Doppler Effect (Sound) - Source moving away from the Observer

[Armenio]

Hi! I was doing some exercises about the doppler effect on sound until I found this problem that I can't find the solution!

"A source of sound of frequency f₀ moves horizontally at constant speed u in the x direction at a distance h above the ground. An observer is situated on the ground at the point x=0 (the source passes over this point at t=0).
Show that the signal received at any time T_r at the ground was emitted by the source at an early time T_s , such that:

[1-(u/v)²]*T_s=T_r-(1/v)√(h²*[1-(u/v)²]+u²*T_r²)

where v is the speed of the sound."

Can someone please help me and explain this exercise to me? I have been trying to solve it for 48 hours and I just can't! -.-

Thank you!

 

[BruceW]

welcome to physicsforums :) Interesting problem. Where are you having difficulties? In this forum, they have a strong policy of homework helping, so we need some more information on what you have tried so far, otherwise I don't really know how to help. For example, I'm guessing that you know the answer for the problem if the source is moving exactly away from the observer? Think about how this problem is different to that problem.

 

[Armenio]

The source at a time t is u*t away from the source at x=0 (t=0).
The distance from the source to the observer at any given t is d=sqrt(h^2+u^2*t^2).
And then I started to work with tetas and angles, but I don't know if that is correct, because I'm using small angles aproximation.

 

[BruceW]

you should be able to do the exercise without small angle approximation. Also, this problem is a bit less straightforward than I first thought. It seems like they want you to assume that the speed of sound is the same according to all observers. Keep this in mind, and try to relate their equation to a relativistic transformation of variables. (remember Tr and Ts are times according to someone, they are not absolute, since this problem is relativistic). Also, are you sure that in the equation
[1-(u/v)²]*Ts=Tr-(1/v)√(h²*[1-(u/v)²]+u²*Tr²)
the last term is Tr² ? It would make more sense to me if it was Ts², I'm not sure though. This problem is a bit vague, I am not surprised that you have been working on it for a while.

 

[TSny]

Let p be the position of the source at time Ts. Let Δt be the time interval it takes the sound to travel from p to the observer. How are Ts, Δt, and Tr related? How can you express Δt in terms of h, u, Ts, and v? (Use your expression for d.) There's no need to use a small angle approximation.

 

[Armenio]

Even my mechanics teacher couldn't solve the problem immediatly (I'm still waiting for an e-mail from him with some hints).

you should be able to do the exercise without small angle approximation. Also, this problem is a bit less straightforward than I first thought. It seems like they want you to assume that the speed of sound is the same according to all observers. Keep this in mind, and try to relate their equation to a relativistic transformation of variables. (remember Tr and Ts are times according to someone, they are not absolute, since this problem is relativistic). Also, are you sure that in the equation
[1-(u/v)²]*Ts=Tr-(1/v)√(h²*[1-(u/v)²]+u²*Tr²)
the last term is Tr² ? It would make more sense to me if it was Ts², I'm not sure though. This problem is a bit vague, I am not surprised that you have been working on it for a while.

Yeah, I know. I tought the same thing but the equation is correct.
I have made some progress and I found out that:

T_s=T_r-(√(h²+u²*T_s²)/v)

I'm still working with angles and have found some curious observations.
I think I'm getting closer but I really don't know how to get to the term (1-[u/v]²)!

 

[barryj]

To verify your equation, you might consider drawing a diagram, assigning some numbers to the problem and then see if the answer works out. For example, let h = say 1000m, let v = 100 m/s, speed of sound 340 m/s. Assume you heard the sound at t = 5 seconds, what would Ts be?

 

[TSny]

I have made some progress and I found out that:

T_s=T_r-(√(h²+u²*T_s²)/v)

I'm still working with angles and have found some curious observations.
I think I'm getting closer but I really don't know how to get to the term (1-[u/v]²)!

This looks good. You just need to solve for Ts. Isolate the square root on one side of the equation and then square both sides. You'll get a quadratic equation for Ts.

 

[Armenio]

This looks good. You just need to solve for Ts. Isolate the square root on one side of the equation and then square both sides. You'll get a quadratic equation for Ts.

Ahahahaha of course!

You are a genius!

I've finally made the exercise and I've sent an e-mail to my teacher to teach him how to do it too.

Thank you all!