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I have come across Doppler shift equation. If we consider the source to be moving toward the observer at a speed $V$ and observer moving toward the source with a speed $U$, for the calculation of the apparent frequency, can't we think of the observer to be at rest from his frame and think that the source is moving with a speed $V+U$?
I have tried to solve a problem doing this but the answers i get are different.
No, assuming you are talking about sound waves, you can't add the velocities of the source and the observer in the equation for effective frequency: $$f_{\text{eff}}=\frac{v'}{\lambda'} $$
The reason is that a source moving toward the observer raises the frequency heard by the observer by decreasing the wavelength of the sound in the denominator (because it is "chasing" its own wave), $$\lambda'=\frac{(v_{\text{sound}}-{V})}{f},$$ but an observer moving toward the source increases the frequency heard by increasing the relative velocity between him and the sound wave in the numerator. $$v'=v_{\text{sound}}-U $$ (where $U$ is negative when toward the source).
That is correct, however when you write apparent frequency w.r.t the observer, the apparent frequency of the sound will also change!
Doppler effect wrt observer! (https://i.stack.imgur.com/5giP1.jpg) Thus if you take that into account it will match with the original method. //Note my sense of direction is different
