Non-symmetric nature of the Doppler Effect with sound

[sora]

While studying the Doppler effect at school, it struck me as strange that the following two problems have different solutions:

a) you are moving at 40 m/s toward a source that is making a sound with a frequency of 1000 Hz. What frequency do you hear? (speed of sound= 340 m/s)
b) The source of a sound is moving toward you at 40 m/s, producing a sound with a frequency of 1000 Hz. What frequency do you hear? (speed of sound = 340m/s)

I thought that they would both have the same observed frequency, but they don't. Could someone explain why?

 

[chogg]

I like to think of the traditional Doppler effect (i.e. waves propagating through a medium) with the following analogy. You know those little wind-up toys that walk in a straight line once you put them on the floor? Imagine a girl (call her "Sara") putting them on the floor at a constant rate, maybe once every $t$ seconds.

• The walking toys represent the wavefronts
• Their speed represents the speed of sound
• The floor represents the medium for the waves
• $1/t$ is the original frequency
• Sara represents the source

Naturally, the point of this analogy is to emphasize that once a wavefront is emitted, it doesn't matter what the source (or receiver) is doing; all that counts is the medium.

If you want to see, qualitatively, why they should be different, I think it's easier to show if the source and receiver are moving away from each other, rather than towards. In fact, let's say their relative speed is faster than the speed of the toys (speed of sound), and see what we get.

1) Moving source: The first toy is dropped, and it travels the initial distance and gets to the receiver. The next toy is dropped from a lot further back, so it has to travel the same distance plus a lot extra. The time between receiving toys is bigger than the time between dropping them, so the frequency is downshifted.

2) Moving receiver: The first toy heads towards the receiver, but the receiver is moving away even faster. The distance only ever increases! The receiver never receives.

Of course, an alternative is to crank the math and notice that you get different answers. But I think the starkness of the above scenario -- shifted frequency versus no waves at all -- makes it easier to grasp that yeah, they really should be different.

 

[nonequilibrium]

Chogg, maybe I'm misreading it, but wouldn't that be a bad analogy, because if the toy is to be analogous with sound, shouldn't it always have the same speed relative to the floor? (which would not be the case if Sara released the toy while moving relative to the floor)

Anyway, to the OP (maybe Chogg was saying the same thing after all, but I'd like to state it succintly): addmitedly, the two situations you sketch look symmetrical, but the symmetry-breaker is the fact that in one scenario the medium is moving, whilst in the other it isn't. (And sound moves at 300 m/s relative to the medium.)

 

[RK1992]

Chogg, maybe I'm misreading it, but wouldn't that be a bad analogy, because if the toy is to be analogous with sound, shouldn't it always have the same speed relative to the floor? (which would not be the case if Sara released the toy while moving relative to the floor)

Anyway, to the OP (maybe Chogg was saying the same thing after all, but I'd like to state it succintly): addmitedly, the two situations you sketch look symmetrical, but the symmetry-breaker is the fact that in one scenario the medium is moving, whilst in the other it isn't. (And sound moves at 300 m/s relative to the medium.)

when you say the medium is moving, do you mean the medium is moving in the observer's frame of reference?

 

[nonequilibrium]

uhu

(it won't allow me to post a three-letter response so i had to add this in)

 

[RK1992]

uhu

(it won't allow me to post a three-letter response so i had to add this in)

:tongue:

thought so.. so the doppler shift is basically the change in the measurement made in the measurement-taker's rest frame?

if you think about the person ("me") who hears the sound, it clears it up especially if i consider "my" rest frame in each scenario

a) I am stationary (this is my rest frame after all! :tongue: ), and the air is rushing past me at 40m/s and it is carrying a ripple moving at 380m/s relative to me (340m/s relative to the source + the source's motion relative to me) oscillating at 1000Hz.

b) I am stationary, the air is stationary and there are waves coming towards me which have a velocity of 340m/s relative to me (speed of sound doesn't depend on the source, only on the air which the vibrations are propagating through) and which have been doppler-shifted so that their frequency is now 1133Hz

is this right?

 

[chogg]

Chogg, maybe I'm misreading it, but wouldn't that be a bad analogy, because if the toy is to be analogous with sound, shouldn't it always have the same speed relative to the floor? (which would not be the case if Sara released the toy while moving relative to the floor)

I think you missed the following line (emphasis in the original):

Naturally, the point of this analogy is to emphasize that once a wavefront is emitted, it doesn't matter what the source (or receiver) is doing; all that counts is the medium.

If the toys walk by themselves, then the speed of the source is completely irrelevant, right? Would this illustration help?

 

[chogg]

When I said, "emphasis in original", I didn't realize that the entire blockquote is in italics. :P

 

[sora]

:tongue:

thought so.. so the doppler shift is basically the change in the measurement made in the measurement-taker's rest frame?

if you think about the person ("me") who hears the sound, it clears it up especially if i consider "my" rest frame in each scenario

a) I am stationary (this is my rest frame after all! :tongue: ), and the air is rushing past me at 40m/s and it is carrying a ripple moving at 380m/s relative to me (340m/s relative to the source + the source's motion relative to me) oscillating at 1000Hz.

b) I am stationary, the air is stationary and there are waves coming towards me which have a velocity of 340m/s relative to me (speed of sound doesn't depend on the source, only on the air which the vibrations are propagating through) and which have been doppler-shifted so that their frequency is now 1133Hz

is this right?

Part b is right, but for part a i got 1117Hz. Shouldn't the frequency be greater in part a since the sound wave is moving at a greater speed relative to you (when you factor in the relative speed of the medium)?

 

[RK1992]

Part b is right, but for part a i got 1117Hz. Shouldn't the frequency be greater in part a since the sound wave is moving at a greater speed relative to you (when you factor in the relative speed of the medium)?

yeah, i don't know why i didnt do that.. i get 1117.6Hz

 

[Acut]

I thought that they would both have the same observed frequency, but they don't. Could someone explain why?

For sound waves, there is a preferred inertial frame of reference - that of the medium, because sound waves always have a definite speed in that frame (e.g. 340 m/s in air). This is the reason why symmetry gets broken. Notice that in the formula for the Doppler Shift, the velocities of the source and the receiver you should plug in are the ones relative to the medium. In general, if you use another frame of reference, you will get wrong answers.

If there is no preferred inertial frame of reference, then the symmetry will not be broken - as is the case of the Relativistic Doppler Effect.

 

[sora]

Thank you, it makes a lot more sense now.