Where did the equation ( (delta)f / f0 ) = ( 2 u / v ) come from or how did they

Question

Where did the equation ( (delta)f / f0 ) = ( 2 u / v ) come from or how did they get it?

The question.

A music student walks in a long hall carrying a vibrating 512Hz tuning fork.The end of the hall is closed so that sound reflects from it. The student hears 4 beats per second. How fast is the student walking and in what direction?

The solution.
Given that,
Frequency of tuning fork f0 = 512 Hz
Number of beats per second = beat frequncy ?f = 4
Here the ?f may be positive or negative.
a) For deltaf = +4
We have the relation ( delta(f) / f0 ) = ( 2 u / v )
Here v = speed of sound = 340 m/s

u = speed of the student

( 4 / 512 ) = ( 2 *u / 340 )

Speed of the student u = 1.3281 m/s towards the wall
b) For ?f = - 4
We have the relation ( Delta(f) / f0 ) = ( 2 u / v )
Here v = speed of sound = 340 m/s

u = speed of the student

( -4 / 512 ) = ( 2 *u / 340 )

Speed of the student u = -1.3281 m/s away from the wall

Explanation / Answer

from doppler effect f = [ (V-vo) / ( V - vs ) ] * fo                       f-fo = { [ (V-vo) / ( V - vs ) ] * fo } -fo                              = fo { [ (V-vo) / ( V - vs ) ] -1}                               = fo { [V-vo-V-vs] / ( V - vs ) }             ( f-fo) / fo = { (-vo-vs) / ( V -vs) }               f / fo = { (-vo-vs) / ( V -vs) } where V = speed of sound          vo = speed of observer          vs = speed of source

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