First slide

Superposition of waves

Question

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

Difficult

A

20, 20/3, 20/5 etc
B

10, 5, 2.5 etc
C

10, 20, 30 etc
D

15, 25, 35 etc

Solution

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 $∠ SMA = 90^{o}$
Path difference between waves $Δx = SAP - SMP$
We have $SAP = SA + AP = 2 (SA)$

$= 2 \sqrt{[{(SM)}^{2} + {(MA)}^{2}]} = 2 \sqrt{(60^{2} + 25^{2})} = 130 m$
$∴$ Path difference = SAP – SMP = 130 -120 = 10 m
Path difference due to reflection from ceiling = $\frac{λ}{2}$
$∴$ Effective path difference $Δx = 10 + \frac{λ}{2}$
For constructive interference
$Δx = 10 + \frac{λ}{2} = nλ \Rightarrow (2 n - 1) \frac{λ}{2} = 10 (n = 1, 2, 3 ....)$
$∴$ Wavelength $λ = \frac{2 \times 10}{(2 n - 1)} = \frac{20}{2 n - 1}$ . The possible wavelength are $λ = 20, \frac{20}{3}, \frac{20}{5}, \frac{20}{7}, \frac{20}{9},$ …..
= 20 m, 6.67 m, 4 m, 2.85 m, 2.22 m, …..

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