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Sound Waves — Physics STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 SciencePhysicsSound Waves3 Marks
Question
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 u.
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Answer

Let the distance between the source and the observer is ‘x’ (initially) So, time taken for the first pulse to reach the observer is $\text{t}_1=\frac{\text{x}}{\text{v}}$ And the second pulse starts after T $\Big($where, $\text{T}=\frac{1}{\text{v}}\Big)$ And it should travel a distance $\Big(\text{x}-\frac{1}{2}\text{a}\text{T}^2\Big).$
So, $\text{t}_2=\text{T}+\frac{\frac{\text{x}-1}{2}\text{a}\text{T}^2}{\text{v}}$$\text{t}_2-\text{t}_1=\text{T}+\frac{\frac{\text{x}-1}{2}\text{aT}^2}{\text{v}}=\frac{\text{x}}{\text{v}}=\text{T}-\frac{1}{2}\frac{\text{aT}^2}{\text{v}}$
Putting = $\text{T}=\frac{1}{\text{v}}.$ we get$\text{t}_2-\text{t}_1=\frac{2\text{uv}-\text{a}}{2\text{vv}^2}$
So, frequency heard $=\frac{2\text{vv}^2}{2\text{uv}-\text{a}}$ $\Big($because, $\text{f}=\frac{1}{\text{t}_2-\text{t}_1}\Big)$

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