Chapter 17

Two speakers, one directly behind the other, are each generating a 245 -Hz sound wave. What is the smallest separation distance between the speakers that will produce destructive interference at a listener standing in front of them? The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\).

Refer to Interactive Solution \(17.817 .8\) at to review a method by which this problem can be solved. Two loudspeakers on a concert stage are vibrating in phase. A listener is \(50.5 \mathrm{~m}\) from the left speaker and \(26.0 \mathrm{~m}\) from the right one. The listener can respond to all frequencies from 20 to \(20000 \mathrm{~Hz}\), and the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What are the two lowest frequencies that can be heard loudly due to constructive interference?

Speakers A and B are vibrating in phase. They are directly facing each other, are \(7.80 \mathrm{~m}\) apart, and are each playing a \(73.0\) - \(\mathrm{Hz}\) tone. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker \(\mathrm{A}\) ?

Speakers \(\mathrm{A}\) and \(\mathrm{B}\) are vibrating in phase. They are directly facing each other, are \(7.80 \mathrm{~m}\) apart, and are each playing a 73.0 -Hz tone. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker \(\mathrm{A} ?\)

Multiple-Concept Example 3 reviews the concepts that are important in this problem. The entrance to a large lecture room consists of two side-by-side doors, one hinged on the left and the other hinged on the right. Each door is \(0.700 \mathrm{~m}\) wide. Sound of frequency \(607 \mathrm{~Hz}\) is coming through the entrance from within the room. The speed of sound is 343 \(\mathrm{m} / \mathrm{s}\). What is the diffraction angle \(\theta\) of the sound after it passes through the doorway when (a) one door is open and (b) both doors are open?

Sound exits a diffraction horn loudspeaker through a rectangular opening like a small doorway. Such a loudspeaker is mounted outside on a pole. In winter, when the temperature is \(273 \mathrm{~K}\), the diffraction angle \(\theta\) has a value of \(15.0^{\circ} .\) What is the diffraction angle for the same sound on a summer day when the temperature is \(311 \mathrm{~K}\) ?

For one approach to problems such as this, see Multiple-Concept Example \(3 .\) Sound emerges through a doorway, as in Figure \(17-11\). The width of the doorway is \(77 \mathrm{~cm}\), and the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). Find the diffraction angle \(\theta\) when the frequency of the sound is (a) \(5.0 \mathrm{kHz}\) and (b) \(5.0 \times 10^{2} \mathrm{~Hz}\).

A row of seats is parallel to a stage at a distance of \(8.7 \mathrm{~m}\) from it. At the center and front of the stage is a diffraction horn loudspeaker. This speaker sends out its sound through an opening that is like a small doorway with a width \(D\) of \(7.5 \mathrm{~cm} .\) The speaker is playing a tone that has a frequency of \(1.0 \times 10^{4} \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the distance between two seats, located near the center of the row, at which the tone cannot be heard?

A \(3.00-\mathrm{kHz}\) tone is being produced by a speaker with a diameter of \(0.175 \mathrm{~m} .\) The air temperature changes from 0 to \(29^{\circ} \mathrm{C}\). Assuming air to be an ideal gas, find the change in the diffraction angle \(\theta\).

Two out-of-tune flutes play the same note. One produces a tone that has a frequency of \(262 \mathrm{~Hz},\) while the other produces \(266 \mathrm{~Hz}\). When a tuning fork is sounded together with the \(262-\mathrm{Hz}\) tone, a beat frequency of \(1 \mathrm{~Hz}\) is produced. When the same tuning fork is sounded together with the 266 -Hz tone, a beat frequency of 3 Hz is produced. What is the frequency of the tuning fork?

Two pianos each sound the same note simultaneously, but they are both out of tune. On a day when the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\), piano A produces a wavelength of \(0.769 \mathrm{~m}\), while piano B produces a wavelength of \(0.776 \mathrm{~m}\). How much time separates successive beats?

When a guitar string is sounded along with a 440 -Hz tuning fork, a beat frequency of \(5 \mathrm{~Hz}\) is heard. When the same string is sounded along with a 436 -Hz tuning fork, the beat frequency is \(9 \mathrm{~Hz}\). What is the frequency of the string?

In Concept Simulation \(17.2\) at you can explore the concepts that are important in this problem. A 440.0-Hz tuning fork is sounded together with an out-of-tune guitar string, and a beat frequency of \(3 \mathrm{~Hz}\) is heard. When the string is tightened, the frequency at which it vibrates increases, and the beat frequency is heard to decrease. What was the original frequency of the guitar string?

A sound wave is traveling in seawater, where the adiabatic bulk modulus and density are \(2.31 \mathrm{X} 10^{9} \mathrm{~Pa}\) and \(1025 \mathrm{~kg} / \mathrm{m}^{3}\), respectively. The wavelength of the sound is \(3.35 \mathrm{~m}\). A tuning fork is struck underwater and vibrates at \(440.0 \mathrm{~Hz} .\) What would be the beat frequency heard by an underwater swimmer?

Two tuning forks \(X\) and \(Y\) have different frequencies and produce an \(8-H z\) beat frequency when sounded together. When \(X\) is sounded along with a \(392-\mathrm{Hz}\) tone, a \(3-\mathrm{Hz}\) beat frequency is detected. When \(Y\) is sounded along with the \(392-\mathrm{Hz}\) tone, a \(5-\mathrm{Hz}\) beat frequency is heard. What are the frequencies \(f_{X}\) and \(f_{Y}\) when \((a) f_{X}\) is greater than \(f_{Y}\) and (b) \(f_{X}\) is less than \(f_{Y}\) ?

The approach to solving this problem is similar to that taken in Multiple- Concept Example 4 . On a cello, the string with the largest linear density \(\left(1.56 \times 10^{-2} \mathrm{~kg} / \mathrm{m}\right)\) is the C string. This string produces a fundamental frequency of \(65.4 \mathrm{~Hz}\) and has a length of \(0.800 \mathrm{~m}\) between the two fixed ends. Find the tension in the string.

A string of length \(0.28 \mathrm{~m}\) is fixed at both ends. The string is plucked and a standing wave is set up that is vibrating at its second harmonic. The traveling waves that make up the standing wave have a speed of \(140 \mathrm{~m} / \mathrm{s}\). What is the frequency of vibration?

Multiple-Concept Example 4 deals with the same concepts as this problem. A \(41-\mathrm{cm}\) length of wire has a mass of \(6.0 \mathrm{~g}\). It is stretched between two fixed supports and is under a tension of \(160 \mathrm{~N}\). What is the fundamental frequency of this wire?

Suppose the strings on a violin are stretched with the same tension and each has the same length between its two fixed ends. The musical notes and corresponding fundamental frequencies of two of these strings are \(\mathrm{G}(196.0 \mathrm{~Hz})\) and \(\mathrm{E}(659.3 \mathrm{~Hz}) .\) The linear density of the E string is \(3.47 \times 10^{-4} \mathrm{~kg} / \mathrm{m}\). What is the linear density of the \(\mathrm{G}\) string?

To review the concepts that play roles in this problem, consult Multiple- Concept Example 4. Sometimes, when the wind blows across a long wire, a low- frequency “moaning" sound is produced. This sound arises because a standing wave is set up on the wire, like a standing wave on a guitar string. Assume that a wire (linear density \(=0.0140\) \(\mathrm{kg} / \mathrm{m}\) ) sustains a tension of \(323 \mathrm{~N}\) because the wire is stretched between two poles that are \(7.60 \mathrm{~m}\) apart. The lowest frequency that an average, healthy human ear can detect is \(20.0 \mathrm{~Hz} .\) What is the lowest harmonic number \(n\) that could be responsible for the “moaning" sound?

A string has a linear density of \(8.5 \times 10^{-3} \mathrm{~kg} / \mathrm{m}\) and is under a tension of \(280 \mathrm{~N}\). The string is \(1.8 \mathrm{~m}\) long, is fixed at both ends, and is vibrating in the standing wave pattern shown in the drawing. Determine the (a) speed, (b) wavelength, and (c) frequency of the traveling waves that make up the standing wave.

The \(\mathrm{E}\) string on an electric bass guitar has a length of \(0.628 \mathrm{~m}\) and, when producing the note E, vibrates at a fundamental frequency of \(41.2 \mathrm{~Hz}\). Players sometimes add to their instruments a device called a "D-tuner." This device allows the E string to be used to produce the note D, which has a fundamental frequency of \(36.7 \mathrm{~Hz}\). The D-tuner works by extending the length of the string, keeping all other factors the same. By how much does a D-tuner extend the length of the \(\mathrm{E}\) string?

The E string on an electric bass guitar has a length of \(0.628 \mathrm{~m}\) and, when producing the note E, vibrates at a fundamental frequency of \(41.2 \mathrm{~Hz}\). Players sometimes add to their instruments a device called a "D-tuner." This device allows the E string to be used to produce the note \(\mathrm{D}\), which has a fundamental frequency of \(36.7 \mathrm{~Hz}\). The D-tuner works by extending the length of the string, keeping all other factors the same. By how much does a D-tuner extend the length of the E string?

The note that is three octaves above middle \(C\) is supposed to have a fundamental frequency of \(2093 \mathrm{~Hz}\). On a certain piano the steel wire that produces this note has a cross-sectional area of \(7.85 \times 10^{-7} \mathrm{~m}^{2}\). The wire is stretched between two pegs. When the piano is tuned properly to produce the correct frequency at \(25.0{ }^{\circ} \mathrm{C},\) the wire is under a tension of \(818.0 \mathrm{~N}\). Suppose the temperature drops to \(20.0{ }^{\circ} \mathrm{C} .\) In addition, as an approximation, assume that the wire is kept from contracting as the temperature drops. Consequently, the tension in the wire changes. What beat frequency is produced when this piano and another instrument (properly tuned) sound the note simultaneously?

An organ pipe is open at both ends. It is producing sound at its third harmonic, the frequency of which is \(262 \mathrm{~Hz}\). The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the length of the pipe?

A tube of air is open at only one end and has a length of \(1.5 \mathrm{~m}\). This tube sustains a standing wave at its third harmonic. What is the distance between one node and the adjacent antinode?

The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of \(343 \mathrm{~m} / \mathrm{s}\) for the speed of sound, what are the lengths of the longest and shortest pipes (open at both ends and producing sound at their fundamental frequencies) that you expect to find in a pipe organ?

Divers working in underwater chambers at great depths must deal with the danger of nitrogen narcosis (the "bends"), in which nitrogen dissolves into the blood at toxic levels. One way to avoid this danger is for divers to breathe a mixture containing only helium and oxygen. Helium, however, has the effect of giving the voice a high-pitched quality, like that of Donald Duck's voice. To see why this occurs, assume for simplicity that the voice is generated by the vocal cords vibrating above a gas-filled cylindrical tube that is open only at one end. The quality of the voice depends on the harmonic frequencies generated by the tube; larger frequencies lead to higher-pitched voices. Consider two such tubes at \(20{ }^{\circ} \mathrm{C}\). One is filled with air, in which the speed of sound is \(343 \mathrm{~m} / \mathrm{s} .\) The other is filled with helium, in which the speed of sound is \(1.00 \times 10^{3} \mathrm{~m} / \mathrm{s}\) To see the effect of helium on voice quality, calculate the ratio of the \(n^{\text {th }}\) natural frequency of the helium- filled tube to the \(n^{\text {th }}\) natural frequency of the air-filled tube.

A tube is open only at one end. A certain harmonic produced by the tube has a frequency of \(450 \mathrm{~Hz}\). The next higher harmonic has a frequency of \(750 \mathrm{~Hz}\). The speed of sound in air is \(343 \mathrm{~m} / \mathrm{s}\). (a) What is the integer \(n\) that describes the harmonic whose frequency is \(450 \mathrm{~Hz} ?\) (b) What is the length of the tube?

Two loudspeakers face each other, vibrate in phase, and produce identical 440 -Hz tones. A listener walks from one speaker toward the other at a constant speed and hears the loudness change (loud-soft-loud) at a frequency of \(3.0 \mathrm{~Hz} .\) The speed of sound is \(343 \mathrm{~m} /\) s. What is the walking speed?

A person hums into the top of a well and finds that standing waves are established at frequencies of \(42,70.0,\) and \(98 \mathrm{~Hz} .\) The frequency of \(42 \mathrm{~Hz}\) is not necessarily the fundamental frequency. The speed of sound is \(343 \mathrm{~m} / \mathrm{s} .\) How deep is the well?

A vertical tube is closed at one end and open to air at the other end. The air pressure is \(1.01 \mathrm{X} 10^{5} \mathrm{~Pa}\). The tube has a length of \(0.75 \mathrm{~m}\). Mercury (mass density \(=13\) \(\left.600 \mathrm{~kg} / \mathrm{m}^{3}\right)\) is poured into it to shorten the effective length for standing waves. What is the absolute pressure at the bottom of the mercury column, when the fundamental frequency of the shortened, air-filled tube is equal to the third harmonic of the original tube?

A tube, open at only one end, is cut into two shorter (nonequal) lengths. The piece that is open at both ends has a fundamental frequency of \(425 \mathrm{~Hz},\) while the piece open only at one end has a fundamental frequency of \(675 \mathrm{~Hz}\). What is the fundamental frequency of the original tube?

Sound enters the ear, travels through the auditory canal, and reaches the eardrum. The auditory canal is approximately a tube open at only one end. The other end is closed by the eardrum. A typical length for the auditory canal in an adult is about \(2.9 \mathrm{~cm} .\) The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the fundamental frequency of the canal? (Interestingly, the fundamental frequency is in the frequency range where human hearing is most sensitive.)

Two ultrasonic sound waves combine and form a beat frequency that is in the range of human hearing. The frequency of one of the ultrasonic waves is \(70 \mathrm{kHz}\). What is (a) the smallest possible and (b) the largest possible value for the frequency of the other ultrasonic wave?

The fundamental frequency of a vibrating system is \(400 \mathrm{~Hz}\). For each of the following systems, give the three lowest frequencies (excluding the fundamental) at which standing waves can occur: (a) a string fixed at both ends, (b) a cylindrical pipe with both ends open, and (c) a cylindrical pipe with only one end open.

The A string on a string bass is tuned to vibrate at a fundamental frequency of 55.0 \(\mathrm{Hz}\). If the tension in the string were increased by a factor of four, what would be the new fundamental freauency?

Two loudspeakers are mounted on a merry-go-round whose radius is \(9.01 \mathrm{~m} .\) When stationary, the speakers both play a tone whose frequency is \(100.0 \mathrm{~Hz}\). As the drawing illustrates, they are situated at opposite ends of a diameter. The speed of sound is 343.00 \(\mathrm{m} / \mathrm{s},\) and the merry-go-round revolves once every \(20.0 \mathrm{~s}\). What is the beat frequency that is detected by the listener when the merry-go- round is near the position shown?

(a) When sound emerges from a loudspeaker, is the diffraction angle determined by the wavelength, the diameter of the speaker, or a combination of these two factors? (b) How is the wavelength of a sound related to its frequency? Explain your answers. The following two lists give diameters and sound frequencies for three loudspeakers. Pair each diameter with a frequency, so that the diffraction angle is the same for each of the speakers. The speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). Find the common diffraction angle. $$ \begin{array}{|l|c|} \hline \text { Diameter, } D & \text { Frequency, } f \\ \hline 0.050 \mathrm{~m} & 6.0 \mathrm{Khz} \\ \hline 0.10 \mathrm{~m} & 4.0 \mathrm{kHz} \\ \hline 0.15 \mathrm{~m} & 12.0 \mathrm{kHz} \\ \hline \end{array} $$

Two cars have identical horns, each emitting a frequency \(f_{s}\). One of the cars is moving toward a bystander waiting at a corner, and the other is parked. The two horns sound simultaneously. (a) From the moving horn, does the bystander hear a frequency that is greater than, less than, or equal to \(f_{\mathrm{s}} ?\) (b) From the stationary horn, does the bystander hear a frequency that is greater than, less than, or equal to \(f_{\mathrm{s}} ?\) (c) Does the bystander hear a beat frequency from the combined sound of the two horns? Account for your answers. The frequency that the horns emit is \(f_{\mathrm{s}}=395 \mathrm{~Hz}\). The speed of the moving car is \(12.0 \mathrm{~m} / \mathrm{s},\) and the speed of sound is \(343 \mathrm{~m} / \mathrm{s}\). What is the beat frequency heard by the bystander?

Two wires are stretched between two fixed supports and have the same length. On wire A there is a second-harmonic standing wave whose frequency is \(660 \mathrm{~Hz}\). However, the same frequency of \(660 \mathrm{~Hz}\) is the third harmonic on wire \(\mathrm{B}\). (a) Is the fundamental frequency of wire A greater than, less than, or equal to the fundamental frequency of wire \(\mathrm{B}\) ? Explain. (b) How is the fundamental frequency related to the length \(L\) of the wire and the speed \(v\) at which individual waves travel back and forth on the wire? (c) Do the individual waves travel on wire A with a greater, smaller, or the same speed as on wire B? Give your reasoning. The common length of the wires is \(1.2 \mathrm{~m}\). Find the speed at which individual waves travel on each wire. Verify that your answer is consistent with your answers to the Concept Questions.

A copper block is suspended in air from a wire in Part 1 of the drawing. A container of mercury is then raised up around the block as in Part \(2 .\) (a) The fundamental frequency of the wire is given by Equation 17.3 with \(n=1: f_{1}=v /(2 L)\). How is the speed \(v\) at which individual waves travel on the wire related to the tension in the wire? (b) Is the tension in the wire in Part 2 less than, greater than, or equal to the tension in Part \(1 ?\) (c) Is the fundamental frequency of the wire in Part 2 less than, greater than, or equal to the fundamental frequency in Part \(1 ?\) Justify each of your answers. In Part 2 of the drawing some \((50.0 \%)\) of the block's volume is submerged in the mercury. The density of copper is \(8890 \mathrm{~kg} / \mathrm{m}^{3},\) and the density of mercury is 13600 \(\mathrm{kg} / \mathrm{m}^{3}\). Find the ratio of the fundamental frequency of the wire in Part 2 to the fundamental frequency in Part \(1 .\) Check to see that your answer is consistent with your answers to the Concept Questions.

One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance \(L\) from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of \(L\) is measured that allows a standing wave to be formed. (a) When a standing wave is formed in the tube, is there a displacement node or antinode at the end of the tube where the tuning fork is, and is there a displacement node or antinode at the plunger? (b) How is the smallest value of \(L\) related to the wavelength of the sound? Explain your answers. The tuning fork produces a \(485-\mathrm{Hz}\) tone, and the smallest value observed for \(L\) is \(0.264 \mathrm{~m}\). What is the speed of the sound in the gas in the tube?

Standing waves are set up on two strings fixed at each end, as shown in the drawing. The tensions in the strings are the same, and each string has the same mass per unit length. However, one string is longer than the other, (a) Do the waves on the longer string have a larger speed, a smaller speed, or the same speed as those on the shorter string? Justify your answer. (b) Will the longer string vibrate at a higher frequency, a lower frequency, or the same frequency as the shorter string? Provide a reason for your answer. (c) Will the beat frequency produced by the two standing waves increase or decrease if the longer string is increased in length? Why? The two strings in the drawing have the same tension and mass per unit length, but they differ in length by \(0.57 \mathrm{~cm} .\) The waves on the shorter string propagate with a speed of \(41.8 \mathrm{~m} / \mathrm{s},\) and the fundamental frequency of the shorter string is \(225 \mathrm{~Hz}\). Determine the beat frequency produced by the two standing waves.