Physics 1240
Homework 6 solutions
Brief solutions to homework set #6. Your numbers will be different, but the idea is the same.
I’m just using sample numbers here.
1. A sailor strikes the side of her ship just below the waterline. She hears the echo of the sound
reflected from the ocean floor directly below 1.99 s later. How deep is the ocean at this point?
Assume the velocity of sound in water is 1560 m/s.
The sound travels from the boat, down to the bottom of the ocean, then back to the sailor.
This means that if the depth of the ocean is d, she hears the sound after it travels a distance
of 2d (down and then back). This is the trick in this question! We can then use that velocity
is distance divided by time: normally we write this as v = d/t. But here the distance the
sound travels is 2d. Rearranging, 2d = vt, so d = 12 vt.
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d = vt = 0.5 × 1560 m/s × 1.99 s = 1552.2 m.
2
2. Change the previous problem to consider sound traveling the same distance, but through air.
Suppose the sailor strikes the side of her ship, and the sound travels the same distance as the
depth of the ocean you calculated in the previous problem. Now the sound travels through
air, reflects (echoes) off a cliff, and returns to the sailor. How much time elapses between
when the sailor produces the sound, and when she hears its echo? Assume the velocity of
sound in air is 344 m/s.
What physical property of liquids and solids do you think causes the speed of sound in them
to be to be so much greater than the speed of sound in air?
This is similar to the previous problem, with a change in velocity. The total time is t = 2d/v,
where d is the distance we found in the previous problem.
t=
2d
2 × 1552.2 m
=
= 9.02 s.
v
344 m/s
The key difference between most liquids and solids (like water and metal) and most gases (like
air) is that liquids and solids typically have much higher density than gases. This means that
more molecules are packed into the same volume—the molecules are therefore much closer
together in a typical liquid or solid than in a gas. For a sound wave to travel, molecules have
to physically collide with each other...when the molecules are closer together, those collisions
can happen more quickly, and the pressure disturbance can propagate more quickly.
3. You are listening to a horn. You know the frequency of the horn is 300 Hz when both you
and the horn are at rest. If you hear a pitch of 330 Hz, there are several possibilities.
For this problem, the key word is “possibility”: you must determine whether or not the type
of motion described could possibly allow you to hear a 330 Hz frequency when the horn has
a 300 Hz frequency at rest. If there is any possible way for this motion to lead to the 330 Hz
perception, that description is true1 .
1
The wording of this question was confusing to several people in the class. If you find wording of a question
unclear, please come to the help room and ask!
1
(a) Both you and the horn can be moving and have the same speed.
This is true/possible: if you and the horn are moving at the same speed but in opposite
directions toward each other, you are getting closer together and you would hear a higher
pitch.
(b) Both you and the horn can be moving, in opposite directions.
This is true/possible: if you and the horn are moving in opposite directions toward each
other, you are getting closer together and you would hear a higher pitch.
(c) The distance between you and the horn is increasing with time.
This is false/not possible: if you and the horn are getting farther apart, you would hear
a lower pitch.
(d) Both you and the horn can be moving and have different speeds.
This is true/possible: if you and the horn are moving with different speeds while moving
toward each other, you are getting closer together and you would hear a higher pitch.
(e) You are moving toward the horn, which is at rest.
This is true/possible: this type of motion means you and the horn are getting closer
together and you would hear a higher pitch.
(f) Both you and the horn cannot be moving in the same direction.
This false, because you and the horn could be moving in the same direction. If you and
the horn are moving in the same direction, but you are moving faster than the horn, you
are getting closer together and you would hear a higher pitch.
4. See comments above.
5. A tractor approaches you from behind one of the dorm buildings on campus. Initially you
can hear it but cannot see it (it’s out of sight behind the building). When it emerges and you
do see it, its sound is suddenly brighter—that is, you hear more of the high-frequency noise.
Explain the reason for this in the space below.
This occurs because of diffraction. When the tractor is out of sight behind the building,
sound can only reach you by diffracting (bending) around the corner. The low frequencies
(long wavelengths) diffract well and reach you, while the high frequencies (short wavelengths)
don’t diffract well and don’t reach you. This makes the engine sound like more of a low roar.
Once the tractor emerges and is visible, all frequencies can reach you in a direct line (without
diffracting). Then you hear all the frequencies, including the high frequencies, so the overall
sound is “brighter.”
6. Traditional methods of protecting the hearing of people who work in areas with very high
noise levels have consisted mainly of efforts to block or reduce noise levels. With a relatively
new technology, noise-canceling headphones are worn that do not block the ambient noise.
Instead, a device is used which detects the noise, inverts it electronically, then feeds it to the
headphones in addition to the ambient noise. How could adding more noise reduce the sound
levels reaching the ears?
Adding more noise can reduce the sound levels reaching the ears if the added noise destructively interferes with the original sound. For this to occur, the added sound must be identical
in frequency to the original sound, but be perfecly out of phase—so that the peaks in pressure of the original sound perfectly correspond to troughs in pressure of the added sound.
Then the pressure fluctuations from the two sounds will cancel, leading to no (or little) sound
reaching the ears.
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