Chapter 14

Earthquakes at fault lines in Earth's crust create seismic waves, which are longitudinal (P-waves) or transverse (S-waves). The P-waves have a speed of about \(7 \mathrm{~km} / \mathrm{s}\). Estimate the average bulk modulus of Earth's crust given that the density of rock is about \(2500 \mathrm{~kg} / \mathrm{m}^{3}\).

that made by the explosion of Krakatoa on August \(26-27,1888\). According to barometric measurements, the sound had a decibel level of \(180 \mathrm{~dB}\) at a distance of \(161 \mathrm{~km}\). Assuming the intensity falls off as the inverse of the distance squared, what was the decibel level on Rodriguez Island, \(4800 \mathrm{~km}\) away?

A sound wave from a siren has an intensity of \(100.0 \mathrm{~W} / \mathrm{m}^{2}\) at a certain point, and a second sound wave from a nearby ambulance has an intensity level \(10 \mathrm{~dB}\) greater than the siren's sound wave at the same point. What is the intensity level of the sound wave due to the ambulance?

BIO A person wears a hearing aid that uniformly increases the intensity level of all audible frequencies of sound by \(30.0 \mathrm{~dB}\). The hearing aid picks up sound having a frequency of \(250 \mathrm{~Hz}\) at an intensity of \(3.0 \times 10^{-11} \mathrm{~W} / \mathrm{m}^{2}\). What is the intensity delivered to the eardrum?

There is evidence that elephants communicate via infrasound, generating rumbling vocalizations as low as \(14 \mathrm{~Hz}\) that can travel up to \(10 \mathrm{~km}\). The intensity level of these sounds can reach \(103 \mathrm{~dB}\), measured a distance of \(5.0 \mathrm{~m}\) from the source. Determine the intensity level of the infrasound \(10 \mathrm{~km}\) from the source, assuming the sound energy radiates uniformly in all directions.

A commuter train passes a passenger platiorm at a constant speed of \(40.0 \mathrm{~m} / \mathrm{s}\). The train horn is sounded at its characteristic frequency of \(320 \mathrm{~Hz}\). (a) What overall change in frequency is detected by a person on the platform as the train moves from approaching to receding? (b) What wavelength is detected by a person

At rest, a car's horn sounds the note A \((440 \mathrm{~Hz})\). The horn is sounded while the car is moving down the street. A bicyclist moving in the same direction with one-third the car's speed hears a frequency of \(415 \mathrm{~Hz}\). (a) Is the cyclist ahead of or behind the car? (b) What is the speed of the car?

A bat flying at \(5.00 \mathrm{~m} / \mathrm{s}\) is chasing an insect flying in the same direction. If the bat emits a \(40.0-\mathrm{kHz}\) chirp and receives back an echo at \(40.4 \mathrm{kHz}\), (a) what is the speed of the insect? (b) Will the bat be able to catch the insect? Explain.

A tuning fork vibrating at \(512 \mathrm{~Hz}\) falls from rest and accelerates at \(9.80 \mathrm{~m} / \mathrm{s}^{2}\). How far below the point of release is the tuning fork when waves of frequency \(485 \mathrm{~Hz}\) reach the release point?

Expectant parents are thrilled to hear their unborn baby's heartbeat, revealed by an ultrasonic motion detector. Suppose the fetus's ventricular wall moves in simple harmonic motion with amplitude \(1.80 \mathrm{~mm}\) and frequency 115 beats per minute. The motion detector in contact with the mother's abdomen produces sound at precisely \(2 \mathrm{MHz}\), which travels through tissue at \(1.50 \mathrm{~km} / \mathrm{s}\). (a) Find the maximum linear speed of the heart wall. (b) Find the maximum frequency at which sound arrives at the wall of the baby's heart. (c) Find the maximum frequency at which reflected sound is received by the motion detector. (By electronically "listening" for echoes at a frequency different from the broadcast frequency, the motion detector can produce beeps of audible sound in synchrony with the fetal heartbeat.)

Two small speakers are driven by a common oscillator at \(8.00 \times 10^{2} \mathrm{~Hz}\). The speakers face each other and are separated by \(1.25 \mathrm{~m}\). Locate the points along a line joining the two speakers where relative minima would be expected. (Use \(v=343 \mathrm{~m} / \mathrm{s}\).)

How far, and in what direction, should a cellist move her finger to adjust a string's tone from an out-of-tune \(449 \mathrm{~Hz}\) to an in-tune \(440 \mathrm{~Hz}\) ? The string is \(68.0 \mathrm{~cm}\) long, and the finger is \(20.0 \mathrm{~cm}\) from the nut for the 449-Hz tone.

A stretched string of length \(L\) is observed to vibrate in five equal segments when driven by a \(630-\mathrm{Hz}\) oscillator. What oscillator frequency will set up a standing wave so that the string vibrates in three segments?

Two pieces of steel wire with identical cross sections have lengths of \(L\) and \(2 L\). The wires are each fixed at both ends and stretched so that the tension in the longer wire is four times greater than in the shorter wire. If the fundamental frequency in the shorter wire is \(60 \mathrm{~Hz}\), what is the frequency of the second harmonic in the longer wire?

A steel wire with mass \(25.0 \mathrm{~g}\) and length \(1.35 \mathrm{~m}\) is strung on a bass so that the distance from the nut to the bridge is \(1.10 \mathrm{~m}\). (a) Compute the linear density of the string. (b) What velocity wave on the string will produce the desired fundamental frequency of the \(\mathrm{E}_{1}\) string, \(41.2 \mathrm{~Hz}\) ? (c) Calculate the tension required to obtain the proper frequency. (d) Calculate the wavelength of the string's vibration. (e) What is the wavelength of the sound produced in air? (Assume the speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\).)

A standing wave is set up in a string of variable length and tension by a vibrator of variable frequency. Both ends of the string are fixed. When the vibrator has a frequency \(f_{A}\), in a string of length \(L_{A}\) and under tension \(T_{A}, n_{A}\) antinodes are set up in the string. (a) Write an expression for the frequency \(f_{A}\) of a standing wave in terms of the number \(n_{A}\), length \(L_{A}\), tension \(T_{A}\), and linear density \(\mu_{A^{*}}\) (b) If the length of the string is doubled to \(L_{B}=2 L_{A}\), what frequency \(f_{B}\) (written as a multiple of \(f_{A}\) ) will result in the same number of antinodes? Assume the tension and linear density are unchanged. Hint: Make a ratio of expressions for \(f_{B}\) and \(f_{A}\). (c) If the frequency and length are held constant, what tension \(T_{B}\) will produce \(n_{A}+1\) antinodes? (d) If the frequency is tripled and the length of the string is halved, by what factor should the tension be changed so that twice as many antinodes are produced?

guitar string under a tension of \(50.000 \mathrm{~N}\) has a mass per unit length of \(0.10000 \mathrm{~g} / \mathrm{cm}\). What is the highest resonant frequency that can be heard by a person capable of hearing frequencies up to \(20000 \mathrm{~Hz}\) ?

\- Standing-wave vibrations are set up in a crystal goblet with four nodes and four antinodes equally spaced around the \(20.0-\mathrm{cm}\) circumference of its rim. If transverse waves move around the glass at \(900 \mathrm{~m} / \mathrm{s}\), an opera singer would have to produce a high harmonic with what frequency in onder to shatter the glass with a resonant vibration?

The overall length of a piccolo is \(32.0 \mathrm{~cm}\). The resonating air column vibrates as in a pipe that is open at both ends. (a) Find the frequency of the lowest note a piccolo can play. (b) Opening holes in the side effectively shortens the length of the resonant column. If the highest note a piccolo can sound is \(4000 \mathrm{~Hz}\), find the distance between adjacent antinodes for this mode of vibration.

The \(\mathrm{G}\) string on a violin has a fundamental frequency of \(196 \mathrm{~Hz}\). It is \(30.0 \mathrm{~cm}\) long and has a mass of \(0.500 \mathrm{~g}\). While this string is sounding, a nearby violinist effectively shortens the \(\mathrm{G}\) string on her identical violin (by sliding her finger down the string) until a beat frequency of \(2.00 \mathrm{~Hz}\) is heard between the two strings. When that occurs, what is the effective length of her string?c

Two train whistles have identical frequencies of \(1.80 \times\) \(10^{2} \mathrm{~Hz}\). When one train is at rest in the station and the other is moving nearby, a commuter standing on the station platform hears beats with a frequency of \(2.00\) beats \(/ \mathrm{s}\) when the whistles operate together. What are the two possible speeds and directions that the moving train can have?

Two pipes of equal length are each open at one end. Each has a fundamental frequency of \(480 \mathrm{~Hz}\) at \(300 \mathrm{~K}\). In one pipe the air temperature is increased to \(305 \mathrm{~K}\). If the two pipes are sounded together, what beat frequency results?

A student holds a tuning fork oscillating at \(256 \mathrm{~Hz}\). He walks toward a wall at a constant speed of \(1.33 \mathrm{~m} / \mathrm{s}\). (a) What beat frequency does he observe between the tuning fork and its echo? (b) How fast must he walk away from the wall to observe a beat frequency of \(5.00 \mathrm{~Hz}\) ?

Assume a 150 -W loudspeaker broadcasts sound equally in all directions and produces sound with a level of \(108 \mathrm{~dB}\) at a distance of \(1.60 \mathrm{~m}\) from its center. (a) Find its sound power output. If a salesperson claims the speaker is rated at \(150 \mathrm{~W}\), he is referring to the maximum electrical power input to the speaker. (b) Find the efficiency of the speaker, that is, the fraction of input power that is converted into useful output power.

M Two ships are moving along a line due east (Fig. P14.68). The trailing vessel has a speed relative to a landbased observation point of \(v_{1}=64.0 \mathrm{~km} / \mathrm{h}\), and the leading ship has a speed of \(v_{2}=45.0 \mathrm{~km} / \mathrm{h}\) relative to that point. The two ships are in a region of the ocean where the current is moving uniformly due west at \(v_{\text {current }}=\) \(10.0 \mathrm{~km} / \mathrm{h}\). The trailing ship transmits a sonar signal at a frequency of \(1200.0 \mathrm{~Hz}\) through the water. What frequency is monitored by the leading ship?

A quartz watch contains a crystal oscillator in the form of a block of quartz that vibrates by contracting and expanding. Two opposite faces of the block, \(7.05 \mathrm{~mm}\) apart, are antinodes, moving alternately toward and away from each other. The plane halfway between these two faces is a node of the vibration. The speed of sound in quartz is \(3.70 \times 10^{3} \mathrm{~m} / \mathrm{s}\). Find the frequency of the vibration. An oscillating electric voltage accompanies the mechanical oscillation, so the quartz is described as piezoelectric. An electric circuit feeds in energy to maintain the oscillation and also counts the voltage pulses to keep time.

A flute is designed so that it plays a frequency of \(261.6 \mathrm{~Hz}\), middle \(\mathrm{C}\), when all the holes are covered and the temperature is \(20.0^{\circ} \mathrm{C}\). (a) Consider the flute to be a pipe open at both ends and find its length, assuming the middle-C frequency is the fundamental frequency. (b) A second player, nearby in a colder room, also attempts to play middle C on an identical flute. A beat frequency of \(3.00\) beats \(/ \mathrm{s}\) is heard. What is the temperature of the room?

A flute is designed so that it plays a frequency of \(261.6 \mathrm{~Hz}\), middle \(\mathrm{C}\), when all the holes are covered and the temperature is \(20.0^{\circ} \mathrm{C}\). (a) Consider the flute to be a pipe open at both ends and find its length, assuming the middle-C frequency is the fundamental frequency. (b) A second player, nearby in a colder room, also attempts to play middle C on an identical flute. A beat frequency of \(3.00\) beats \(/ \mathrm{s}\) is heard. What is the temperature of the room?

A student stands several meters in front of a smooth reflecting wall, holding a board on which a wire is fixed at each end. The wire, vibrating in its third harmonic, is \(75.0 \mathrm{~cm}\) long, has a mass of \(2.25 \mathrm{~g}\), and is under a tension of \(400 \mathrm{~N}\). A second student, moving toward the wall, hears \(8.30\) beats per second. What is the speed of the student approaching the wall?

By proper excitation, it is possible to produce both longitudinal and transverse waves in a long metal rod. In a particular case, the rod is \(150 \mathrm{~cm}\) long and \(0.200 \mathrm{~cm}\) in radius and has a mass of \(50.9 \mathrm{~g}\). Young's modulus for the material is \(6.80 \times 10^{10} \mathrm{~Pa}\). Determine the required tension in the rod so that the ratio of the speed of longitudinal waves to the speed of transverse waves is 8 .