Question

A source emitting sound of frequency 165 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 distance between the source and the detector for which the detector detects phase difference of 21 t between the direct and reflected wave. Speed of sound in air = 330 m/s :

Answer 1

let the detector be x meter away from source therefore one which travels directly from source to detector travels x m and the other which will travel to reflective wall and then reach to the detector is,

$p_2=2\times (2^2+(\frac{x}{2}{)}^2{)}^{1/2}$

therefore the path difference

$\Delta p=p_2-x$

$n\lambda =p_2-x$

$\lambda =\frac{v}{f}=\frac{330}{165}=2m$

$2=p_2-x$

$x+2=p_2$

$x+2=2\times (2^2+(\frac{x}{2}{)}^2{)}^{1/2}$

$1+\frac{x}{2}=(2^2+(\frac{x}{2}{)}^2{)}^{1/2}$

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

$x=3m$

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