KCET · Physics · Waves and Sound

A stationary point source of sound emits sound uniformly in all directions in a non-absorbing medium. Two points \(P\) and \(Q\) are at a distance of \(4 \mathrm{~m}\) and \(9 \mathrm{~m}\) respectively from the source. The ratio of amplitudes of the waves at \(P\) and \(Q\) is

A \(\frac{3}{2}\)
B \(\frac{4}{9}\)
C \(\frac{2}{3}\)
D \(\frac{9}{4}\)

Verified Solution

Answer & Solution

Correct Answer

(D) \(\frac{9}{4}\)

Step-by-step Solution

Detailed explanation

For an isotropic point source of power P, intensity I at a distance \(\mathrm{r}\) from it is
\(\mathrm{I}=\frac{\mathrm{P}}{4 \pi \mathrm{r}^{2}}\)
Since power P remains the same,
\(\therefore \frac{\mathrm{I}_{1}}{\mathrm{I}_{2}}=\left(\frac{\mathrm{r}_{2}}{\mathrm{r}_{1}}\right)^{2}=\left(\frac{9}{4}\right)^{2}\)
\(\because I \propto \mathrm{A}^{2}\)
where \(A\) is the amplitude of a wave \(\therefore \frac{\mathrm{A}_{1}}{\mathrm{~A}_{2}}=\sqrt{\frac{\mathrm{I}_{1}}{\mathrm{I}_{2}}}=\frac{9}{4}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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