Waves/Acoustics

Adjacent antinodes of a standing wave on a string are \(15\;{\rm{cm}}\) apart. A particle at an antinode oscillates in simple harmonic motion with amplitude \(0.850\;{\rm{cm}}\) and period \(0.0750\;{\rm{s}}\). The string lies along the \( + x\)-axis and is fixed at \(x = 0\). 

(a) How far apart are the adjacent nodes? 

(b) What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern? 

(c) Find the maximum and minimum transverse speeds of a point at an antinode. (d) What is the shortest distance along the string between a node and an antinode?

Standing waves on a wire are described by Eq. (15.28), with \({A_{sw}} = 2.5\;{\rm{mm}}\), \(\omega  = 942\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}\), and \(k = 0.750\pi \;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{m}}}} \right. \\} {\rm{m}}}\). The left end of the wire is at \(x = 0\). At what distances from the left end are (a) the nodes of the standing wave and (b) the antinodes of the standing wave

 \(1.50\;{\rm{m}}\)-long rope is stretched between two supports with a tension that makes the speed of transverse waves \(62.0\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. \\} {\rm{s}}}\). What are the wavelength and frequency of 

(a) the fundamental;

 (b) the second overtone;

 (c) the fourth harmonic?

A wire with mass \(40.0\;{\rm{g}}\) is stretched so that its ends are tied down at points \(80.0\;{\rm{cm}}\) apart. The wire vibrates in its fundamental mode with frequency \(60.0\;{\rm{Hz}}\) and with an amplitude at the antinodes of \(0.300\;{\rm{cm}}\). 

(a) What is the speed of propagation of transverse waves in the wire? 

(b) Compute the tension in the wire. 

(c) Find the maximum transverse velocity and acceleration of particles in the wire.

A piano tuner stretches a steel piano wire with a tension of \(800\;{\rm{N}}\). The steel wire is \(0.400\;{\rm{m}}\) long and has a mass of \(3.00\;{\rm{g}}\). 

(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 \(10,000\;{\rm{Hz}}\)?

A thin, taut string tied at both ends and oscillating in its third harmonic has its shape described by the equation \(y\left( {x,t} \right) = \left( {5.6\;{\rm{cm}}} \right)\sin \left[ {\left( {0.0340\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {{\rm{cm}}}}} \right. \\} {{\rm{cm}}}}} \right)x} \right]\sin \left[ {\left( {50.0\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\), where the origin is at the left end of the string, the x-axis is along the string, and the y-axis is perpendicular to the string.

 (a) Draw a sketch that shows the standing-wave pattern.

 (b) Find the amplitude of the two traveling waves that make up this standing wave.

 (c) What is the length of the string?

 (d) Find the wavelength, frequency, period, and speed of the traveling waves.

 (e) Find the maximum transverse speed of a point on the string.

 (f) What would be the equation \(y\left( {x,t} \right)\)for this string if it were vibrating in its eighth harmonic?

Waves on a Stick. A flexible stick \(2.0\;{\rm{m}}\) long is not fixed in any way and is free to vibrate. Make clear drawings of this stick vibrating in its first three harmonics, and then use your drawings to find the wavelengths of each of these harmonics. (Hint: Should the ends be nodes or antinodes?)

One string of a certain musical instrument is \(75.0\;{\rm{cm}}\) long and has a mass of \(8.75\;{\rm{g}}\). It is being played in a room where the speed of sound is \(344\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. \\} {\rm{s}}}\). (a) To what tension must you adjust the string so that, when vibrating in its second overtone, it produces sound of wavelength \(0.765\;{\rm{m}}\)? (Assume that the breaking stress of the wire is very large and isn’t exceeded.) 

(b) What frequency sound does this string produce in its fundamental mode of vibration?

The portion of the string of a certain musical instrument between the bridge and upper end of the finger board (that part of the string that is free to vibrate) is \(60.0\;{\rm{cm}}\) long, and this length of the string has mass \(2.00\;{\rm{g}}\). The string sounds an \({A_4}\) note (\(440\;{\rm{Hz}}\)) when played. 

(a) Where must the player put a finger (what distance x from the bridge) to play a \({D_5}\) note (\(587\;{\rm{Hz}}\))? (See Fig. E15.45.) For both the \({A_4}\) and \({D_5}\) notes, the string vibrates in its fundamental mode. 

(b) Without retuning, is it possible to play a \({G_4}\) note (\(392\;{\rm{Hz}}\)) on this string? Why or why not?

(a) A horizontal string tied at both ends is vibrating in its fundamental mode. The traveling waves have speed \(v\), frequency \(f\), amplitude \(A\), and wavelength \(\lambda \). Calculate the maximum transverse velocity and maximum transverse acceleration of points located at (i) \(x = \frac{\lambda }{2}\), (ii) \(x = \frac{\lambda }{4}\), and (iii) \(x = \frac{\lambda }{8}\), from the left-hand end of the string. 

(b) At each of the points in part (a), what is the amplitude of the motion? 

(c) At each of the points in part (a), how much time does it take the string to go from its largest upward displacement to its largest downward displacement?

Guitar String. One of the \(63.5\;{\rm{cm}}\)-long strings of an ordinary guitar is tuned to produce the note \({B_3}\) (frequency \(245\;{\rm{Hz}}\)) 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 \(344\;{{\rm{m}} \mathord{\left/ {\vphantom {{\rm{m}} {\rm{s}}}} \right. \\} {\rm{s}}}\), find the frequency and wavelength of the sound wave produced in the air by the vibration of the \({B_3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

A transverse wave on a rope is given by

\(y\left( {x,t} \right) = \left( {0.750\;{\rm{cm}}} \right)\cos \pi \left[ {\left( {0.400\;{\rm{c}}{{\rm{m}}^{ - 1}}} \right)x + \left( {250\;{{\rm{s}}^{ - 1}}} \right)t} \right]\)

(a) Find the amplitude, period, frequency, wavelength, and speed of propagation. 

(b) Sketch the shape of the rope at these values of \(t\): \(0,0.0005\;s,0.0010\;s\).

(c) Is the wave travelling in the \( + x\;{\rm{or}} - x\)-direction?

(d) The mass per unit length of the rope is \(0.0500\;{\rm{kg/m}}\). Find the tension.

(e) Find the average power of this wave.

A transverse sine wave with an amplitude of 2.50 mm and a wavelength of 1.80 m travels from left to right along a long, horizontal, stretched string with a speed of 36.0 m/s. Take the origin at the left end of the string has its maximum upward displacement.

(a) What are the frequency, angular frequency, and wave number of the wave? 

(b) What is the function \(y\left( {x,t} \right)\) that describes the wave?

(c) What is \(y\left( t \right)\)for a particle at the left end of the string?

(d) What is \(y\left( t \right)\) for a particular 1.35 m to right of the origin? 

(e) What is the maximum magnitude of transverse velocity of any particle of the string?

(f) Find the transverse displacement and the transverse velocity of a particle 1.35 m to the right of the origin at time \(t = 0.0625\;s\).

A 1750-N irregular beam is hanging horizontally by its ends from the ceiling by two vertical wires (A and B), each 1.25 m long and weighing 0.290 N. The center of gravity of this beam is one-third of the way along the beam from the end where wire A is attached. If you pluck both strings at the same time at  the beam, what is the time delay between the arrival of the two pulses at the ceiling? Which pulse arrives first?(Ignore the effect of the weight of the wires on the tension in the wires).

Three pieces of string, each of length \(L\), are joined together end to end, to make a combined string of length \(3L\). The has mass per unit length \({\mu _2} = 4{\mu _1}\), and the third piece has mass per unit length \({\mu _3} = \frac{{{\mu _1}}}{4}\). (a) If the combined string is under tension \(F\), how much time does it take a transverse wave to travel the entire length \(3L\)? Give your answer in terms of \(L,F,\;{\rm{and}}\;{\mu _1}\).

(b) Does your answer to part (a) depend on the order in which the three pieces are joined together? Explain.

An ant with mass \(m\) is standing peacefully on top of a horizontal, stretched rope. The rope has mass per unit length \(\mu \) and is under tension \(F\).Without warning, Cousin Throckmorton starts a sinusoidal transverse wave of wave-length \(\lambda \) propagating along the rope. The motion of the rope is in a vertical plane. What minimum wave amplitude will make the ant become momentarily weightless? Assume that \(m\) is so small that the presence of the ant has no effect on the propagation of the wave.

A 5.00 m, 0.732 kg wire is used to support two uniform 235 N posts of equal length (Fig. P15.55), Assume that the wire is essentially horizontal and that the speed of sound is 344 m/s. A strong wind is blowing, causing the wire to vibrate in its 5^th overtone. What are the frequency and wavelength of the sound this wire produces?

You are exploring a newly discovered planet. The radius of the planet is \(7.20 \times {10^7}\;{\rm{m}}\). You suspend a lead weight from the lower end of a light string that is 4.00 m long and has mass 0.0280 kg. You measure that it takes 0.0685 s for a transverse pulse to travel the length of the string. The weight of the string is small enough that you ignore its effect on the tension in the string. Assuming that the mass of the planet is distributed with spherical symmetry, what is its mass?

A vertical, \(1.20\;{\rm{m}}\) length of 18 gauge (diameter of 1.024 mm) copper wire has a 100.0 N ball hanging from it.

(a) What is the wavelength of the third harmonic for this wire?

(b) A 500.0 N ball now replaces the original ball. What is the change in the wavelength of the third harmonic caused by replacing the light ball with the heavy one? (Hint: Se Table 11.1 for Young’s modulus.)

A sinusoidal transverse wave travel on a string. The string has length \(8.00\;{\rm{m}}\) and mass \(6.00\;{\rm{g}}\). The wave speed is \(30.0\;{\rm{m/s}}\) and the wavelength is \(0.200\;{\rm{m}}\).

(a) If the wave is have an average power of \(50.0\;{\rm{W}}\), what must be the amplitude of the wave?

(b) For this same string, if the amplitude and wavelength are the same as in part (a), what is the average power for the wave if the tension is increased such that the wave speed is doubled?

A strong string of mass 3.00 g and length 2.20 m is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is \(9.00\;{\rm{m/s}}\). The tension in the string is 330 N.

(a) What is the amplitude of the standing wave at its antinode?

(b) What is the magnitude of the maximum transverse acceleration of a point at the antinode?

A thin string 2.50 m in length is stretched with a tension of 90.0 N between two supports. When the string vibrates in its first overtone, a point at an antinode of the standing wave on the string has an amplitude of 3.50 cm and a maximum transverse speed of \(28.0\;{\rm{m/s}}\).

(a) What is the string’s mass?

(b) What is the magnitude of the maximum transverse acceleration of this point on the string?

A guitar string is vibrating in its fundamental mode, with nodes at each end. The length of the segment of the string that is free to vibrate is 0.386 m. The maximum transverse acceleration of a point at the middle of the segment is \(8.40 \times {10^3}\;{\rm{m/}}{{\rm{s}}^{\rm{2}}}\) and the maximum transverse velocity is \(3.80\;{\rm{m/s}}\).

(a) What is the amplitude of this standing wave?

(b) What is the wave speed for the transverse travelling waves on this string?

The wave function of a standing wave is \(y\left( {x,t} \right) = 4.44\;{\rm{mm}}\sin \left[ {\left( {32.5\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{m}}}} \right. \\} {\rm{m}}}} \right)x} \right]\sin \left[ {\left( {754\;{{{\rm{rad}}} \mathord{\left/ {\vphantom {{{\rm{rad}}} {\rm{s}}}} \right. \\} {\rm{s}}}} \right)t} \right]\). For the two traveling waves that make up this standing wave, find the (a) amplitude;

(b) wavelength; 

(c) frequency; 

(d) wave speed;

(e) wave functions. 

(f) From the information given, can you determine which harmonic this is? Explain.

When sound travels from air into water, does the frequency of the wave change? The speed? The wavelength? Explain your reasoning.

The hero of a western movie listens for an oncoming train by putting his ear to the track. Why does this method give an earlier warning of the approach of a train than just listening in the usual way?

Would you expect the pitch (or frequency) of an organ pipe to increase or decrease with increasing temperature? Explain.

In most modern wind instruments the pitch is changed by 

using keys or valves to change the length of the vibrating air column.

The bugle, however, has no valves or keys, yet it can play many notes. How might this be possible? Are there restrictions on what notes a bugle can play?

Symphonic musicians always “warm up” their wind instruments by blowing into them before a performance. What purpose

does this serve?

In a popular and amusing science demonstration, a person inhales helium and then his voice becomes high and squeaky. Why does this happen? (Warning: Inhaling too much helium can cause

Lane dividers on highways sometimes have regularly

spaced ridges or ripples. When the tires of a moving car roll along

such a divider, a musical note is produced. Why? Explain how this

phenomenon could be used to measure the car’s speed.

Question: (a) Does a sound level of 0 dB mean that there is no

sound? (b) Is there any physical meaning to a sound having a negative

intensity level? If so, what is it? (c) Does a sound intensity of

zero mean that there is no sound? (d) Is there any physical meaning

to a sound having a negative intensity? Why?

Does an aircraft make a sonic boom only at the instant its speed exceeds Mach 1? Explain.

If you are riding in a supersonic aircraft, what do you hear? Explain. In particular, do you hear a continuous sonic boom? Why or why not?

Question: Example 16.1 (Section 16.1) showed that for sound waves in air with frequency 1000 Hz, a displacement amplitude of \(1.2 \times {10^{ - 8}}\;{\rm{m}}\) produces a pressure amplitude of \(3.0 \times {10^{ - 2}}\;{\rm{Pa}}\). ( a) What is the wavelength of these waves? (b) For 1000-Hz waves in air, what displacement amplitude would be needed for the pressure amplitude to be at the pain threshold, which is 30 Pa? (c) For what wavelength and frequency will waves with a displacement amplitude of  \(1.2 \times {10^{ - 8}}\;{\rm{m}}\)produce a pressure amplitude of \(1.5 \times {10^{ - 3}}\;{\rm{Pa}}\)?

For sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 x 108 m produces a pressure amplitude of 3.0 × 10-² Pa. Water at 20°C has a bulk modulus of 2.2 x 10⁹ Pa, and the speed of sound in water at this temperature is 1480 m/s. For 1000-Hz sound waves in 20°C water, what displacement amplitude is produced if the pressure amplitude is 3.0 x 102 Pa? Explain why your answer is much less than 1.2 x 10-8 m.

Consider a sound wave in air that has displacement amplitude 0.0200 mm. Calculate the pressure amplitude for frequencies of (a) 150 Hz; (b) 1500 Hz; (c) 15,000 Hz. In each case compare the result to the pain threshold, which is 30 Pa.

A loud factory machine produces sound having a displacement amplitude of 1.00 μ-m, but the frequency of this sound can be adjusted. In order to prevent ear damage to the workers, the maxi mum pressure amplitude of the sound waves is limited to 10.0 Pa. Under the conditions of this factory, the bulk modulus of air is 1.42 x 105 Pa. What is the highest-frequency sound to which this machine can be adjusted without exceeding the prescribed limit? Is this frequency audible to the workers?

Question: BIO Ultrasound and Infrasound. (a) Whale communication. Blue whales apparently communicate with each other using sound of frequency 17 Hz, which can be heard nearly 1000 km away in the ocean. What is the wavelength of such a sound in seawater, where the speed of sound is 1531 m/s? (b) Dolphin clicks. One type of sound that dolphins emit is a sharp click of wavelength 1.5 cm in the ocean. What is the frequency of such clicks? (c) Dog whistles. One brand of dog whistles claims a frequency of 25 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 kHz and 78 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 mm across if the speed of sound in the tissue is 1550 m/s?

A large church has part of the organ in the front of the church and part in the back. A person walking rapidly down the aisle while both segments are playing at once reports that the two segments sound out of tune. Why?

Can you think of circumstances in which a Doppler effect would be observed for surface waves in water? For elastic waves propagating in a body of water deep below the surface? If so, describe the circumstances and explain your reasoning. If not, explain why not

Stars other than our sun normally appear featureless when viewed through telescopes. Yet astronomers can readily use the light from these stars to determine that they are rotating and even measure the speed of their surface. How do you think they can do this?

If you wait at a railroad crossing as a train approach and passes, you hear a Doppler shift in its sound. But if you listen closely, you hear that the change in frequency is continuous; it does not suddenly go from one high frequency to another low frequency. Instead the frequency smoothly (but rather quickly) changes from high to low as the train passes. Why does this smooth change occur?

In case 1, a source of sound approaches a stationary observer at speed u. In case 2, the observer moves toward the stationary source at the same speed u. If the source is always producing the same frequency sound, will the observer hear the same frequency in both cases, since the relative speed is the same each time? Why or why not?

For sound waves in air with frequency 1000 Hz, a displacement amplitude of 1.2 x 108 m produces a pressure amplitude of 3.0 × 10-² Pa. Water at 20°C has a bulk modulus of 2.2 x 10⁹ Pa, and the speed of sound in water at this temperature is 1480 m/s. For 1000-Hz sound waves in 20°C water, what displacement amplitude is produced if the pressure amplitude is 3.0 x 102 Pa? Explain why your answer is much less than 1.2 x 10-8 m.

25 A jet airplane is flying at a constant altitude at a steady speed v_s greater than the speed of sound. Describe what observers at points A, B, and C hear at the instant shown in the figure, the shock wave has just reached point B. Explain.

Question: (a) In a liquid with density \({\bf{1300}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\), longitudinal waves with frequency 400 Hz are found to have wavelength 8.00 m. Calculate the bulk modulus of the liquid. (b) A metal bar with a length of 1.50 m has density  \({\bf{6400}}\;{\bf{kg/}}{{\bf{m}}^{\bf{3}}}\). Longitudinal sound waves take \({\bf{3}}{\bf{.90}} \times {\bf{1}}{{\bf{0}}^{ - {\bf{4}}}}\;{\bf{s}}\) to travel from one end of the bar to the other. What is Young’s modulus for this metal?

Question: A submerged scuba diver hears the sound of a boat horn directly above her on the surface of the lake. At the same time, a friend on dry land 22.0 m from the boat also hears the horn (Fig. E16.7). The horn is 1.2 m above the surface of the water. What is the distance (labeled “?”) from the horn to the diver? Both air and water are at 20° C.

Question: At a temperature of 27.0° C, what is the speed of longitudinal waves in (a) hydrogen (molar mass 2.02 g/mol); (b) helium (molar mass 4.00 g/mol); (c) argon (molar mass 39.9 g/mol)? See Table 19.1 for values of \(\gamma \). (d) Compare your answers for parts (a), (b), and (c) with the speed in air at the same temperature.

Question: An oscillator vibrating at 1250 Hz produces a sound wave that travels through an ideal gas at 325 m/s when the gas temperature is 22.0° C. For a certain experiment, you need to have the same oscillator produce sound of wavelength 28.5 cm in this gas. What should the gas temperature be to achieve this wavelength?