Chapter 12

\(\cdot\) The electromagnetic spectrum. Electromagnetic waves, which include light, consist of vibrations of electric and magnetic fields, and they all travel at the speed of light. (a) FM radio. Find the wavelength of an FM radio station signal broadcasting at a frequency of 104.5 \(\mathrm{MHz}\) . (b) \(\mathrm{X}\) rays. X rays have a wavelength of about 0.10 \(\mathrm{nm}\) . What is their frequency? (c) The Big Bang. Microwaves with a wavelength of 1.1 \(\mathrm{mm}\) , left over from soon after the Big Bang, have been detected. What is their frequency? (d) Sunburn. Sunburn (and skin cancer) are caused by ultraviolet light waves having a frequency of around \(10^{16} \mathrm{Hz}\) . What is their wavelength? (e) SETI. It has been suggested that extraterrestrial civilizations (if they exist) might try to communicate by using electromagnetic waves having the same frequency as that given off by the spin flip of the electron in hydrogen, which is 1.43 GHz. To what wave-length should we tune our telescopes in order to search for such signals? (f) Microwave ovens. Microwave ovens cook food with electromagnetic waves of frequency around 2.45 \(\mathrm{GHz}\) . What wavelength do these waves have?

\(\bullet\) If an earthquake wave having a wavelength of 13 \(\mathrm{km}\) causes the ground to vibrate 10.0 times each minute, what is the speed of the wave?

\(\bullet\) A fisherman notices that his boat is moving up and down periodically, owing to waves on the surface of the water. It takes 2.5 s for the boat to travel from its highest point to its lowest, a total distance of 0.62 \(\mathrm{m} .\) The fisherman sees that the wave crests are spaced 6.0 \(\mathrm{m}\) apart. (a) How fast are the waves traveling? (b) What is the amplitude of each wave? (c) If the total vertical distance traveled by the boat were \(0.30 \mathrm{m},\) but the other data remained the same, how would the answers to parts (a) and (b) be affected?

A stecl wire 4.00 \(\mathrm{m}\) long has a mass of 0.0600 \(\mathrm{kg}\) and is stretched with a tension of 1000 \(\mathrm{N}\) . What is the speed of prop- agation of a transverse wave on the wire?

\(\cdot\) With what tension must a rope with length 2.50 \(\mathrm{m}\) and mass 0.120 \(\mathrm{kg}\) be stretched for transverse waves of frequency 40.0 \(\mathrm{Hz}\) to have a wavelength of 0.750 \(\mathrm{m} ?\)

\(\bullet\) One end of a horizontal rope is attached to a prong of an electrically driven tuning fork that vibrates at 120 \(\mathrm{Hz}\) . The other end passes over a pulley and supports a 1.50 \(\mathrm{kg}\) mass. The linear mass density of the rope is 0.0550 \(\mathrm{kg} / \mathrm{m}\) . (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would your answers to parts (a) and (b) change if the mass were increased to 3.00 \(\mathrm{kg}\) ?

\(\bullet\) (a) If the amplitude in a sound wave is doubled, by what factor does the intensity of the wave increase? (b) By what fac- tor must the amplitude of a sound wave be increased in order to increase the intensity by a factor of 9\(?\)

\(\bullet\) A certain transverse wave is described by the equation $$y(x, t)=(6.50 \mathrm{mm}) \sin 2 \pi\left(\frac{t}{0.0360 \mathrm{s}}-\frac{x}{0.280 \mathrm{m}}\right)$$ Determine this wave's (a) amplitude, (b) wavelength, (c) frequency, (d) speed of propagation, and (e) direction of propagation.

Transverse waves on a string have wave speed \(8.00 \mathrm{m} / \mathrm{s},\) amplitude \(0.0700 \mathrm{m},\) and wavelength 0.320 \(\mathrm{m} .\) These waves travel in the \(x\) direction, and at \(t=0\) the \(x=0\) end of the string is at \(y=0\) and moving downward. (a) Find the frequency, period, and wave number of these waves. (b) Write the equation for \(y(x, t)\) describing these waves. (c) Find the transverse displacement of a point on the string at \(x=0.360 \mathrm{m}\) at time \(t=0.150 \mathrm{s}\) .

\(\bullet\) The equation describing a transverse wave on a string is $$y(x, t)=(1.50 \mathrm{mm}) \sin \left[\left(157 \mathrm{s}^{-1}\right) t-\left(41.9 \mathrm{m}^{-1}\right) x\right]$$ Find (a) the wavelength, frequency, and amplitude of this wave, (b) the speed and direction of motion of the wave, and (c) the transverse displacement of a point on the string when \(t=0.100\) s and at a position \(x=0.135 \mathrm{m} .\)

\(\bullet\) Transverse waves are traveling on a long string that is under a tension of 4.00 \(\mathrm{N}\) . The equation describing these waves is $$y(x, t)=(1.25 \mathrm{cm}) \sin \left[\left(415 \mathrm{s}^{-1}\right) t-\left(44.9 \mathrm{m}^{-1}\right) x\right]$$ Find the linear mass density of this string.

\(\cdot\) Mapping the ocean floor. The ocean floor is mapped by sending sound waves (sonar) downward and measuring the time it takes for their echo to return. From this information, the ocean depth can be calculated if one knows that sound travels at 1531 \(\mathrm{m} / \mathrm{s}\) in seawater. If a ship sends out sonar pulses and records their echo 3.27 s later, how deep is the ocean floor at that point, assuming that the speed of sound is the same at all depths?

A piano tuner stretches a steel piano wire with a tension of 800 N. The wire is 0.400 m long and has a mass of 3.00 g. (a) What is the frequency of its fundamental mode of vibration? (b) What is the number of the highest harmonic that could be heard by a person who is capable of hearing frequencies up to 10,000 Hz?

A wire with mass 40.0 g is stretched so that its ends are tied down at points 80.0 cm apart. The wire vibrates in its fundamental mode with frequency 60.0 Hz and with an amplitude of 0.300 cm at the antinodes. (a)What is the speed of propagation of transverse waves in the wire? (b) Compute the tension in the wire.

The portion of string between the bridge and upper end of the fingerboard (the part of the string that is free to vibrate) of a certain musical instrument is 60.0 cm long and has a mass of 2.00 g. The string sounds an \(\mathrm{A}_{4}\) note \((440 \mathrm{Hz})\) when played. (a) Where must the player put a finger (at what distance \(x\) from the bridge) to play a \(\mathrm{D}_{5}\) note \((587 \mathrm{Hz}) ?\) (See Figure \(12.40 . )\) For both notes, the string vibrates in its fundamental mode. (b) Without retuning, is it possible to play a G \(_{4}\) note \((392\) Hz \()\) on this string? Why or why not?

\(\bullet\) Voiceprints. In this chapter, we have been concentrating on sinusoidal waves. But most waves in the real world are far more complicated. However, many complicated waves can be created by adding together sine waves of varying amplitude and frequency. When a singer, for example, sings a note, the pitch we hear is the fundamental frequency at which his or her larynx is vibrating. But the larynx also vibrates in other frequencies (the overtones) at the same time. So the sound we hear is a superposition of the fundamental frequency plus all the overtones. This set of all the frequencies (with their respective amplitudes) is called the person's voice print. (a) To see how this works, carefully graph a sine wave of frequency 440 Hz (concert \(A ),\) with time on the horizontal axis and displacement on the vertical axis. Let the amplitude be 1 unit. On the same set of axes, graph the first overtone of 880 \(\mathrm{Hz}\) , but with an amplitude of \(\frac{1}{2}\) unit. (b) Now add the two waves to find their superposition. Notice that the shape is no longer a Isine wave.

\(\bullet\) Guitar string. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) frequency 245 Hz \()\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 \(\mathrm{m} / \mathrm{s}\) , find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

Standing sound waves are produced in a pipe that is 1.20 \(\mathrm{m}\) long. For the fundamental frequency and the first two over-tones, determine the locations along the pipe (measured from the left end) of the displacement nodes if (a) the pipe is open atboth ends; (b) the pipe is closed at the left end and open at the right end.

\(\cdot\) Find the fundamental frequency and the frequency of the first three overtones of a pipe 45.0 \(\mathrm{cm}\) long (a) if the pipe is open at both ends; (b) if the pipe is closed at one end. (c) For each of the preceding cases, what is the number of the highest harmonic that may be heard by a person who can hear frequencies from 20 \(\mathrm{Hz}\) to \(20,000 \mathrm{Hz}\) ?

\(\bullet\) The fundamental frequency of a pipe that is open at both ends is 594 Hz. (a) How long is this pipe? If one end is now closed, find (b) the wavelength and (c) the frequency of the new fundamental.

\(\cdot\) The role of the mouth in sound. The production of sound during speech or singing is a complicated process. Let's concentrate on the mouth. A typical depth for the human mouth is about \(8.0 \mathrm{cm},\) although this number can vary. (Check it against your own mouth.) We can model the mouth as an organ pipe that is open at the back of the throat. What are the wavelengths and frequencies of the first four harmonics you can produce if your mouth is (a) open, (b) closed? Use \(v=354 \mathrm{m} / \mathrm{s}\) .

\(\bullet\) You blow across the open mouth of an empty test tube and produce the fundamental standing wave of the air column inside the test tube. The speed of sound in air is 344 \(\mathrm{m} / \mathrm{s}\) and the test tube acts as a stopped pipe. (a) If the length of the air column in the test tube is \(14.0 \mathrm{cm},\) what is the frequency of this standing wave? (b) What is the frequency of the fundamental standing wave in the air column if the test tube is half filled with water?

Ultrasound and infrasound. (a) Whale communication. Blue whales apparently communicate with each other using sound of frequency 17 \(\mathrm{Hz}\) , which can be heard nearly 1000 \(\mathrm{km}\) away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 \(\mathrm{m} / \mathrm{s} ?\) (b) Dolphin clicks. One type of sound that dolphins emit is a sharp click of wavelength 1.5 \(\mathrm{cm}\) in the ocean. What is the frequency of such clicks? (c) Dog whistles. One brand of dog whistles claims a frequency of 25 \(\mathrm{kHz}\) for its product. What is the wavelength of this sound? (d) Bats. While bats emit a wide variety of sounds, one type emits pulses of sound having a frequency between 39 \(\mathrm{kHz}\) and 78 \(\mathrm{kHz}\) . What is the range of wavelengths of this sound? (e) Sonograms. Ultrasound is used to view the interior of the body, much as x rays are utilized. For sharp imagery, the wavelength of the sound should be around one-fourth (or less) the size of the objects to be viewed. Approximately what frequency of sound is needed to produce a clear image of a tumor that is 1.0 \(\mathrm{mm}\) across if the speed of sound in the tissue is 1550 \(\mathrm{m} / \mathrm{s} ?\)

A \(\mathrm{A} 75.0 \mathrm{cm}\) wire of mass 5.625 \(\mathrm{g}\) is tied at both ends and adjusted to a tension of 35.0 \(\mathrm{N} .\) When it is vibrating in its sec- ond overtone, find (a) the frequency and wavelength at which it is vibrating and (b) the frequency and wavelength of the sound waves it is producing.

\(\cdot\) Find the intensity \(\left(\) in \(W / m^{2}\right)\) of (a) a 55.0 dB sound, (b) a 92.0 dB sound, (c) a \(-2.0\) dB sound.

\(\bullet\) Find the noise level (in dB) of a sound having an intensity of (a) 0.000127 \(\mathrm{W} / \mathrm{m}^{2}\)(b) \(6.53 \times 10^{-10} \mathrm{W} / \mathrm{cm}^{2},\) (c) \(1.5 \times\) \(10^{-14} \mathrm{W} / \mathrm{m}^{2}\)

\(\bullet\) (a) By what factor must the sound intensity be increased to raise the sound intensity level by 13.0 dB? (b) Explain why you don't need to know the original sound intensity.

Eavesdropping! You are trying to overhear a juicy conversation, but from your distance of \(15.0 \mathrm{m},\) it sounds like only an average whisper of 20.0 \(\mathrm{dB}\) . So you decide to move closer to give the conversation a sound level of 60.0 \(\mathrm{dB}\) instead. How close should you come?

\(\bullet\) The intensity due to a number of independent sound sources is the sum of the individual intensities. (a) When four quadruplets cry simultaneously, how many decibels greater is the sound intensity level than when a single one cries? (b) To increase the sound intensity level again by the same number of decibels as in part (a), how many more crying babies are required?

\(\bullet(\) a) What is the sound intensity level in a car when the sound intensity is 0.500\(\mu \mathrm{W} / \mathrm{m}^{2} ?\) (b) What is the sound intensity in the air near a jackhammer when the sound intensity level is 103 \(\mathrm{dB} ?\)

A trumpet player is tuning his instrument by playing an A note simultaneously with the first-chair trumpeter, who has perfect pitch. The first-chair player's note is exactly 440 Hz, and 2.8 beats per second are heard. What are the two possible frequencies of the other player's note?

\(\bullet\) Two tuning forks are producing sounds of wavelength 34.40 \(\mathrm{cm}\) and 33.94 \(\mathrm{cm}\) simultaneously. How many beats do you hear each second?

\(\cdot\) Two guitarists attempt to play the same note of wavelength 6.50 \(\mathrm{cm}\) at the same time, but one of the instruments is slightly out of tune and plays a note of wavelength 6.52 \(\mathrm{cm}\) instead. What is the frequency of the beat these musicians hear when they play together?

\(\bullet\) Tuning a violin. A violinist is tuning her instrument to concert \(\mathrm{A}(440 \mathrm{Hz}) .\) She plays the note while listening to an electronically generated tone of exactly that frequency and hears a beat of frequency 3 \(\mathrm{Hz}\) , which increases to 4 \(\mathrm{Hz}\) when she tightens her violin string slightly.(a) What was the frequency of her violin when she heard the 3 -Hz beat? (b) To get her violin perfectly tuned to concert \(\mathrm{A}\) , should she tighten or loosen her string from what it was when she heard the 3 -Hz beat?

\(\cdot\) On the planet Arrakis, a male ornithoid is flying toward his stationary mate at 25.0 \(\mathrm{m} / \mathrm{s}\) while singing at a frequency of 1200 \(\mathrm{Hz} .\) If the female hears a tone of 1240 \(\mathrm{Hz}\) , what is the speed of sound in the atmosphere of Arrakis?

\(\cdot\) A car alarm is emitting sound waves of frequency 520 \(\mathrm{Hz}\) . You are on a motorcycle, traveling directly away from the car. How fast must you be traveling if you detect a frequency of 490 \(\mathrm{Hz} ?\)

\(\bullet\) The siren of a fire engine that is driving northward at 30.0 \(\mathrm{m} / \mathrm{s}\) emits a sound of frequency 2000 \(\mathrm{Hz}\) . A truck in front of this fire engine is moving northward at 20.0 \(\mathrm{m} / \mathrm{s}\) . (a) What is the frequency of the siren's sound that the fire engine's driver hears reflected from the back of the truck? (b) What wavelength would this driver measure for these reflected sound waves?

A stationary police car emits a sound of frequency 1200 \(\mathrm{Hz}\) that bounces off of a car on the highway and returns with a frequency of 1250 \(\mathrm{Hz}\) . The police car is right next to the highway, so the moving car is traveling directly toward or away from it. (a) How fast was the moving car going? Was it moving towards or away from the police car? (b) What frequency would the police car have received if it had been traveling toward theother car at 20.0 \(\mathrm{m} / \mathrm{s} ?\)

A container ship is traveling westward at a speed of 5.00 \(\mathrm{m} / \mathrm{s} .\) The waves on the surface of the ocean have a wave-length of 40.0 \(\mathrm{m}\) and are traveling eastward at a speed of 16.5 \(\mathrm{m} / \mathrm{s} .\) (a) At what time intervals does the ship encounter the crest of a wave? (b) At what time intervals will the ship encounter wave crests if it turns around and heads eastward?

While sitting in your car by the side of a country road, you see your friend, who happens to have an identical car with an identical horn, approaching you. You blow your horn, which has a frequency of \(260 \mathrm{Hz} ;\) your friend begins to blow his horn as well, and you hear a beat frequency of 6.0 \(\mathrm{Hz} .\) How fast is your friend approaching you?

Moving source vs. moving listener. (a) A sound source producing 1.00 kHz waves moves toward a stationary listener at one-half the speed of sound. What frequency will the listener hear? (b) Suppose instead that the source is stationary and the listener moves toward the source at one-half the speed of sound. What frequency does the listener hear? How does your answer compare with that in part (a)? Did you expect to get the same answer in both cases? Explain on physical grounds why the two answers differ.

\(\bullet\) How fast (as a percentage of light speed) would a star have to be moving so that the frequency of the light we receive from it is 10.0\(\%\) higher than the frequency of the light it is emitting? Would it be moving away from us or toward us? (Assume it is moving either directly away from us or directly toward us.)

A very noisy chain saw operated by a tree surgeon emits a total acoustic power of 20.0 \(\mathrm{W}\) uniformly in all directions. At what distance from the source is the sound level equal to (a) 100 \(\mathrm{dB}\) . (b) 60 \(\mathrm{dB} ?\)

Tuning a cello. A cellist tunes the C-string of her instrument to a fundamental frequency of 65.4 Hz. The vibrating portion of the string is 0.600 \(\mathrm{m}\) long and has a mass of 14.4 \(\mathrm{g} .\) (a) With what tension must she stretch that portion of the string? (b) What percentage increase in tension is needed to increase the frequency from 65.4 Hz to 73.4 Hz, corresponding to a rise in pitch from \(\mathrm{C}\) to \(\mathrm{D} ?\)

\(\bullet\) A person is playing a small flute 10.75 \(\mathrm{cm}\) long, open at one end and closed at the other, near a taut string having a fundamental frequency of 600.0 \(\mathrm{Hz}\) . If the speed of sound is \(344.0 \mathrm{m} / \mathrm{s},\) for which harmonics of the flute will the string res- onate? In each case, which harmonic of the string is in resonance?

A bat flies toward a wall, emitting a steady sound of frequency 2000 \(\mathrm{Hz}\) . The bat hears its own sound, plus the sound reflected by the wall. How fast should the bat fly in order to hear a beat frequency of 10.0 \(\mathrm{Hz}\) ? (Hint: Break this problem into two parts, first with the bat as the source and the wall as the listener and then with the wall as the source and the bat as the listener.)

You're standing between two speakers that are driven by the same amplifier and are emitting sound waves with frequency 229 Hz. The two speakers are facing each other, 15 meters apart. (a) You begin walking away from one speaker toward the other one, and as you walk, you hear what sounds like beats, with a frequency of 2.50 Hz. How fast are you walking? (b) If the frequency of the sound emitted by the speakers increases to 573 Hz and you continue to walk at the same speed, what frequency of beats will you hear? [Hint: You can model this situation as a tube open at both ends; alternatively, you can treat it as a Doppler effect problem.]

\(\bullet\) The sound source of a ship's sonar system operates at a frequency of 22.0 \(\mathrm{kHz}\) . The speed of sound in water (assumed to be at a uniform \(20^{\circ} \mathrm{C}\) ) is 1482 \(\mathrm{m} / \mathrm{s}\) . (a) What is the wavelength of the waves emitted by the source? (b) What is the difference in frequency between the directly radiated waves andthe waves reflected from a whale traveling straight toward the ship at 4.95 \(\mathrm{m} / \mathrm{s} ?\) The ship is at rest in the water.

\(\bullet\) The range of human hearing. A young person with normal hearing can hear sounds ranging from 20 Hz to 20 \(\mathrm{kHz}\) .How many octaves can such a person hear? (Recall that if two tones differ by an octave, the higher frequency is twice the lower frequency.)

A person leaning over a 125 -m-deep well accidentally drops a siren emitting sound of frequency 2500 Hz. Just before this siren hits the bottom of the well, find the frequency and wavelength of the sound the person hears (a) coming directly from the siren, \((\) b) reflected off the bottom of the well. (c) What beat frequency does this person perceive?