# Displacement Wave , Pressure Wave

### Displacement wave & Pressure waves:

A longitudinal sound wave can be expressed either in terms of the longitudinal displacement of the particles of the medium or in terms of excess pressures produced due to compression or rarefection.

The first type is called the displacement wave and the second type the pressure wave.

The displacement wave can be described by the equation :

y = A sin (ωt − kx)

### Pressure Wave

Consider the element of medium which is confined within x and x + Δx in the undisturbed state.

If S is the cross section, the volume of the element in undisturbed state will be V = S Δx.

As the wave passes, the ends at x and x + Δx are displaced by amount y and y + Δy.

⇒ Increase in volume ΔV = SΔy

⇒ Volumetric Strain , $\displaystyle \frac{\Delta V}{V} = \frac{S\Delta y}{S \Delta x}= \frac{\Delta y}{\Delta x}$

⇒ Excess pressure , $\displaystyle P = -B\frac{\Delta V}{V} = -B\frac{\Delta y}{\Delta x}$

Here B is the bulk modulus of elasticity or we can write-

$\displaystyle P = -B\frac{dy}{dx}$

if y = A sin (ωt − kx )

Then , $\large \frac{dy}{dx} = – A k cos(\omega t – k x)$ at a certain time t.

⇒ P = A k B cos (ωt − k x)

⇒ P =  Po cos (ωt − k x) ; Where Po = A k B

So, the following conclusions may be drawn from the above result.

(a) If the displacement wave is represented by

Y = A sin (ωt − kx) , then the corresponding pressure wave will be represented by

P = Po cos (ωt − kx) ;

Where Po = ABk and $\displaystyle P = -B\frac{dy}{dx}$

(b) Pressure wave is (π/2) out of phase with displacement wave. i.e. pressure is maximum when displacement is minimum and vice-versa.

(c) The amplitude of the pressure wave is Po = A k B

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