Doppler Effect (Proving equal wavelengths )

[Peter G.]

Hi,

I am doing doppler effect and this question wants me to prove that the wavelength perceived by the observer and that of the wave emitted is the same.

I tried doing it but couldn't, so I checked out what they had as an answer:

They said that the wavelength of the emitted wave is: v / f, which I agree with

They said the perceived wavelength is equal to: u + v / f (1 + u/v), which I also agree with.

Now, how they proved that those are equal I don't understand... I tried but I just can't see it. Can anyone help me with the steps involved?

Thanks,
Peter

 

[PeterO]

Hi,

I am doing doppler effect and this question wants me to prove that the wavelength perceived by the observer and that of the wave emitted is the same.

I tried doing it but couldn't, so I checked out what they had as an answer:

They said that the wavelength of the emitted wave is: v / f, which I agree with

They said the perceived wavelength is equal to: u + v / f (1 + u/v), which I also agree with.

Now, how they proved that those are equal I don't understand... I tried but I just can't see it. Can anyone help me with the steps involved?

Thanks,
Peter

it is just transformation of the expression, using common denominator.

(1 + u/v) = (v/v + u/v) = (v + u)/v = (u + v)/v [I reversed the v+u so it was identical to the numerator]

Put that in the expression and the (u + v) terms cancel, and the v ends up in the numerator to give v/f