In a large room,a person receives direct soun | vedclass

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(A) Let $S$ be the source of sound and $P$ be the person or listener.The waves from $S$ reach point $P$ directly following the path $SP$ and also afte

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$20, 20/3, 20/5$ etc.

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(A) Let $S$ be the source of sound and $P$ be the person or listener.The waves from $S$ reach point $P$ directly following the path $SP$ and also after being reflected from the ceiling at point $A$ following the path $SAP$.$M$ is the midpoint of $SP$ (i.e.,$SM = MP = 60 \ m$) and $\angle SMA = 90^\circ$.The path length of the direct wave is $SP = 120 \ m$.The path length of the reflected wave is $SAP = SA + AP = 2 	imes SA$.In $	riangle SMA$,$SA = \sqrt{SM^2 + MA^2} = \sqrt{60^2 + 25^2} = \sqrt{3600 + 625} = \sqrt{4225} = 65 \ m$.So,$SAP = 2 	imes 65 = 130 \ m$.The geometric path difference is $\Delta x_{geom} = SAP - SP = 130 - 120 = 10 \ m$.Since the wave is reflected from a rigid ceiling,there is an additional phase shift of $\pi$,which corresponds to a path difference of $\lambda/2$.Thus,the effective path difference is $\Delta x = 10 + \lambda/2$.For constructive interference,the effective path difference must be an integer multiple of $\lambda$,i.e.,$\Delta x = n\lambda$ (where $n = 1, 2, 3, \dots$).$10 + \lambda/2 = n\lambda \Rightarrow 10 = (n - 1/2)\lambda = (2n - 1)\lambda/2$.$\lambda = 20 / (2n - 1)$.For $n = 1, 2, 3, \dots$,the wavelengths are $\lambda = 20/1, 20/3, 20/5, \dots$ or $20, 20/3, 20/5, \dots \ m$.

In a large room,a person receives direct sound waves from a source $120 \ m$ away from him. He also receives waves from the same source which reach him after being reflected from the $25 \ m$ high ceiling at a point halfway between them. The two waves interfere constructively for wavelengths of

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