Waves/Acoustics

A sinusoidal wave is propagating along a stretched string that lies along the x-axis. The displacement of the string as a function of time is graphed in Fig. E15.11 for particles at \({\bf{x}}{\rm{ }} = {\rm{ }}{\bf{0}}\) and at\(x  =   0.0900 m\). (a) What is the amplitude of the wave? (b) What is the period of the wave? (c) You are told that the two points \({\bf{x}}{\rm{ }} = {\rm{ }}{\bf{0}}\) and \(x  =  0.0900 m\) are within one wavelength of each other. If the wave is moving in the +x direction, determine the wavelength and the wave speed. (d) If instead the wave is moving in the –x direction, determine the wavelength and the wave speed. (e) Would it be possible to determine definitively the wavelengths in parts (c) and (d) if you were not told that the two points were within one wavelength of each other? Why or why not?

A wave on a string is described by\(y\left( {x,t} \right) = Acos\left( {kx - \omega t} \right)\). (a) Graph \(y,\,{v_y}\,and\,{a_y}\)as functions of \(x\) for time\(t  =  0\). (b) Consider the following points on the string:\(\left( i \right) x  =   0;\left( {ii} \right) x  = {\pi  \mathord{\left/

 {\vphantom {\pi  {4K}}} \right.

\\} {4K}}; \left( {iii} \right)  x  = {\pi   \mathord{\left/

 {\vphantom {\pi  {2K}}} \right.

\\} {2K}}; \left( {iv} \right) x  = 3{\pi  \mathord{\left/

 {\vphantom {\pi  {4K}}} \right.

\\} {4K}}; \left( v \right) x  = {\pi  \mathord{\left/

 {\vphantom {\pi  K}} \right.

\\} K};\)\(\left( {vi} \right) x  = {{5\pi } \mathord{\left/

 {\vphantom {{5\pi } {4k}}} \right.

 \nulldelimiterspace} {4k}}; \left( {vii} \right)x  = {{3\pi } \mathord{\left/

 {\vphantom {{3\pi } {2k}}} \right.

\\} {2k}}; \left( {viii} \right)x  = {{7\pi } \mathord{\left/

 {\vphantom {{7\pi } {4k}}} \right.

\\} {4k}}\). For a particle at each of these points at\(t  =  0\), describe in words whether the particle is moving and in what direction, and whether the particle is speeding up, slowing down, or instantaneously not accelerating.

Two waves travel on the same string. Is it possible for them to have (a) different frequencies; (b) different wavelengths; (c) different speeds; (d) different amplitudes; (e) the same frequency but different wavelengths? Explain your reasoning.

Under a tension F, it takes 2.00 s for a pulse to travel the length of a taut wire. What tension is required (in terms of F) for the pulse to take 6.00 s instead? Explain how you arrive at your answer.

What kinds of energy are associated with waves on a stretched string? How could you detect such energy experimentally?

The amplitude of a wave decreases gradually as the wave travels down a long, stretched string. What happens to the energy of the wave when this happens?

For the wave motions discussed in this chapter, does the speed of propagation depend on the amplitude? What makes you say this?

The speed of ocean waves depends on the depth of the water; the deeper the water, the faster the wave travels. Use this to explain why ocean waves crest and “break” as they near the shore.

Is it possible to have a longitudinal wave on a stretched string? Why or why not? Is it possible to have a transverse wave on a steel rod? Again, why or why not? If your answer is yes in either case, explain how you would create such a wave.

For transverse waves on a string, is the wave speed the same as the speed of any part of the string? Explain the difference between these two speeds. Which one is constant?

The four strings on a violin have different thicknesses but are all under approximately the same tension. Do waves travel faster on the thick strings or the thin strings? Why? How does the fundamental vibration frequency compare for the thick versus the thin strings?

A sinusoidal wave can be described by a cosine function, which is negative just as often as positive. So why isn’t the average power delivered by this wave zero?

Two strings of different mass per unit length ${\mathbf{\mu }}_{\mathbf{1}}$ and ${\mathbf{\mu }}_{\mathbf{2}}$ are tied together and stretched with a tension  F. A wave travels along the string and passes the discontinuity in ${\mathbf{\mu }}_{}$ . Which of the following wave properties will be the same on both sides of the discontinuity, and which will change: speed of the wave; frequency; wavelength? Explain the physical reasoning behind each answer.

A long rope with mass m is suspended from the ceiling and hangs vertically. A wave pulse is produced at the lower end of the rope, and the pulse travels up the rope. Does the speed of the wave pulse change as it moves up the rope, and if so, does it increase or decrease? Explain.

In a transverse wave on a string, the motion of the string is perpendicular to the length of the string. How, then, is it possible for energy to move along the length of the string?

BIO Audible Sound. Provided the amplitude is sufficiently great, the human ear can respond to longitudinal waves over a range of frequencies from about 20.0 Hz to about 20.0 kHz  . (a) If you were to mark the beginning of each complete wave pattern with a red dot for the long-wavelength sound and a blue dot for the short-wavelength sound, how far apart would the red dots be, and how far apart would the blue dots be? (b) In reality would adjacent dots in each set be far enough apart for you to easily measure their separation with a meter stick? (c) Suppose you repeated part (a) in water, where sound travels at 1480 m/s . How far apart would the dots be in each set? Could you readily measure their separation with a meter stick?

Tsunami! On December 26, 2004 , a great earthquake occurred off the coast of Sumatra and triggered immense waves (tsunami) that killed some 200,000 people. Satellites observing these waves from space measured 800 km from one wave crest to the next and a period between waves of 1.0 hour . What was the speed of these waves in m/s and in  km/hr? Does your answer help you understand why the waves caused such devastation?

15.4. BIO Ultrasound Imaging. Sound having frequencies above the range of human hearing (about 20,000 Hz) is called ultrasound. Waves above this frequency can be used to penetrate the body and to produce images by reflecting from surfaces. In a typical ultrasound scan, the waves travel through body tissue with a speed of 1500 m/s . For a good, detailed image, the wavelength should be no more than 1.0 mm. What frequency sound is required for a good scan?

(a) Audible wavelengths. The range of audible frequencies is from about\(20 Hz to 20,000 Hz\) . What is the range of the wavelengths of audible sound in air? (b) Visible light. The range of visible light extends from\(380 nm to 750 nm\) . What is the range of visible frequencies of light? (c) Brain surgery. Surgeons can remove brain tumors by using a cavitron ultrasonic surgical aspirator, which produces sound waves of frequency\(23 kHz\). What is the wavelength of these waves in air? (d) Sound in the body. What would be the wavelength of the sound in part (c) in bodily fluids in which the speed of sound is \(1480 {m \mathord{\left/

 {\vphantom {m s}} \right.

\\s}\) but the frequency is unchanged?

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\(0.53 m\). The fisherman sees that the wave crests are spaced \(4.8 m\)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 \(0.30 m\)but the other data remained the same, how would the answers to parts (a) and (b) change?

Transverse waves on a string have wave speed  8 m/s, amplitude 0.07 m, and wavelength 0.32 m . The waves travel in the  -x-direction, and at t=0  the x=0 end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write a wave function describing the wave. (c) Find the transverse displacement of a particle at x=36 m and time  t=0.15 s. (d) How much time must elapse from the instant in part (c) until the particle at x=0.36 m  next has maximum upward displacement?

A certain transverse wave is described by

\(y\left( {x,t} \right) = \left( {6.50\,mm} \right)cos2\pi \left( {\frac{x}{{28.0\,{\kern 1pt} cm}} - \frac{t}{{0.0360\,s}}} \right)\) 

Determine the wave’s (a) amplitude; (b) wavelength; (c) frequency; (d) speed of propagation; (e) direction of propagation.

Which of the following wave functions satisfies the wave equation, Eq. (15.12)? (a)\(y\left( {x,t} \right) = Acos\left( {kx + \omega t} \right)\) ; (b)\(y\left( {x,t} \right) = Asin\left( {kx + \omega t} \right)\) ; (c) \(y\left( {x,t} \right) = A\left( {coskx + cos\omega t} \right)\) (d) For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point x.

Energy can be transferred along a string by wave motion. However, in a standing wave on a string, no energy can ever be transferred past a node. Why not?

Can a standing wave be produced on a string by superposing two waves traveling in opposite directions with the same frequency but different amplitudes? Why or why not? Can a standing wave be produced by superposing two waves traveling in opposite directions with different frequencies but the same amplitude? Why or why not?

If you stretch a rubber band and pluck it, you hear a (somewhat) musical tone. How does the frequency of this tone change as you stretch the rubber band further? (Try it!) Does this agree with Eq. (15.35) for a string fixed at both ends? Explain.

A musical interval of an octave corresponds to a factor of 2 in frequency. By what factor must the tension in a guitar or violin string be increased to raise its pitch one octave? To raise it two octaves? Explain your reasoning. Is there any danger in attempting these changes in pitch?

By touching a string lightly at its center while bowing, a violinist can produce a note exactly one octave above the note to which the string is tuned—that is, a note with exactly twice the frequency. Why is this possible?

As we discussed in Section 15.1, water waves are a combination of longitudinal and transverse waves. Defend the following statement: “When water waves hit a vertical wall, the wall is a node of the longitudinal displacement but an antinode of the transverse displacement.”

Violins are short instruments, while cellos and basses are long. In terms of the frequency of the waves they produce, explain why this is so.

What is the purpose of the frets on a guitar? In terms of the frequency of the vibration of the strings, explain their use.

Speed of Propagation vs. Particle Speed. (a) Show that Eq. (15.3) may be written as 

                    \(y\left( {x,t} \right) = Acos\left[ {\frac{{2\pi }}{\lambda }\left( {x - vt} \right)} \right]\) 

(b) Use \(y\left( {x,t} \right)\) to find an expression for the transverse velocity \({v_y}\)of a particle in the string on which the wave travels. (c) Find the maximum speed of a particle of the string. Under what circumstances is this equal to the propagation speed \(v\) ? Less than\(v\)? Greater than\(v\)?

A transverse wave on a string has amplitude 0.300 cm, wavelength   12.0 cm, and speed 6.00 cm/s. It is represented by y(x,t) as given in Exercise 15.12. 

(a) At time t = 0, compute y at 1.5-cm intervals of x (that is, at x = 0, x = 1.5 cm, x = 3.0 cm, and so on) from x = 0 to x = 12.0 cm. Graph the results. This is the shape of the string at time t = 0. 

(b) Repeat the calculations for the same values of x at times t = 0.400 s and t = 0.800 s. Graph the shape of the string at these instants. In what direction is the wave traveling?

One end of a horizontal rope is attached to a prong of another end passes over a pulley and supports a 1.50-kg mass. The linear mass density of the rope is 0.0480 kg/m. (a) What is the speed of a transverse wave on the rope? (b) What is the wavelength? (c) How would you answer to parts (a) and (b) change if the mass were increased to 3.00 kg?

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

The upper end of a 3.80mlong steel wire is fastened to the ceiling, and a 54.0kg object is suspended from the lower end of the wire. You observe that it takes a transverse pulse 0.0492s to travel from the bottom to the top of the wire. What is the mass of the wire?

18 A 1.50m string of weight 0.0125N is tied to the ceiling at its upper end, and the lower end supports a weight W. Ignore the very small variation in tension along the length of the string that is produced by the weight of the string. When you pluck the string slightly, the waves traveling up the string obey the equation 
 
 y (x, t) = (8.50 mm) cos (172 rad/mx 4830 rad/s t) 
 
Assume that the tension of the string is constant and equal to W. (a) How much time does it take a pulse to travel the full length of the string? (b) What is the weight W? (c) How many wavelengths are on the string at any instant of time? (d) What is the equation for waves traveling down the string?

A thin, 75.0cm wire has a mass of 16.5g. One end is tied to a nail, and the other end is attached to a screw that can be adjusted to vary the tension in the wire. (a) To what tension (in newtons) must you adjust the screw so that a transverse wave of wavelength 3.33 cm makes 625 vibrations per second? (b) How fast would this wave travel?

A heavy rope 6.00 m long and weighing 29.4 N is attached at one end to a ceiling and hangs vertically. A 0.500-kg mass is suspended from the lower end of the rope. What is the speed of transverse waves on the rope at the (a) bottom of the rope, (b) middle of the rope, and (c) top of the rope? (d) Is the tension in the middle of the rope the average of the tensions at the top and bottom of the rope? Is the wave speed at the middle of the rope the average of the wave speeds at the top and bottom? Explain.

A simple harmonic oscillator at the point x = 0 gener ates a wave on a rope. The oscillator operates at a frequency of 40.0 Hz and with an amplitude of 3.00 cm. The rope has a linear mass density of 50.0 g/m and is stretched with a tension of 5.00 N. (a) Determine the speed of the wave. (b) Find the wavelength. (c) Write the wave function y(x, t) for the wave. Assume that the oscillator has its maximum upward displacement at time t = 0. (d) Find the maximum transverse acceleration of points on the rope. (e) In the discussion of transverse waves in this chapter, the force of gravity was ignored. Is that a reasonable approximation for this wave? Explain.

A piano wire with mass 3.00 g and length 80.0 cm is stretched with a tension of 25.0 N. A wave with frequency 120.0 Hz and amplitude 1.6 mm travels along the wire. (a) Calculate the average power carried by the wave. (b) What happens to the average power if the wave amplitude is halved?

A horizontal wire is stretched with a tension of 94.0 N, and the speed of transverse waves for the wire is 406 m/s. What must the amplitude of a traveling wave of frequency 69.0 Hz be for the average power carried by the wave to be 0.365 W?

A light wire is tightly stretched with tension F. Transverse traveling waves of amplitude A and wavelength A₁ carry average power P_avg = 0.400 W. If the wavelength of the waves is doubled, so A₂ = 2A₁, while the tension F and amplitude A are not altered, what then is the average power Pav,2 carried by the waves?

Threshold of Pain. You are investigating the report of a UFO landing in an isolated portion of New Mexico, and you encounter a strange object that is radiating sound waves uniformly in all directions. Assume that the sound comes from a point source and that you can ignore reflections. You are slowly walking toward the source. When you are \({\bf{7}}.{\bf{5}}{\rm{ }}{\bf{m}}\)from it, you measure its intensity to be\(0.11 {W \mathord{\left/

 {\vphantom {W {{m^2}}}} \right.

 \kern-\nulldelimiterspace} {{m^2}}}\). An intensity of \(1.0 {W \mathord{\left/

 {\vphantom {W {{m^2}}}} \right.

 \kern-\nulldelimiterspace} {{m^2}}}\) is often used as the “threshold of pain.” How much closer to the source can you move before the sound intensity reaches this threshold?

Energy Output. By measurement you determine that sound waves are spreading out equally in all directions from a point source and that the intensity is \(0.026 {W \mathord{\left/

 {\vphantom {W {{m^2}}}} \right.

 \kern-\nulldelimiterspace} {{m^2}}}\) at a distance of \(4.3 m\) from the source. (a) What is the intensity at a distance of \(3.1 m\) from the source? (b) How much sound energy does the source emit in one hour if its power output remains constant?

At a distance of \(7.00 \times {10^{12}}\;{\rm{m}}\) from a star, the intensity of the radiation from the star is \(15.4\;{{\rm{W}} \mathord{\left/

 {\vphantom {{\rm{W}} {{{\rm{m}}^{\rm{2}}}}}} \right.

 \kern-\nulldelimiterspace} {{{\rm{m}}^{\rm{2}}}}}\). Assuming that the star radiates uniformly in all directions, what is the total power output of the star?

Interference of Triangular Pulses. Two triangular wave pulses are traveling toward each other on a stretched string as shown in Fig. E15.32. Each pulse is identical to the other and travels at \(2.00\;{{{\rm{cm}}} \mathord{\left/ {\vphantom {{{\rm{cm}}} {\rm{s}}}} \right. \\} {\rm{s}}}\). The leading edges of the pulses are \(1.00\;{\rm{cm}}\) apart at \(t = 0\). Sketch the shape of the string at \(t = 0.250\;{\rm{s}}\), \(t = 0.500\;{\rm{s}}\), \(t = 0.750\;{\rm{s}}\), \(t = 1.000\;{\rm{s}}\), and \(t = 1.250\;{\rm{s}}\)

Suppose that the left-traveling pulse in Exercise 15.32 is below the level of the unstretched string instead of above it. Make the same sketches that you did in that exercise

Two pulses are moving in opposite directions at \(1.0\;{{{\rm{cm}}} \mathord{\left/ {\vphantom {{{\rm{cm}}} {\rm{s}}}} \right. \\} {\rm{s}}}\) on a taut string, as shown in Fig. E15.34. Each square is \(1.0\;{\rm{cm}}\). Sketch the shape of the string at the end of 

(a) \(6.0\;{\rm{s}}\); 

(b) \(7.0\;{\rm{s}}\); 

(c) \(8.0\;{\rm{s}}\)

Interference of Rectangular Pulses. Figure E15.35 shows two rectangular wave pulses on a stretched string traveling toward each other. Each pulse is traveling with a speed of \(1.00\;{{{\rm{mm}}} \mathord{\left/ {\vphantom {{{\rm{mm}}} {\rm{s}}}} \right. \\} {\rm{s}}}\) and has the height and width shown in the figure. If the leading edges of the pulses are \(8.00\;{\rm{mm}}\) apart at \(t = 0\), sketch the shape of the string at \(t = 4.00\;{\rm{s}}\), \(t = 6.00\;{\rm{s}}\), and \(t = 10.0\;{\rm{s}}\).