This is a part of a sound wave that moving to the right (positive x-direction).

Question

This is a part of a sound wave that moving to the right (positive x-direction). A difference in pressure will cause it to accelerate. The molecules on the left side have a different velocity than the molecules on the right. Divide each side of the equation (as ma = -A Delta P, so ea = -Delta P/Delta x ; where e is density) a by the length of the small cylinder of air to obtain a relationship between a spatial variation in pressure to a variation of velocityover time: where U is the velocity of the air.

Explanation / Answer

The mass of small element = . A .dx

so m. a =- (p(x + x,t ) - p (x,t)) * A

now acceleration is differential of velocity wrt time

so a = U /t

so . U/t =- (p(x+ x,t) - p(x,t))/dx

so U/t = - p/x

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