The Doppler Effect: Deducing an Expression for Frequency

[jono90one]

Homework Statement

A sound source moves at a constant velocity. A listener is standing at a distance L away from it. Given that the source moves in a straight line at a right angle to the listener and starts closest to the listener (ie at t=0) deduce an expression for the frequency heard by the listener in relation to time.

Homework Equations

f_l=f_s(v/(v-v_s))
where l is listener and s is source

The Attempt at a Solution

It's obviously Doppler effect related.
So far I’ve done T=1/f_s
And gotten
f_l=v/T(v-v_s)

But unsure how to get L into the equation.
Surely VT is the distance traveled by the wave and v_sT is λ_s. Is there any orientation where L can be introduced? Or is this version correct?

 

[abhishek ghos]

take in the velocity component of the source away from the observer at each instant of time in the governing equation.

 

[jono90one]

(edit) So you mean f=v/T(vs-v)??

 

[abhishek ghos]

No
O(t=0)----------------------->vt(point A)(source)
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|L
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Point B(you)
I want you to find the component of source’s(A) velocity
Along the line joining point A & point B at an instant t
And plug that velocity into your equation for vs.

 

[rwisz]

I would approach this problem by first understanding the principles of the Doppler effect. The Doppler effect is the change in frequency of a wave (in this case, sound waves) due to the relative motion between the source and the observer. This change in frequency is caused by the compression or stretching of the wave as the source moves towards or away from the observer.

In this scenario, we have a sound source moving at a constant velocity and a listener standing at a fixed distance L away from the source. The key to deducing an expression for the frequency heard by the listener is to understand the relationship between the wavelength and the frequency of the sound wave.

We know that the wavelength (λ) is equal to the velocity (v) divided by the frequency (f). So, we can rewrite our equation as:

fl=fs(v/(v-vs)) = v/fs = v/(v/T) = vT/v

Now, we can use the Pythagorean theorem to relate the distance L to the velocity v and the time T. We know that the distance traveled by the sound wave is equal to the distance traveled by the source (vs) plus the distance traveled by the wave (vT). This can be represented as:

L = vs + vT

Rearranging this equation, we get:

vT = L - vs

Substituting this into our previous equation, we get:

fl = vT/v = (L - vs)/v

This is the final expression for the frequency heard by the listener in relation to time. It incorporates both the velocity of the source and the distance between the source and the listener.