A source emitting a sound of frequency v is placed at a large distance from an observer. The source starts moving towards the observer with a uniform acceleration a. Find the frequency heard by the observer corresponding to the wave emitted just after the source starts. The speed of sound in the medium is v.

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Solution

Let d be the initial distance between the source and the observer.
If v is the speed of sound emitted by the observer, then the time taken by the sound to reach the observer is given by:
T₁ = d/v
The source is also moving. Therefore, at t = T, it moves a distance of (s) and is given by:
$s = 0 \times T + \frac{1}{2} a T^{2}$

Time taken by the pulse to reach the observer:
$\frac{(d - \frac{1}{2} a T^{2})}{v}$

Time difference $(∆ t)$ between the two pulses:

$(T + (\frac{d - \frac{1}{2} a T^{2}}{v})) - \frac{d}{v}$ = $T - \frac{a T^{2}}{2 v}$

On replacing u = $\frac{1}{T}$ ,

the apparent frequency will be:

$\frac{1}{∆ t}$ = $\frac{2 u v^{2}}{2 u v - a}$ .

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