer that the laws of physics prevented you from knowing that the light was red. In fact, as you drove toward it, the light was Doppler shifted to where it appeared green to you. “OK,” says the officer, “Then I’ll ticket you for speeding. The fine is $1 for every $1\text{km}\text{/}\text{h}$ over the posted speed limit of $50\text{km}\text{/}\text{h}$.” How big is your fine? Use 650 nm as the wavelength of red light and 540 nm as the wavelength of green light.

||| An avant-garde composer wants to use the Doppler effect in his new opera. As the soprano sings, he wants a large bat to fly toward her from the back of the stage. The bat will be outfitted with a microphone to pick up the singer’s voice and a loudspeaker to rebroadcast the sound toward the audience. The composer wants the sound the audience hears from the bat to be, in musical terms, one half-step higher in frequency than the note they are hearing from the singer. Two notes a half-step apart have a frequency ratio of $2^{1/12}=1.059$. With what speed must the bat fly toward the singer?

||| A loudspeaker, mounted on a tall pole, is engineered to emit 75% of its sound energy into the forward hemisphere, 25% toward the back. You measure an 85 dB sound intensity level when standing 3.5 m in front of and 2.5 m below the speaker. What is the speaker’s power output?

||| CALC A rope of mass m and length L hangs from a ceiling.

1. Show that the wave speed on the rope a distance y above the lower end is $v=\sqrt{gy}$.

2. Show that the time for a pulse to travel the length of the string is $\text{Δ}t=2\sqrt{L\text{/}g}$.

||| CALC A communications truck with a 44-cm-diameter dish receiver on the roof starts out 10 km from its base station. It drives directly away from the base station at 50 km/h for 1.0 h, keeping the receiver pointed at the base station. The base station antenna broadcasts continuously with 2.5 kW of power, radiated uniformly in all directions. How much electromagnetic energy does the truck’s dish receive during that 1.0 h?

||| CALC A water wave is a shallow-water wave if the water depth d is less than $\approx \text{λ}\text{/}10$. It is shown in hydrodynamics that the speed of a shallow-water wave is $v=\sqrt{gd}$, so waves slow down as they move into shallower water. Ocean waves, with wavelengths of typically 100 m, are shallow-water waves when the water depth is less than $\approx 10\text{m}$. Consider a beach where the depth increases linearly with distance from the shore until reaching a depth of 5.0 m at a distance of 100 m. How long does it take a wave to move the last 100 m to the shore? Assume that the waves are so small that they don’t break before reaching the shore.