Question

A source emitting sound of frequency 180 Hz is placed in front of a wall at a distance of 2 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 $m{s}^{-1}$

• A. 1 m
• B. 2 m
• C. 3 m
• D. 4 m

Answer 1

The correct option is C

3 m

The situation is shown in figure. suppose the detector is placed at a distance of x meter from the source. The direct wave received from the source travels a distance of x meter. The wave reaching the detector after reflection from the wall has travelled a distance of 2 $\left[\right(2{)}^2+\frac{x^2}{4}{]}^{\frac{1}{2}}$ meter. The path difference between the two waves is

$△=\left\{2\right[(2{)}^2+\frac{x^2}{4}{]}^{\frac{1}{2}}-x\}$ meter

Constructive interference will take place when $△=\lambda ,2\lambda$ . . . . . . The minimum distance x for a maximum corresponds to

$△=\lambda$ - - - - - - - - (i)

The wavelength is $\lambda =\frac{v}{f}=\frac{360m{s}^{-1}}{180s^{-1}}=2m$

Thus,by (i), 2 $\left[\right(2{)}^2+\frac{x^2}{4}{]}^{\frac{1}{2}}-x=2$

or, $[4+\frac{x^2}{4}{]}^{\frac{1}{2}}=1+\frac{x}{2}$

or, $4+\frac{x^2}{4}=1+\frac{x^2}{4}+x$

or, $3=x$

Thus, the detector should be placed at a distance of 3 m from the source.