The two speakers in the drawing are vibrating in phase,

In Figure 17.7, suppose that the separation between speakers A and B is 5.00 m and the speakers are vibrating in phase. They are playing identical 125-Hz tones, and the speed of sound is 343 m/s. What is the largest possible distance between speaker B and the observer at C, such that he observes destructive interference?

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 m/s.

The drawing graphs a string on which two rectangular pulses are traveling at a constant speed of 1 cm/s at time t 5 0 s. Using the principle of linear superposition, draw the shape of the string at t 5 1 s, 2 s, 3 s, and 4 s.

Loudspeakers A and B are vibrating in phase and are playing the same tone, which has a frequency of 250 Hz. They are set up as in Figure 17.7, and point C is located as shown there. However, the distance between the speakers and the distance between speaker B and point C have the same value d. The speed of sound is 343 m/s. What is the smallest value of d, such that constructive interference occurs at point C?

Two waves are traveling in opposite directions on the same string. The displacements caused by the individual waves are given by y1 5 (24.0 mm)sin(9.00pt 2 1.25px) and y2 5 (35.0 mm)sin(2.88pt 1 0.400px). Note that the phase angles (9.00pt 2 1.25px) and (2.88pt 1 0.400px) are in radians, t is in seconds, and x is in meters. At t 5 4.00 s, what is the net displacement (in mm) of the string at (a) x 5 2.16 m and (b) x 5 2.56 m? Be sure to include the algebraic sign (1 or 2) with your answers.

Both drawings show the same square, each of which has a side of length L 5 0.75 m. An observer O is stationed at one corner of each square. Two loudspeakers are located at corners of the square, as in either drawing 1 or drawing 2. The speakers produce the same single-frequency tone in either drawing and are in phase. The speed of sound is 343 m/s. Find the single smallest frequency that will produce both constructive interference in drawing 1 and destructive interference in drawing 2.

The drawing shows a loudspeaker A and point C, where a listener is positioned. A second loudspeaker B is located somewhere to the right of loudspeaker A. Both speakers vibrate in phase and are playing a 68.6-Hz tone. The speed of sound is 343 m/s. What is the closest to speaker A that speaker B can be located, so that the listener hears no sound?

Suppose that the two speakers in Figure 17.7 are separated by 2.50 m and are vibrating exactly out of phase at a frequency of 429 Hz. The speed of sound is 343 m/s. Does the observer at C observe constructive or destructive interference when his distance from speaker B is (a)1.15m and (b) 2.00 m?

Two loudspeakers on a concert stage are vibrating in phase. A listener is 50.5 m from the left speaker and 26.0 m from the right one. The listener can respond to all frequencies from 20 to 20 000 Hz, and the speed of sound is 343 m/s. What are the two lowest frequencies that can be heard loudly due to constructive interference?

A listener is standing in front of two speakers that are producing sound of the same frequency and amplitude, except that they are vibrating out of phase. Initially, the distance between the listener and each speaker is the same (see the drawing). As the listener moves sideways, the sound intensity gradually changes. When the distance x in the drawing is 0.92 m, the change reaches the maximum amount (either loud to soft, or soft to loud). Using the data shown in the drawing and 343 m/s for the speed of sound, determine the frequency of the sound coming from the speakers.

Speakers A and B are vibrating in phase. They are directly facing each other, are 7.80 m apart, and are each playing a 73.0-Hz tone. The speed of sound is 343 m/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 A?

Consult Multiple-Concept Example 3 for background pertinent to this problem. A speaker has a diameter of 0.30 m. (a) Assuming that the speed of sound is 343 m/s, find the diffraction angle u for a 2.0-kHz tone. (b) What speaker diameter D should be used to generate a 6.0-kHz tone whose diffraction angle is as wide as that for the 2.0-kHz tone in part (a)?

Sound exits a diff raction 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 K, the diff raction angle u has a value of 15.08. What is the diff raction angle for the same sound on a summer day when the temperature is 311 K?

For one approach to problems such as this, see Multiple-Concept Example 3. Sound emerges through a doorway, as in Figure 17.10. The width of the doorway is 77 cm, and the speed of sound is 343 m/s. Find the diff raction angle u when the frequency of the sound is (a) 5.0 kHz and (b) 5.0 3 102 Hz

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 m wide. Sound of frequency 607 Hz is coming through the entrance from within the room. The speed of sound is 343 m/s. What is the diff raction angle u of the sound after it passes through the doorway when (a) one door is open and (b) both doors are open?

The following two lists give the diameters and sound frequencies for three loudspeakers. Pair each diameter with a frequency, so that the diff raction angle is the same for each of the speakers, and then fi nd the common diff raction angle. Take the speed of sound to be 343 m/s.

A 3.00-kHz tone is being produced by a speaker with a diameter of 0.175 m. The air temperature changes from 0 to 29 8C. Assuming air to be an ideal gas, fi nd the change in the diff raction angle u.

Sound (speed 5 343 m/s) exits a diff raction horn loudspeaker through a rectangular opening like a small doorway. A person is sitting at an angle a off to the side of a diff raction horn that has a width D of 0.060 m. This individual does not hear a sound wave that has a frequency of 8100 Hz. When she is sitting at an angle a/2, the frequency that she does not hear is diff erent. What is this frequency?

Two pure tones are sounded together. The drawing shows the pressure variations of the two sound waves, measured with respect to atmospheric pressure. What is the beat frequency?

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 m/s, piano A produces a wavelength of 0.769 m, while piano B produces a wavelength of 0.776 m. How much time separates successive beats?

A 440.0-Hz tuning fork is sounded together with an outof-tune guitar string, and a beat frequency of 3 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?

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

Two out-of-tune fl utes play the same note. One produces a tone that has a frequency of 262 Hz, while the other produces 266 Hz. When a tuning fork is sounded together with the 262-Hz tone, a beat frequency of 1 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 cars have identical horns, each emitting a frequency of fs 5 395 Hz. One of the cars is moving with a speed of 12.0 m/s toward a bystander waiting at a corner, and the other car is parked. The speed of sound is 343 m/s. What is the beat frequency heard by the bystander?

A sound wave is traveling in seawater, where the adiabatic bulk modulus and density are 2.31 3 109 Pa and 1025 kg/m3 , respectively. The wavelength of the sound is 3.35 m. A tuning fork is struck under water and vibrates at 440.0 Hz. What would be the beat frequency heard by an underwater swimmer?

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

The fundamental frequency of a string fi xed at both ends is 256 Hz. How long does it take for a wave to travel the length of this string?

A string that is fi xed at both ends has a length of 2.50 m. When the string vibrates at a frequency of 85.0 Hz, a standing wave with fi ve loops is formed. (a) What is the wavelength of the waves that travel on the string? (b) What is the speed of the waves? (c) What is the fundamental frequency of the string?

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 (1.56 3 1022 kg/m) is the C string. This string produces a fundamental frequency of 65.4 Hz and has a length of 0.800 m between the two fi xed ends. Find the tension in the string.

Two wires, each of length 1.2 m, are stretched between two fi xed supports. On wire A there is a second-harmonic standing wave whose frequency is 660 Hz. However, the same frequency of 660 Hz is the third harmonic on wire B. Find the speed at which the individual waves travel on each wire.

Suppose that the strings on a violin are stretched with the same tension and each has the same length between its two fi xed ends. The musical notes and corresponding fundamental frequencies of two of these strings are G (196.0 Hz) and E (659.3 Hz). The linear density of the E string is 3.47 3 1024 kg/m. What is the linear density of the 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 5 0.0140 kg/m) sustains a tension of 323 N because the wire is stretched between two poles that are 7.60 m apart. The lowest frequency that an average, healthy human ear can detect is 20.0 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 3 1023 kg/m and is under a tension of 280 N. The string is 1.8 m long, is fi xed 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.

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

A copper block is suspended from a wire, as in part 1 of the drawing. A container of mercury is then raised up around the block, as in part 2, so that 50.0% of the blocks volume is submerged in the mercury. The density of copper is 8890 kg/m3 , and that of mercury is 13 600 kg/m3 . Find the ratio of the fundamental frequency of the wire in part 2 to the fundamental frequency of the wire in part 1.

The drawing shows two strings that have the same length and linear density. The left end of each string is attached to a wall, while the right end passes over a pulley and is connected to objects of different weights (WA and WB). Different standing waves are set up on each string, but their frequencies are the same. If WA 5 44 N, what is WB?

The E string on an electric bass guitar has a length of 0.628 m and, when producing the note E, vibrates at a fundamental frequency of 41.2 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 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?

Standing waves are set up on two strings fi xed at each end, as shown in the drawing. The two strings have the same tension and mass per unit length, but they diff er in length by 0.57 cm. The waves on the shorter string propagate with a speed of 41.8 m/s, and the fundamental frequency of the shorter string is 225 Hz. Determine the beat frequency produced by the two standing waves.

The note that is three octaves above middle C is supposed to have a fundamental frequency of 2093 Hz. On a certain piano the steel wire that produces this note has a cross-sectional area of 7.85 3 1027 m2 . The wire is stretched between two pegs. When the piano is tuned properly to produce the correct frequency at 25.0 8C, the wire is under a tension of 818.0 N. Suppose the temperature drops to 20.0 8C. 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?

Review Conceptual Example 5 before attempting this problem. As the drawing shows, the length of a guitar string is 0.628 m. The frets are numbered for convenience. A performer can play a musical scale on a single string because the spacing between the frets is designed according to the following rule: When the string is pushed against any fret j, the fundamental frequency of the shortened string is larger by a factor of the twelfth root of two ( 121 2 ) than it is when the string is pushed against the fret j 2 1. Assuming that the tension in the string is the same for any note, fi nd the spacing (a) between fret 1 and fret 0 and (b) between fret 7 and fret 6.

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 cm. The speed of sound is 343 m/s. What is the fundamental frequency of the canal? (Interestingly, the fundamental frequency is in the frequency range where human hearing is most sensitive.)

A tube with a cap on one end, but open at the other end, has a fundamental frequency of 130.8 Hz. The speed of sound is 343 m/s (a) If the cap is removed, what is the new fundamental frequency of the tube? (b) How long is the tube?

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

The range of human hearing is roughly from twenty hertz to twenty kilohertz. Based on these limits and a value of 343 m/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 fi nd in a pipe organ?

The fundamental frequencies of two air columns are the same. Column A is open at both ends, while column B is open at only one end. The length of column A is 0.70 m. What is the length of column B?

A piccolo and a fl ute can be approximated as cylindrical tubes with both ends open. The lowest fundamental frequency produced by one kind of piccolo is 587.3 Hz, and that produced by one kind of fl ute is 261.6 Hz. What is the ratio of the piccolos length to the fl utes length?

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

A thin 1.2-m aluminum rod sustains a longitudinal standing wave with vibration antinodes at each end of the rod. There are no other antinodes. The density and Youngs modulus of aluminum are, respectively, 2700 kg/m3 and 6.9 3 1010 N/m2 . What is the frequency of the rods vibration?

A person hums into the top of a well and fi nds that standing waves are established at frequencies of 42, 70.0, and 98 Hz. The frequency of 42 Hz is not necessarily the fundamental frequency. The speed of sound is 343 m/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 3 105 Pa. The tube has a length of 0.75 m. Mercury (mass density 5 13 600 kg/m3 ) is poured into it to shorten the eff ective length for standing waves. What is the absolute pressure at the bottom of the mercury column, when the fundamental frequency of the shortened, air-fi lled tube is equal to the third harmonic of the original tube?

Two ideal gases have the same temperature and the same value for g (the ratio of the specifi c heat capacities at constant pressure and constant volume). A molecule of gas A has a mass of 7.31 3 10226 kg, and a molecule of gas B has a mass of 1.06 3 10225 kg. When gas A (speed of sound 5 259 m/s) fi lls a tube that is open at both ends, the fi rst overtone frequency of the tube is 386 Hz. Gas B fi lls another tube open at both ends, and this tube also has a fi rst overtone frequency of 386 Hz. What is the length of the tube fi lled with gas B?

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 Hz, while the piece open only at one end has a fundamental frequency of 675 Hz. What is the fundamental frequency of the original tube?

A string is fi xed at both ends and is vibrating at 130 Hz, which is its third harmonic frequency. The linear density of the string is 5.6 3 1023 kg/m, and it is under a tension of 3.3 N. Determine the length of the string.

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 fi xed frequency, the plunger is moved until the smallest value of L is measured that allows a standing wave to be formed. Suppose that the tuning fork produces a 485-Hz tone, and that the smallest value observed for L is 0.264 m. What is the speed of sound in the gas in the tube?

Review Example 1 in the text. Speaker A is moved further to the left, while ABC remains a right triangle. What is the separation between the speakers when constructive interference occurs again at point C?

A pipe open only at one end has a fundamental frequency of 256 Hz. A second pipe, initially identical to the fi rst pipe, is shortened by cutting off a portion of the open end. Now, when both pipes vibrate at their fundamental frequencies, a beat frequency of 12 Hz is heard. How many centimeters were cut off the end of the second pipe? The speed of sound is 343 m/s.

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 eff ect of giving the voice a highpitched quality, like that of Donald Ducks voice. To see why this occurs, assume for simplicity that the voice is generated by the vocal cords vibrating above a gas-fi lled 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 8C. One is fi lled with air, in which the speed of sound is 343 m/s. The other is fi lled with helium, in which the speed of sound is 1.00 3 103 m/s. To see the eff ect of helium on voice quality, calculate the ratio of the nth natural frequency of the helium-fi lled tube to the nth natural frequency of the air-fi lled tube.

The drawing graphs a string on which two pulses (half up and half down) are traveling at a constant speed of 1 cm/s at t 5 0 s. Using the principle of linear superposition, draw the shape of the string at t 5 1 s, 2 s, 3 s, and 4 s.

The A string on a string bass vibrates at a fundamental frequency of 55.0 Hz. If the strings tension were increased by a factor of four, what would be the new fundamental frequency?

Two strings have diff erent lengths and linear densities, as the drawing shows. They are joined together and stretched so that the tension in each string is 190.0 N. The free ends of the joined string are fi xed in place. Find the lowest frequency that permits standing waves in both strings with a node at the junction. The standing wave pattern in each string may have a diff erent number of loops.

The two speakers in the drawing are vibrating in phase, and a listener is standing at point P. Does constructive or destructive interference occur at P when the speakers produce sound waves whose frequency is (a) 1466 Hz and (b) 977 Hz? Justify your answers with appropriate calculations. Take the speed of sound to be 343 m/s.

Two carpenters are hammering at the same time, each at a diff erent hammering frequency. The hammering frequency is the number of hammer blows per second. Every 4.6 s, both carpenters strike at the same instant, producing an eff ect very similar to a beat frequency. The fi rst carpenter strikes a blow every 0.75 s. How many seconds elapse between the second carpenters blows if the second carpenter hammers (a) more rapidly than the fi rst carpenter, and (b) less rapidly than the fi rst carpenter?

The arrangement in the drawing shows a block (mass 5 15.0 kg) that is held in position on a frictionless incline by a cord (length 5 0.600 m). The mass per unit length of the cord is 1.20 3 1022 kg/m, so the mass of the cord is negligible compared to the mass of the block. The cord is being vibrated at a frequency of 165 Hz (vibration source not shown in the drawing). What are the values of the angle u between 15.08 and 90.08 at which a standing wave exists on the cord?

A fl autist is playing a fl ute as discussed in Example 6, but now the temperature is 305 K instead of 293 K. As a result, the speed of sound is no longer 343 m/s. Therefore, with the length calculated in Example 6, the note middle C does not have the proper fundamental frequency of 261.6 Hz. In other words, the fl ute is out of tune. To adjust the tuning, the fl autist can alter the fl utes length by changing the extent to which the head joint (see Figure 17.21) is inserted into the main stem of the instrument. To what length must the fl ute be adjusted to play middle C at its proper frequency?

A sound wave with a frequency of 15 kHz emerges through a circular opening that has a diameter of 0.20 m. Concepts: (i) The diff raction angle for a wave emerging through a circular opening is given by sin u 5 1.22 l/D, where l is the wavelength of the sound and D is the diameter of the opening. What is meant by the diff raction angle? (ii) How is the wavelength related to the frequency of the sound? (iii) Is the wavelength of the sound in air greater than, smaller than, or equal to the wavelength in water? Why? (Note: The speed of sound in air is 343 m/s and the speed of sound in water is 1482 m/s.) (iv) Is the diff raction angle of the sound in air greater than, * smaller than, or equal to the diff raction angle in water? Explain. Calculations: Find the diff raction angle u when the sound travels (a) in air and (b) in water

Two tubes of gas are identical and are open only at one end. One tube contains neon (Ne) and the other krypton (Kr). Both are monatomic gases, have the same temperature, and may be assumed to be ideal gases. The fundamental frequency of the tube containing neon is 481 Hz. Concepts: (i) For a gas-fi lled tube open only at one end, the fundamental frequency (n 5 1) is f1 5 v/(4L), where v is the speed of sound and L is the length of the tube. How is the speed related to the properties of the gas? (ii) All of the factors that aff ect the speed of sound in this problem are the same except for the atomic masses, which are given by 20.180 u for neon, and 83.80 u for krypton. Is the speed of sound in krypton greater than, smaller than, or equal to the speed of sound in neon? Why? (iii) Is the fundamental frequency of the tube containing krypton greater than, less than, or equal to the fundamental frequency of the tube containing neon? Explain. Calculations: What is the fundamental frequency of the tube containing krypton?