|| A string vibrates at its third-harmonic frequency. The amplitude at a point 30 cm from one end is half the maximum amplitude. How long is the string?

| A 2.0-m-long string vibrates at its second-harmonic frequency with a maximum amplitude of 2.0 cm. One end of the string is at $x=0\text{cm}$. Find the oscillation amplitude at $x=10$, 20, 30, 40, and 50 cm.

|| BIO Tendons are, essentially, elastic cords stretched between two fixed ends. As such, they can support standing waves. A woman has a 20-cm-long Achilles tendon—connecting the heel to a muscle in the calf—with a cross-section area of $90{\text{mm}}^2$. The density of tendon tissue is $1100\text{kg}\text{/}{\text{m}}^3$. For a reasonable tension of 500 N, what will be the fundamental frequency of her Achilles tendon?

||| BIO Biologists think that some spiders “tune” strands of their web to give enhanced response at frequencies corresponding to those at which desirable prey might struggle. Orb spider web silk has a typical diameter of $20\mu \text{m,}$ and spider silk has a density of $1300\text{kg}\text{/}{\text{m}}^3$. To have a fundamental frequency at 100 Hz, to what tension must a spider adjust a 12-cm-long strand of silk?

|| A violinist places her finger so that the vibrating section of a $1.0\text{g}\text{/}\text{m}$ string has a length of 30 cm, then she draws her bow across it. A listener nearby in a $20\text{°}\text{C}$ room hears a note with a wavelength of 40 cm. What is the tension in the string?

|| A particularly beautiful note reaching your ear from a rare Stradivarius violin has a wavelength of 39.1 cm. The room is slightly warm, so the speed of sound is $344\text{m}\text{/}\text{s}$. If the string’s linear density is $0.600\text{g}\text{/}\text{m}$ and the tension is 150 N, how long is the vibrating section of the violin string?

|| INT One end of a 75-cm-long, 2.5 g guitar string is attached to a spring. The other end is pulled, which stretches the spring. The guitar string’s second harmonic occurs at 550 Hz when the spring has been stretched by 5.0 cm. What is the value of the spring constant?

||| Astronauts visiting Planet X have a 250-cm-long string whose mass is 5.00 g. They tie the string to a support, stretch it horizontally over a pulley 2.00 m away, and hang a 4.00 kg mass on the free end. Then the astronauts begin to excite standing waves on the horizontal portion of the string. Their data are as follows:

Use the best-fit line of an appropriate graph to determine the value of g, the free-fall acceleration on Planet X.

||| INT A 75 g bungee cord has an equilibrium length of 1.20 m. The cord is stretched to a length of 1.80 m, then vibrated at 20 Hz. This produces a standing wave with two antinodes. What is the spring constant of the bungee cord?

|| INT In a laboratory experiment, one end of a horizontal string is tied to a support while the other end passes over a frictionless pulley and is tied to a 1.5 kg sphere. Students determine the frequencies of standing waves on the horizontal segment of the string, then they raise a beaker of water until the hanging 1.5 kg sphere is completely submerged. The frequency of the fifth harmonic with the sphere submerged exactly matches the frequency of the third harmonic before the sphere was submerged. What is the diameter of the sphere?

| A string under tension has a fundamental frequency of 220 Hz. What is the fundamental frequency if the tension is doubled?

||| INT A 750 g mass hung from a 5.0 g elastic cord stretches the cord to twice its unstretched length. If the mass is pulled down slightly and released, it oscillates vertically at a frequency of 2.0 Hz. If the stretched cord is plucked while the mass is hanging from it, what is the cord’s fundamental frequency?

|| The two strings in FIGURE P17.48 are of equal length and are being driven at equal frequencies. The linear density of the left string is 5.0 g/m. What is the linear density of the right string?

FIGURE P17.48

|| INT What is the fundamental frequency of the steel wire in FIGURE P17.49?

FIGURE P17.49

| An open-open organ pipe is 78.0 cm long. An open-closed pipe has a fundamental frequency equal to the third harmonic of the open-open pipe. How long is the open-closed pipe?

|| Western music uses a musical scale with equal temperament tuning, which means that any two adjacent notes have the same frequency ratio r. That is, notes n and $n+1$ are related by $f_{n+1}=r{f}_n$ for all n. In this system, the frequency doubles every 12 notes—an interval called an octave.

1. What is the value of r?

2. Orchestras tune to the note A, which has a frequency of 440 Hz. What is the frequency of the next note of the scale (called A-sharp)?

|| BIO Deep-sea divers often breathe a mixture of helium and oxygen to avoid getting the “bends” from breathing high-pressure nitrogen. The helium has the side effect of making the divers’ voices sound odd. Although your vocal tract can be roughly described as an open-closed tube, the way you hold your mouth and position your lips greatly affects the standing-wave frequencies of the vocal tract. This is what allows different vowels to sound different. The “ee” sound is made by shaping your vocal tract to have standing-wave frequencies at, normally, 270 Hz and 2300 Hz. What will these frequencies be for a helium-oxygen mixture in which the speed of sound at body temperature is $750\text{m/s}$? The speed of sound in air at body temperature is $350\text{m/s}$.

|| In 1866, the German scientist Adolph Kundt developed a technique for accurately measuring the speed of sound in various gases. A long glass tube, known today as a Kundt’s tube, has a vibrating piston at one end and is closed at the other. Very finely ground particles of cork are sprinkled in the bottom of the tube before the piston is inserted. As the vibrating piston is slowly moved forward, there are a few positions that cause the cork particles to collect in small, regularly spaced piles along the bottom. FIGURE P17.53 shows an experiment in which the tube is filled with pure oxygen and the piston is driven at 400 Hz. What is the speed of sound in oxygen?

FIGURE P17.53

|| The 40-cm-long tube of FIGURE P17.54 has a 40-cm-long insert that can be pulled in and out. A vibrating tuning fork is held next to the tube. As the insert is slowly pulled out, the sound from the tuning fork creates standing waves in the tube when the total length L is 42.5 cm, 56.7 cm, and 70.9 cm. What is the frequency of the tuning fork? Assume $v_{\text{sound}}=343\text{m}\text{/}\text{s}$.

FIGURE P17.54

||| A 280 Hz sound wave is directed into one end of the trombone slide seen in FIGURE P17.55. A microphone is placed at the other end to record the intensity of sound waves that are transmitted through the tube. The straight sides of the slide are 80 cm in length and 10 cm apart with a semicircular bend at the end. For what slide extensions s will the microphone detect a maximum of sound intensity?

FIGURE P17.55

||| A 1.0-m-tall vertical tube is filled with $20\text{°}\text{C}$ water. A tuning fork vibrating at 580 Hz is held just over the top of the tube as the water is slowly drained from the bottom. At what water heights, measured from the bottom of the tube, will there be a standing wave in the tube above the water?

|| A 25-cm-long wire with a linear density of $20\text{g}\text{/}\text{m}$ passes across the open end of an 85-cm-long open-closed tube of air. If the wire, which is fixed at both ends, vibrates at its fundamental frequency, the sound wave it generates excites the second vibrational mode of the tube of air. What is the tension in the wire? Assume $v_{\text{sound}}=340\text{m}\text{/}\text{s}$.

|| An old mining tunnel disappears into a hillside. You would like to know how long the tunnel is, but it’s too dangerous to go inside. Recalling your recent physics class, you decide to try setting up standing-wave resonances inside the tunnel. Using your subsonic amplifier and loudspeaker, you find resonances at 4.5 Hz and 6.3 Hz, and at no frequencies between these. It’s rather chilly inside the tunnel, so you estimate the sound speed to be $335\text{m}\text{/}\text{s}$. Based on your measurements, how far is it to the end of the tunnel?

|| A long tube that is open at one end has the other end sealed by a movable piston that can change the length of the open-closed tube. There is a 100 Hz standing wave in the tube when the length is 255 cm and when it is 425 cm but at no lengths in between. For what tube length is 100 Hz the fundamental frequency?

|| Two in-phase loudspeakers emit identical 1000 Hz sound waves along the x-axis. What distance should one speaker be placed behind the other for the sound to have an amplitude 1.5 times that of each speaker alone?

|| Two loudspeakers emit sound waves of the same frequency along the x-axis. The amplitude of each wave is a. The sound intensity is minimum when speaker 2 is 10 cm behind speaker 1. The intensity increases as speaker 2 is moved forward and first reaches maximum, with amplitude 2a, when it is 30 cm in front of speaker 1. What is

1. The wavelength of the sound?

2. The phase difference between the two loudspeakers?

3. The amplitude of the sound (as a multiple of a) if the speakers are placed side by side?

|| Two loudspeakers emit sound waves along the x-axis. A listener in front of both speakers hears a maximum sound intensity when speaker 2 is at the origin and speaker 1 is at $x=0.50\text{m}$. If speaker 1 is slowly moved forward, the sound intensity decreases and then increases, reaching another maximum when speaker 1 is at $x=0.90\text{m}$.

1. What is the frequency of the sound? Assume $v_{\text{sound}}=340\text{m}\text{/}\text{s}$.

2. What is the phase difference between the speakers?

| A sheet of glass is coated with a 500-nm-thick layer of oil $(n=1.42)$.

1. For what visible wavelengths of light do the reflected waves interfere constructively?

2. For what visible wavelengths of light do the reflected waves interfere destructively?

3. What is the color of reflected light? What is the color of transmitted light?

|| Microwaves with a frequency of 10.5 GHz are aimed downward into a flat-bottomed beaker that contains sunflower oil. A microwave detector above the beaker finds that there are strong reflections when the oil depth is 2.76 cm and 3.68 cm but at no depths in between. What is the index of refraction of sunflower oil at microwave frequencies?

|| Engineers are testing a new thin-film coating whose index of refraction is less than that of glass. They deposit a 560-nm-thick layer on glass, then shine lasers on it. A red laser with a wavelength of 640 nm has no reflection at all, but a violet laser with a wavelength of 400 nm has a maximum reflection. How the coating behaves at other wavelengths is unknown. What is the coating’s index of refraction?

|| A soap bubble is essentially a very thin film of water $(n=1.33)$ surrounded by air. The colors that you see in soap bubbles are produced by interference.

1. Derive an expression for the wavelengths ${\text{λ}}_{\text{C}}$ for which constructive interference causes a strong reflection from a soap bubble of thickness d.

   Hint: Think about the reflection phase shifts at both boundaries.

2. What visible wavelengths of light are strongly reflected from a 390-nm-thick soap bubble? What color would such a soap bubble appear to be?

||| Scientists are testing a transparent material whose index of refraction for visible light varies with wavelength as $n=30.0{\text{nm}}^{1/2}/\text{λ}{}^{1/2}$, where $\text{λ}$ is in nm. If a 295-nm-thick coating is placed on glass $(n=1.50)$ for what visible wavelengths will the reflected light have maximum constructive interference?

|| Two loudspeakers in a plane, 5.0 m apart, are playing the same frequency. If you stand 12.0 m in front of the plane of the speakers, centered between them, you hear a sound of maximum intensity. As you walk parallel to the plane of the speakers, staying 12.0 m in front of them, you first hear a minimum of sound intensity when you are directly in front of one of the speakers. What is the frequency of the sound? Assume a sound speed of $340\text{m}\text{/}\text{s}$.

|| You are standing 2.5 m directly in front of one of the two loudspeakers shown in FIGURE P17.69. They are 3.0 m apart and both are playing a 686 Hz tone in phase. As you begin to walk directly away from the speaker, at what distances from the speaker do you hear a minimum sound intensity? The room temperature is $20\text{°}\text{C}$.

FIGURE P17.69

|| Two identical loudspeakers separated by distance $\text{Δ}x$ each emit sound waves of wavelength $\text{λ}$ and amplitude a along the x-axis. What is the minimum value of the ratio $\text{Δ}x\text{/}\text{λ}$ for which the amplitude of their superposition is also a?

|| The three identical loudspeakers in FIGURE P17.71 play a 170 Hz tone in a room where the speed of sound is $340\text{m}\text{/}\text{s}$. You are standing 4.0 m in front of the middle speaker. At this point, the amplitude of the wave from each speaker is a.

1. What is the amplitude at this point?

2. How far must speaker 2 be moved to the left to produce a maximum amplitude at the point where you are standing?

3. When the amplitude is maximum, by what factor is the sound intensity greater than the sound intensity from a single speaker?

FIGURE P17.71

| Piano tuners tune pianos by listening to the beats between the harmonics of two different strings. When properly tuned, the note A should have a frequency of 440 Hz and the note E should be at 659 Hz.

1. What is the frequency difference between the third harmonic of the A and the second harmonic of the E?

2. A tuner first tunes the A string very precisely by matching it to a 440 Hz tuning fork. She then strikes the A and E strings simultaneously and listens for beats between the harmonics. What beat frequency indicates that the E string is properly tuned?

3. The tuner starts with the tension in the E string a little low, then tightens it. What is the frequency of the E string when she hears four beats per second?

|| A flutist assembles her flute in a room where the speed of sound is $342\text{m}\text{/}\text{s}$. When she plays the note A, it is in perfect tune with a 440 Hz tuning fork. After a few minutes, the air inside her flute has warmed to where the speed of sound is $346\text{m}\text{/}\text{s}$.

1. How many beats per second will she hear if she now plays the note A as the tuning fork is sounded?

2. How far does she need to extend the “tuning joint” of her flute to be in tune with the tuning fork?

|| INT A 2000 Hz sound generator is attached to the end of a large spring that oscillates with an amplitude of 45 cm toward and away from you. A second 2000 Hz sound generator is at rest next to the spring. As you listen, the beat frequency between the two sounds ranges between zero and 2.0 beats per second. What is the period of the oscillator?

|| INT Two loudspeakers emit 400 Hz notes. One speaker sits on the ground. The other speaker is in the back of a pickup truck. You hear eight beats per second as the truck drives away from you. What is the truck’s speed?

||| Two radio antennas are separated by 2.0 m. Both broadcast identical 750 MHz waves. If you walk around the antennas in a circle of radius 10 m, how many maxima will you detect?

||| CALC You have two small, identical boxes that generate 440 Hz notes. While holding one, you drop the other from a 20-m-high balcony. How many beats will you hear before the falling box hits the ground? You can ignore air resistance.

||| As the captain of the scientific team sent to Planet Physics, one of your tasks is to measure g. You have a long, thin wire labeled $1.00\text{g}\text{/}\text{m}$ and a 1.25 kg weight. You have your accurate space cadet chronometer but, unfortunately, you seem to have forgotten a meter stick. Undeterred, you first find the midpoint of the wire by folding it in half. You then attach one end of the wire to the wall of your laboratory, stretch it horizontally to pass over a pulley at the midpoint of the wire, then tie the 1.25 kg weight to the end hanging over the pulley. By vibrating the wire, and measuring time with your chronometer, you find that the wire’s second-harmonic frequency is 100 Hz. Next, with the 1.25 kg weight still tied to one end of the wire, you attach the other end to the ceiling to make a pendulum. You find that the pendulum requires 314 s to complete 100 oscillations. Pulling out your trusty calculator, you get to work. What value of g will you report back to headquarters?

||| When mass M is tied to the bottom end of a long, thin wire suspended from the ceiling, the wire’s second-harmonic frequency is 200 Hz. Adding an additional 1.0 kg to the hanging mass increases the second-harmonic frequency to 245 Hz. What is M?

||| A water wave is called a deep-water wave if the water’s depth is more than one-quarter of the wavelength. Unlike the waves we’ve considered in this chapter, the speed of a deep-water wave depends on its wavelength:

Longer wavelengths travel faster. Let’s apply this to standing waves. Consider a diving pool that is 5.0 m deep and 10.0 m wide. Standing water waves can set up across the width of the pool. Because water sloshes up and down at the sides of the pool, the boundary conditions require antinodes at $x=0$ and $x=L$. Thus a standing water wave resembles a standing sound wave in an open-open tube.

1. What are the wavelengths of the first three standing-wave modes for water in the pool? Do they satisfy the condition for being deep-water waves?

2. What are the wave speeds for each of these waves?

3. Derive a general expression for the frequencies $f_m$ of the possible standing waves. Your expression should be in terms of m, g, and L.

4. What are the oscillation periods of the first three standing wave modes?

||| BIO Ultrasound has many medical applications, one of which is to monitor fetal heartbeats by reflecting ultrasound off a fetus in the womb.

1. Consider an object moving at speed $v_{\text{o}}$ toward an at-rest source that is emitting sound waves of frequency $f_0$. Show that the reflected wave (i.e., the echo) that returns to the source has a Doppler-shifted frequency

   $$f_{\text{echo}}=\left(\frac{v+v_{\text{o}}}{v-v_{\text{o}}}\right)f_0$$

   where v is the speed of sound in the medium.

2. Suppose the object’s speed is much less than the wave speed: $v_{\text{o}}\ll v$. Then $f_{\text{echo}}\approx f_0$, and a microphone that is sensitive to these frequencies will detect a beat frequency if it listens to $f_0$ and $f_{\text{echo}}$ simultaneously. Use the binomial approximation and other appropriate approximations to show that the beat frequency is $f_{\text{beat}}\approx (2v_{\text{o}}\text{/}v)f_0$.

3. The reflection of 2.40 MHz ultrasound waves from the surface of a fetus’s beating heart is combined with the 2.40 MHz wave to produce a beat frequency that reaches a maximum of 65 Hz. What is the maximum speed of the surface of the heart? The speed of ultrasound waves within the body is $1540\text{m}\text{/}\text{s}$.

4. Suppose the surface of the heart moves in simple harmonic motion at $90\text{beats/min}$. What is the amplitude in mm of the heartbeat?