Two coherent sources of sound, S _ 1 and S _ 2 , produce sound...

Given,

Two sources od same wavelength, $\lambda = 1m$

Distance between two sources,${S}_{1}{S}_{2} = 1 . 5m$

Distance of listener from the source ${S}_{2}$, ${S}_{2}L = 2m$

Condition for minimum intensity, at $2m$ is

Path difference, $P . D . = ( 2n + 1 ) \dfrac{\lambda}{2} . . . ( 1 )$

$⇒ P . D . = \sqrt{{2}^{2} + ( 1 . 5{)}^{2}} = ( 2n + 1 ) \dfrac{\lambda}{2}$
$⇒ 2 . 5 = ( 2n + 1 ) \dfrac{\lambda}{2} . . . ( 1 )$

Now, the path difference between the rays is so that the adjacent maxima is obtained,

$P . D . = d - 2 = n\lambda . . . ( 2 )$

The path difference between adjacent maxima and minima is $\dfrac{\lambda}{2}$, form equation (1) and (2) we get,

$( d - 2 ) - 2 . 5 = \dfrac{\lambda}{2} . . . ( 3 )$
$⇒ d - 2 = 2 . 5 + \dfrac{1}{2}$

$⇒ d = 5m$