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Mechanical Properties Of Fluids

Question

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A source emitting sound of wavelength 2m is placed in front of a wall 2m from it. A detector is also placed in front of the wall at a distance 2m on the same side. Find the minimum distance between source and detector to hear the maximum sound.

Solution

The question is illustrated in the figure below.
Let the minimum distance between source and detector to hear the minimum sound be 2x.

The path travelled by reflected wave is AO + OB.
The path travelled by direct wave is AB.
After reflection, the wave suffers a phase change of π radian which is equivalent to an extra path of λ/2.
The net path difference between two waves reaching the detector is,

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equals 2 AO minus 2 straight x plus straight lambda over 2 equals 2 left parenthesis AO minus straight x right parenthesis plus straight lambda over 2

equals 2 left parenthesis square root of straight x squared plus 4 end root minus straight x right parenthesis plus straight lambda over 2

For maximum intensity at B, the value of

increment

should be

nλ.

For AB to be minimum

increment

should be

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i.e.

increment equals 2 left parenthesis square root of straight x squared plus 4 end root minus straight x right parenthesis plus straight lambda over 2 equals straight lambda

rightwards double arrow

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rightwards double arrow

space space space 2 left parenthesis square root of straight x squared plus 4 end root minus straight x right parenthesis space equals space 1

rightwards double arrow

2 straight x space equals space 7.5 straight m

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