# Doppler effect Formula manipulation

• Trec93
In summary, the conversation is about the Doppler effect formula and how to derive it. The formula is presented and the individual explains their attempt at solving it and how they got stuck. Another person then provides a step by step solution, combining terms on the right hand side with a common denominator to arrive at the final formula of v_{s}=\frac{v(f-f_{s})}{f}. The individual expresses their gratitude and confirms the correctness of the solution.

## Homework Statement

I have this Doppler effect formula, but I don't know how it was derived, I can't repeat the process myself to solve for speed of the source, I would really appreciate if someone could mathematicly solve this in steps, thank you very much.

## Homework Equations

$$f=f_{s}\frac{v}{v-v_{s}} \Rightarrow v_{s}=\frac{v(f-f_{s})}{f}$$

## The Attempt at a Solution

I checked and this is the same thing as the equation above, but mine is messy and ugly, I don't know how to prepare "neaty" formulas like the one above, this often confuses me and forces me to do checks whether my formula is right or not.
$$v_{s}=\frac{-f_{s}v}{f}+v$$

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Show, step by step, what you've tried and where you've become stuck.

gneill said:
Show, step by step, what you've tried and where you've become stuck.
Ok here's how I did it:

$$f=f_{s}\frac{v}{v-v_{s}}$$
I multiplied this by: $$(v-v_{s}) \Rightarrow f(v-v_{s})=f_{s}v$$
Then I divided by F and subtracted v
$$-v_{s}=\frac{f_{s}v}{f}-v$$

Finally I multiplied by the negative sign, that's my result:
$$v_{s}=\frac{-f_{s}v}{f}+v$$

I know they both are equal because I checked, but I don't have the skill to make my formula "neat" I often don't understand how people derive their formulas, I hope you know what I mean, I can't transform my formula into one above.
$$\frac{v(f-f_{s})}{f} = \frac{-f_{s}v}{f}+v$$

Combine the terms on the RHS with a common denominator.

Trec93
gneill said:
Combine the terms on the RHS with a common denominator.
Like this?
$$v_{s}=\frac{-f_{s}v}{f}+\frac{vf}{f}$$
Wait I see where this is going..
$$v_{s}=\frac{-f_{s}v+vf}{f}$$
$$v_{s}=\frac{v(f-f_{s})}{f}$$
Is this right?

Yup.

Trec93
gneill said:
Yup.
Wow thank you.