hw-3_2

Your Name : Your ID Number : Due Date : 16.10.12

Consider a longitudinal (compressional) wave of wave-length λ traveling with speed v along the x direction through a medium of density ρ. The displacement of the molecules of the medium from their equilibrium position is

Show that the pressure variation in the medium is given by

542 C H A P T E R 1 7 Sound Waves

wave fronts both to the east and to the west of thepolice car. (b) What would be the wavelength in air ofthe siren sound if the police car were at rest? (c) Whatis the wavelength in front of the car? (d) What is thewavelength behind the police car? (e) What frequencyis heard by the driver being chased?

58. A copper bar is given a sharp compressional blow at oneend. The sound of the blow, traveling through air at0°C, reaches the opposite end of the bar 6.40 ms laterthan the sound transmitted through the metal of thebar. What is the length of the bar? (Refer to Table17.1.)

59. The power output of a certain public address speaker is6.00 W. Suppose it broadcasts equally in all directions.(a) Within what distance from the speaker would thesound be painful to the ear? (b) At what distance fromthe speaker would the sound be barely audible?

60. A jet flies toward higher altitude at a constant speed of1 963 m/s in a direction that makes an angle ! with thehorizontal (Fig. P17.60). An observer on the groundhears the jet for the first time when it is directly over-head. Determine the value of ! if the speed of sound inair is 340 m/s.

63. A meteoroid the size of a truck enters the Earth’s atmos-phere at a speed of 20.0 km/s and is not significantlyslowed before entering the ocean. (a) What is the Machangle of the shock wave from the meteoroid in theatmosphere? (Use 331 m/s as the sound speed.) (b) Assuming that the meteoroid survives the impactwith the ocean surface, what is the (initial) Mach angleof the shock wave that the meteoroid produces in thewater? (Use the wave speed for sea water given in Table 17.1.)

64. Consider a longitudinal (compressional) wave of wave-length " traveling with speed v along the x directionthrough a medium of density #. The displacement of themolecules of the medium from their equilibrium posi-tion is

Show that the pressure variation in the medium is givenby

65. By proper excitation, it is possible to produce both lon-gitudinal and transverse waves in a long metal rod. Aparticular metal rod is 150 cm long and has a radius of0.200 cm and a mass of 50.9 g. Young’s modulus for thematerial is 6.80 $ 1010 N/m2. What must the tension inthe rod be if the ratio of the speed of longitudinal wavesto the speed of transverse waves is 8.00?

66. An interstate highway has been built through a neigh-borhood in a city. In the afternoon, the sound level in arented room is 80.0 dB as 100 cars per minute pass out-side the window. Late at night, the traffic flow on thefreeway is only five cars per minute. What is the averagelate-night sound level in the room?

67. A siren creates a sound level of 60.0 dB at a location 500 m from the speaker. The siren is powered by a bat-tery that delivers a total energy of 1.00 kJ. Assumingthat the efficiency of the siren is 30.0% (that is, 30.0%of the supplied energy is transformed into sound en-ergy), determine the total time the siren can sound.

68. A siren creates a sound level % at a distance d from thespeaker. The siren is powered by a battery that delivers atotal energy E. Assuming that the efficiency of the sirenis e (that is, e is equal to the output sound energy di-vided by the supplied energy), determine the total timethe siren can sound.

69. The Doppler equation presented in the text is validwhen the motion between the observer and the sourceoccurs on a straight line, so that the source and ob-server are moving either directly toward or directly awayfrom each other. If this restriction is relaxed, one mustuse the more general Doppler equation

f & ' ! v ( vO cos !O

v ) vS cos !S" f

*P ' )! 2+#v2

" smax"

cos(kx ) ,t)

s ' smax sin(kx ) ,t)

WEB

61. Two ships are moving along a line due east. The trailingvessel has a speed of 64.0 km/h relative to a land-basedobservation point, and the leading ship has a speed of45.0 km/h relative to that point. The two ships are in aregion of the ocean where the current is moving uni-formly due west at 10.0 km/h. The trailing ship trans-mits a sonar signal at a frequency of 1 200.0 Hz. Whatfrequency is monitored by the leading ship? (Use 1 520 m/s as the speed of sound in ocean water.)

62. A microwave oven generates a sound with intensity level40.0 dB everywhere just outside it, when consuming1.00 kW of power. Find the fraction of this power that is converted into the energy of sound waves. Assume the dimensions of the oven are 40.0 cm $ 40.0 cm $50.0 cm.

!

Figure P17.60














f & ' ! v ( vO cos !O

v ) vS cos !S" f

*P ' )! 2+#v2

" smax"

cos(kx ) ,t)


WEB



!

Figure P17.60

PHYS 211

Department of Physics

Bilkent University

1

Homework 3


A long spring such as a SlinkyTM (see figure) is often used to demonstrate longitudinal waves. (a) Show that if a spring that obeys Hooke’s law has mass m, length L, and force constant k’, the speed of longitudinal waves on the spring is v = L(k’/m)1/2. (b) Evaluate v for a spring with m = 0.250 kg, L = 2.00 m, and k’ = 1.50 N/m.

PHYS 211


Bilkent University

2

Homework 3


The commonly used Doppler equation is valid when the motion between the observer and the source occurs on a straight line, so that the source and observer are moving either directly toward or directly away from each other. If this restriction is relaxed, one must use the more general Doppler equation














f & ' ! v ( vO cos !O

v ) vS cos !S" f

*P ' )! 2+#v2

" smax"

cos(kx ) ,t)


WEB



!

Figure P17.60

where θo and θs are defined in the figure. (a) Show that if the observer and source are moving away from each other, the preceding equation reduces to usual equation.

Answers to Quick Quizzes 543

where !O and !S are defined in Figure P17.69a. (a) Show that if the observer and source are movingaway from each other, the preceding equation reducesto Equation 17.17 with lower signs. (b) Use the preced-ing equation to solve the following problem. A trainmoves at a constant speed of 25.0 m/s toward the inter-section shown in Figure P17.69b. A car is stopped nearthe intersection, 30.0 m from the tracks. If the train’shorn emits a frequency of 500 Hz, what frequency isheard by the passengers in the car when the train is 40.0 m from the intersection? Take the speed of soundto be 343 m/s.

70. Figure 17.5 illustrates that at distance r from a pointsource with power , the wave intensity is

/4"r2. Study Figure 17.11a and prove that at dis-tance r straight in front of a point source with power

moving with constant speed vS , the wave intensity is

71. Three metal rods are located relative to each other asshown in Figure P17.71, where The den-L1 # L2 $ L3 .

I $!av

4"r 2 ! v % vS

v "!av ,

!av

I $!av

sity values and Young’s moduli for the three materialsare kg/m3, N/m2;

kg/m3, N/m2;kg/m3, N/m2.

(a) If m, what must the ratio L1/L 2 be if asound wave is to travel the combined length of rods 1 and 2 in the same time it takes to travel the length ofrod 3? (b) If the frequency of the source is 4.00 kHz,determine the phase difference between the wave trav-eling along rods 1 and 2 and the one traveling alongrod 3.

L3 $ 1.50Y3 $ 11.0 & 1010'3 $ 8.80 & 103Y2 $ 1.60 & 1010'2 $ 11.3 & 103

Y1 $ 7.00 & 1010'1 $ 2.70 & 103

72. The volume knob on a radio has what is known as a“logarithmic taper.” The electrical device connected tothe knob (called a potentiometer) has a resistance Rwhose logarithm is proportional to the angular positionof the knob: that is, log If the intensity of thesound I (in watts per square meter) produced by thespeaker is proportional to the resistance R, show thatthe sound level ( (in decibels) is a linear function of !.

73. The smallest wavelength possible for a sound wave in airis on the order of the separation distance between airmolecules. Find the order of magnitude of the highest-frequency sound wave possible in air, assuming a wavespeed of 343 m/s, a density of 1.20 kg/m3, and an aver-age molecular mass of 4.82 & 10%26 kg.

R ) !.

fSvS

fOvO

(b)

25.0 m/s

(a)

!O!

!S!

Figure P17.69

1 23

L3

L2L1

Figure P17.71

ANSWERS TO QUICK QUIZZES

17.3 Because the bottom of the bottle does not allow molecu-lar motion, the displacement in this region is at its mini-mum value. Because the pressure variation is a maxi-mum when the displacement is a minimum, thepressure variation at the bottom is a maximum.

17.4 (a) 10 dB. If we call the intensity of each violin I, the to-tal intensity when all the violins are playing is

Therefore, the addition of the nine violinsincreases the intensity of the sound over that of one vio-lin by a factor of 10. From Equation 17.7 we see that anincrease in intensity by a factor of 10 increases thesound level by 10 dB. (b) 13 dB. The intensity is now in-creased by a factor of 20 over that of a single violin.

17.5 The Mach number is the ratio of the plane’s speed(which does not change) to the speed of sound, which isgreater in the warm air than in the cold, as we learned

I # 9I $ 10I.

17.1 The only correct answer is (c). Although the speed of awave is given by the product of its wavelength and fre-quency, it is not affected by changes in either one. Forexample, if the sound from a musical instrument in-creases in frequency, the wavelength decreases, and thus

remains constant. The amplitude of a soundwave determines the size of the oscillations of air mole-cules but does not affect the speed of the wave throughthe air.

17.2 The ground tremor represents a sound wave movingthrough the Earth. Sound waves move faster throughthe Earth than through air because rock and otherground materials are much stiffer against compression.Therefore—the vibration through the ground and thesound in the air having started together—the vibrationthrough the ground reaches the observer first.

v $ *f

PHYS 211


Bilkent University

3

Homework 3


A train whistle (f = 400 Hz) sounds higher or lower in pitch depending on whether it is approaching or receding. (a) Prove that the difference in frequency between the approaching and receding train whistle is

Problems 541

by Zerbinetta in the original version of Richard Strauss’sopera Ariadne auf Naxos. (a) Find the wavelength of thissound in air. (b) Suppose that the people in the fourthrow of seats hear this note with a level of 81.0 dB. Findthe displacement amplitude of the sound. (c) In re-sponse to complaints, Strauss later transposed the notedown to F above high C, 1.397 kHz. By what incrementdid the wavelength change?

48. A sound wave in a cylinder is described by Equations 17.2 through 17.4. Show that

49. On a Saturday morning, pickup trucks carrying garbageto the town dump form a nearly steady procession on acountry road, all traveling at 19.7 m/s. From this direc-tion, two trucks arrive at the dump every three minutes.A bicyclist also is traveling toward the dump at 4.47 m/s. (a) With what frequency do the trucks passhim? (b) A hill does not slow the trucks but makes theout-of-shape cyclist’s speed drop to 1.56 m/s. How oftendo the noisy trucks whiz past him now?

50. The ocean floor is underlain by a layer of basalt thatconstitutes the crust, or uppermost layer, of the Earth inthat region. Below the crust is found denser peridotiterock, which forms the Earth’s mantle. The boundary be-tween these two layers is called the Mohorovicic discon-tinuity (“Moho” for short). If an explosive charge is setoff at the surface of the basalt, it generates a seismicwave that is reflected back out at the Moho. If the speedof the wave in basalt is 6.50 km/s and the two-way traveltime is 1.85 s, what is the thickness of this oceanic crust?

51. A worker strikes a steel pipeline with a hammer, gener-ating both longitudinal and transverse waves. Reflectedwaves return 2.40 s apart. How far away is the reflectionpoint? (For steel, vlong ! 6.20 km/s and vtrans !3.20 km/s.)

52. For a certain type of steel, stress is proportional to strainwith Young’s modulus as given in Table 12.1. The steelhas the density listed for iron in Table 15.1. It bends per-manently if subjected to compressive stress greater thanits elastic limit, " ! 400 MPa, also called its yield strength.A rod 80.0 cm long, made of this steel, is projected at12.0 m/s straight at a hard wall. (a) Find the speed ofcompressional waves moving along the rod. (b) After thefront end of the rod hits the wall and stops, the back endof the rod keeps moving, as described by Newton’s firstlaw, until it is stopped by the excess pressure in a soundwave moving back through the rod. How much timeelapses before the back end of the rod gets the message?(c) How far has the back end of the rod moved in thistime? (d) Find the strain in the rod and (e) the stress.(f) If it is not to fail, show that the maximum impactspeed a rod can have is given by the expression

53. To determine her own speed, a sky diver carries abuzzer that emits a steady tone at 1 800 Hz. A friend atthe landing site on the ground directly below the skydiver listens to the amplified sound he receives from thebuzzer. Assume that the air is calm and that the speed

"/!#Y .

$P ! % #v&!s2max ' s2.

of sound is 343 m/s, independent of altitude. While thesky diver is falling at terminal speed, her friend on theground receives waves with a frequency of 2 150 Hz. (a) What is the sky diver’s speed of descent? (b) Sup-pose the sky diver is also carrying sound-receiving equip-ment that is sensitive enough to detect waves reflectedfrom the ground. What frequency does she receive?

54. A train whistle Hz) sounds higher or lower inpitch depending on whether it is approaching or reced-ing. (a) Prove that the difference in frequency betweenthe approaching and receding train whistle is

where u is the speed of the train and v is the speed ofsound. (b) Calculate this difference for a train movingat a speed of 130 km/h. Take the speed of sound in airto be 340 m/s.

55. A bat, moving at 5.00 m/s, is chasing a flying insect. Ifthe bat emits a 40.0-kHz chirp and receives back anecho at 40.4 kHz, at what relative speed is the bat mov-ing toward or away from the insect? (Take the speed ofsound in air to be m/s.)v ! 340

$f !2(u/v)

1 ' (u2/v2) f

( f ! 400

56. A supersonic aircraft is flying parallel to the ground.When the aircraft is directly overhead, an observer onthe ground sees a rocket fired from the aircraft. Tenseconds later the observer hears the sonic boom, whichis followed 2.80 s later by the sound of the rocket en-gine. What is the Mach number of the aircraft?

57. A police car is traveling east at 40.0 m/s along a straightroad, overtaking a car that is moving east at 30.0 m/s.The police car has a malfunctioning siren that is stuckat 1 000 Hz. (a) Sketch the appearance of the wavefronts of the sound produced by the siren. Show the

Figure P17.55

where u is the speed of the train and v is the speed of sound. (b) Calculate this difference for a train moving at a speed of 130 km/h. Take the speed of sound in air to be 340 m/s.

PHYS 211


Bilkent University

4

Homework 3