In a large room, a person receives direct sound waves from a source $120$ metres away from him. He also receives waves from the same source which reach him, being reflected from the $25$ metre high ceiling at a point halfway between them. The two waves interfere constructively for wavelength of
Diffcult
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(a) Let $S$ be source of sound and $P$ the person or listner.
The waves from $S$ reach point $P$ directly following the path $SMP$ and being reflected from the ceiling at point $A$ following the path $SAP$.
$M$ is mid-point of $SP$ (i.e. $SM = MP)$ and $\angle \,SMA = {90^o}$
Path difference between waves $\Delta x = SAP - SMP$
We have $SAP = SA + AP = 2(SA)$
$ = $$2\sqrt {[{{(SM)}^2} + {{(MA)}^2}]} = 2\sqrt {({{60}^2} + {{25}^2})} =130m$
$\therefore $ Path difference $= SAP -SMP = 130 - 120 = 10 m$
Path difference due to reflection from ceiling = $\frac{\lambda }{2}$
$\therefore $ Effective path difference $\Delta x = 10 + \frac{\lambda }{2}$
For constructive interference
$\Delta x = 10 + \frac{\lambda }{2} = n\lambda \Rightarrow (2n - 1)\frac{\lambda }{2} = 10(n = 1,\,\,2,\,\,3....)$
$\therefore $ Wavelength $\lambda = \frac{{2 \times 10}}{{(2n - 1)}} = \frac{{20}}{{2n - 1}}$.
The possible wavelength are ?$ = $$20,\,\,\frac{{20}}{3},\,\,\frac{{20}}{5}\,,\,\,\frac{{20}}{7}\,,\,\,\frac{{20}}{9}\,,$…..
$ = $ $20$$m$, $6.67$$m$, $4m,$$2.85\,m,$ $2.22$$m,$…..
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