DILACEMENT WAVE AND PRESSURE WAVE

A longitudinal wave in a fluid (liquid or gas) can be described either in terms of the longitudinal dilacement suffered by the particles of the medium or in terms of the excess pressure generated due to the compression or rarefaction. Let us see how the two representations are related to each other.

Consider a wave going in the x-direction in a fluid. Suppose that at a time t, the particle at the undisturbed position x suffers a dilacement s in the x-direction. The wave can then be described by the equation.

s = s₀ sin$\omega$(t – x / v) ….(i)

Consider the element of the material which is contained within x and x + $\Delta$x (figure) in the undisturbed state. Considering a cross-sectional area A, the volume of the element in the undisturbed state is A, the volume of the element in the undisturbed state is A$\Delta$x and its mass is $\rho$ A$\Delta$x. As the wave passes, the ends at x and x + $\Delta$x are dilaced by amounts s and s + $\Delta$s according to equation (xv) above. The increase in volume of this element at time t is

$\Delta V=A\Delta s=A\frac{\delta s}{\delta x}.\Delta x$

= As₀(–$\omega$/v) cos(t–x/v)$\Delta$x,

Where $\Delta$s has been obtained by differentiating equation (xv) with reect to x. The element is, therefore, under a volume strain.

$\frac{\Delta V}{V}=\frac{-A{x}_0\omega \cos \omega \left(t-x/v\right)\Delta x}{vA\Delta x}$

$=\frac{-s_0\omega }{v}\cos \omega \left(t-x/v\right)$

The correonding stress i.e., the excess pressure developed in the element at x at time t is,

$p=B\left(\frac{-\Delta V}{V}\right)$,

Where B is the bulk modulus of the material. Thus,

$p=B\frac{s_0\omega }{v}\cos \omega \left(t-x/v\right)$ …… (xvi)

Comparing with (xv), we see that the pressure amplitude P₀ and the dilacement amplitude s₀ are related as

$P_0=\frac{B\omega }{v}{s}_0=Bk{s}_0$,

Where k is the wave number. Also, we see from (xv) and (xvi) that the pressure wave differs in phase by $\pi$/2 from the dilacement wave. The pressure maxima occur where the dilacement is zero and dilacement maxima occur where the pressure is at its normal level.

The fact that, dilacement is zero where the pressure-change is maximum and vice versa, puts the two descriptions on different footings. The human ear or an electronic detector reonds to the change in pressure and not to the dilacement in a straight forward way. Suppose two audio eakers are driven by the same amplifier and are placed facing each other. A detector is placed midway between them.

The dilacement of the air particles near the detector will be zero as the two sources drive these particles in opposite directions. However, both send compression waves and rarefaction waves together. As a result, pressure increases at the detector simultaneously due to both the sources. Accordingly, the pressure amplitude will be doubled, although the dilacement remains zero here. A detector detects maximum intensity in such a condition. Thus, the description in terms of pressure wave is more appropriate than the description in