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Two in-phase sources of waves are separated by a distance of 3.83 m. These sources produce...

Question:

Two in-phase sources of waves are separated by a distance of 3.83 m. These sources produce identical waves that have a wavelength of 5.90 m. On the line between them, there are two places at which the same type of interference occurs. (a) Is it constructive or destructive interference? (b) and (c) Where are these places located (the smaller distance should be the answer to (b)?

Superposition of Waves:

When two of more waves travelling simultaneously in a medium, the resultant displacement of a particle of medium is the vector sum of the displacement produces by each wave. Thus amplitude of resultant wave becomes maximum at some position and minimum at others. This phenomenon is called interference of waves. The maximum amplitude formation is know as constructive interference and the minimum amplitude formation is called destructive interference.

Answer and Explanation:

let

{eq}l_1 {/eq} and {eq}l_2 {/eq} be the distances between the source 1 and 2 respectively.

{eq}d=l_2-l_1=m \lambda \\ here\ \lambda=wavelength \\ m=0,1,2....... {/eq}

Condition of destructive interference .

{eq}d=l_2-l_1= (2m+1) \dfrac{ \lambda }{2} \\ here\ \lambda=wavelength \\ m=0,1,2....... {/eq}

Since the separation between the two sources is 3.83 m, value of d cannot be greater than 3.83 m. Therefore for wavelength of 5.9 m, the only way to have an constructive interference is at a place where

{eq}m=0 \\ \Rightarrow l_1=l_2 {/eq} which is possible at only one point. But it is given that there are two places at which the same type of interference occurs. Therefore we conclude that only destructive interference is occurring.

b)

separation between the two sources

{eq}D=l_2+l_1= 3.83 m ........................[1] {/eq}

Condition of destructive interference .

{eq}d=l_2-l_1= (2m+1) \dfrac{ \lambda }{2} \\ for\ m=0 \\ l_2-l_1=\dfrac{ \lambda }{2} =0.5*5.9=2.95\ m..........................[2] {/eq}

adding [1] and [2] we get

{eq}l_2=3.39\ m \\ l_1=0.44\ m {/eq}

so the two places are 0.44 m and 3.39 m from the source.


Learn more about this topic:

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Interference Patterns of Sound Waves

from MTEL Physics (11): Practice & Study Guide

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