Question

A source emitting sound of frequency $90\text{Hz}$ is placed in front of a wall at a distance $3\text{m}$ from it. A detector is also placed in front of the wall at the same distance from it. Find the minimum distance between the source and the detector for which the detector detects a maximum of sound. Speed of sound in air = $360{\text{ms}}^{-1}$.

• A. $2\text{m}$
• B. $4\text{m}$
• C. $6\text{m}$
• D. $8\text{m}$

Answer 1

The correct option is D $8\text{m}$


For obtaining maxima, the sound waves from the source and the reflector should interfere constructively at $\text{D}$.
Condition for constructive interference,
Path difference $\Delta l=n\lambda$ , where $n=1,2,3,\mathrm{4....}$

Let $x$ be the minimum seperation between source and detector, for which two sound waves interfere constructively.
Then, the maxima corresponds to path difference,
$\Delta l=\lambda$
From the given figure,

$\{2\times {\left[\right(3{)}^2+\frac{x^2}{4}]}^{1/2}-x\}=\lambda ..........\left(1\right)$

From the data given in the question,

$\lambda =\frac{v}{f}=\frac{360}{90}=4\text{m}..........\left(2\right)$
Substituting $\left(2\right)$ in $\left(1\right)$, we get

${[36+x^2]}^{1/2}-x=4\Rightarrow [36+x^2]=(x+4{)}^2$

$\Rightarrow 36+x^2=x^2+16+8x$
$\Rightarrow 8x=20\Rightarrow x=2.5\text{m}$

Thus, option (b) is the correct answer.