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A plane longitudinal wave with a frequency of $ 1000{\text{ }}{\sec ^{ - 1}} $ is travelling along the positive x-direction in a homogeneous gaseous medium of density $ \rho = 1\,kg/{m^3} $ . The intensity of the wave is $ \;I{\text{ }} = {\text{ }}{10^{ - 10}}Wm $ and the maximum change in pressure is $ {(\Delta P)_m} = 2 \times {10^{ - 4}}\,N/{m^2} $ . Find the velocity of the sound wave.

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Last updated date: 16th Jun 2024
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Answer
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Hint: The intensity of the sound wave is related to the pressure amplitude of the longitudinal sound waves. The speed depends on the density of the medium and the speed of sound in the medium.

Formula used: In this solution, we will use the following formula
The intensity of a sound wave: $ I = \dfrac{{{{(\Delta P)}^2}}}{{2\rho v}} $ where $ \Delta P $ is the pressure amplitude, $ \rho $ is the density of the medium, and $ v $ is the velocity of the sound wave in the medium.

Complete step by step answer
Sound waves are longitudinal waves which mean they travel in regions of high and low pressure which is also known as pressure amplitudes. The regions of high pressures are called compressions of the medium and the regions of low pressure are called rarefactions.
The intensity of the sound wave is related to its velocity as
 $ I = \dfrac{{{{(\Delta P)}^2}}}{{2\rho v}} $
We can rewrite the above equation to find the velocity of sounds wave as
 $ v = \dfrac{{{{(\Delta P)}^2}}}{{2\rho I}} $
Now substituting the value of $ (\Delta P) = 2 \times {10^{ - 4}}\,N/{m^2} $ , $ \rho = 1\,kg/{m^3} $ and $ \;I{\text{ }} = {\text{ }}{10^{ - 10}}Wm $ , we get
 $ v = \dfrac{{{{(2 \times {{10}^{ - 4}}\,)}^2}}}{{2(1) \times {{10}^{ - 10}}}} $
Which gives us
 $ v = 200\,m/s $
Hence the velocity of the sound wave will be $ v = 200\,m/s $ .

Note
To answer such questions, we should be aware of the relation between the sound wave intensity and its pressure amplitude. The typical speed of sound in the air is $ 343\,m/s $ . The frequency of the sound wave is inconsequential to us since it won’t affect the velocity of the sound wave. Sound waves like all other longitudinal waves need a medium to travel and their velocity will change in different mediums, unlike transverse waves which don’t need a medium to travel.