Moving Loudspeakers on a Railroad Car

[e(ho0n3]

[SOLVED] Moving Loudspeakers on a Railroad Car

Homework Statement

Two loudspeakers are at opposite ends of a railroad car as it moves past a stationary observer at u meters per second. If they have identical sound frequencies of f, what is the beat frequency heard by the observer when (a) he listens from the position A, in front of the car, (b) he is between the speakers, at B and (c) he hears the speakers after they have passed him, at C?

Note: There is a figure accompanying this problem showing that the observer is besides the railroad tracks.

Homework Equations

I will need the equations for calculating the frequency heard by an observer of a moving sound source. Also, beat frequency = difference in the two wave frequencies.

The Attempt at a Solution

Let's concentrate on just (a) as the others are probably similar in nature. The observer measures a beat frequency f_b = f₁ - f₂, where f₁ is the largest frequency the observer hears from one of the two speakers and f₂ is the heard frequency of the other speaker. What I don't understand is why is one frequency larger than the other. It would make sense if the observer is a non-negligible distance away from the tracks. However, the observer is close to the tracks so the heard frequency from both speakers should be the same.

Also, the distance between the speakers is not mentioned. I would imagine that this plays an important role in the resultant frequency of the sound wave heard by the observer.

 

[chaoseverlasting]

No. Do you know about the Doppler Effect? Thats what happens here. The actual frequency is different from the apparent frequency and that is what causes the beats. Also, this is only possible if the observer is a finite distance away from the railroad tracks because then the speeds of the two ends of the train will also be different at different times (think Pythagorean Theorem). Look it up, its pretty simple.

 

[e(ho0n3]

Yes, I know about the Doppler effect. You say that the beat frequency would be the difference between the apparent frequency and the actual frequency. But you did not mention anything with the two speakers. If I eliminate one of the speakers, will the observer still hear beats?

 

[Mentz114]

If I eliminate one of the speakers, will the observer still hear beats?

You need both speakers on to get a beat frequency. If the train was stationary they would both be heard at the same frequency so here's no beat. Any beat will be between the shifted frequencies.

 

[e(ho0n3]

The formula I have for calculating the frequency f' heard by an observer of a sound source moving toward him at velocity u is f' = f/(1 - u/v) where v is the speed of sound in air and f is the actual frequency. Since the actual frequency and the speed of the speakers are the same for both of them, I would get the same f' for both of them and hence the difference, the beat frequency, is zero.

Am I wrong?

 

[Mentz114]

It depends where the carriage is in relation to the listener. If one speaker is moving away and one is moving towards the listener - what then ?

 

[e(ho0n3]

The picture accompanying the problem statement shows the observer close to the tracks. Both speakers are moving towards position A. The problem does not mention anything about the distance of the observer from the speakers or the distance between the speakers.

 

[Mentz114]

(a) he listens from the position A, in front of the car, (b) he is between the speakers, at B and (c) he hears the speakers after they have passed him, at C?

It gives 3 relative positions to consider.

 

[e(ho0n3]

Yes. And I am only considering position A. Why would the observer at A hear a beat frequency?

 

[Mentz114]

So you are. Sorry it's a bit late here. In fact I'm off after this.

In my judgement you are right, as long as the listener is in line with the speakers the frequencies should stay equal. The same for position C.

But B is different. Good luck with it.

 

[e(ho0n3]

Yes. The situation for B is certainly different as one speaker is approaching the observer and the other is moving away from him. I'm a bit more confident now that the observer at A and C will not hear any beats. Thanks.