Waves in Fluids



Waves in Fluids. WWM Chapter 6, etc. January 8, 2006

First a derivation of the acoustic wave equation based on bulk modulus, and stress and strain.

We first recall that the bulk modulus is the change in pressure over the fractional decrease in volume

B = (P/(-(V/V) . (An increase in pressure decreases the volume)

(P = P-Po is the acoustic pressure p. Next from our study of 1-D waves on bars that strain = lim (x(0 ((xs - (x)/(x) = ((/(x, where ( is the displacement in x. This could be further written as

strain = (Lx - Lxo)/Lxo = ((/(x , or

Lx = Lxo (1+((/(x) .

(Fluids do not support shear stresses, so we do not have to worry about stresses in y and z causing further strain in x. )

Then by considering a displacement in y of (, we would find

Ly = Lyo (1+((/(y) .

Doing the same thing for a displacement ( in z, we would arrive at

LxLyLz = LxoLyoLzo (1 + ((/(x)(1 +((/(y)(1+((/(z) , or (since the partials are small)

V ( Vo (1 + ((/(x+((/(y+((/(z), and from here that

(V/Vo = (((/(x+((/(y+((/(z)

Combining this with the definition of p and B at the top of the page we find

(((/(x+((/(y+((/(z) = p/B.

Then we apply F = ma in x to arrive at -(p/(x = ( (2(/(t2 , and by taking the x-derivative

-(2p/(x2 = ( (2/(t2 (((/(x) .

Combining this with y and z directions, and applying (((/(x+((/(y+((/(z) = p/B we get

(2p = (/B (2p/(t2, the acoustic wave equation .

On the next page we will derive the acoustic wave equation in terms of fluid concepts involving flow, gradient, divergence, and so on. This is patterned after a treatment in Kinsler, Frey, Coppens, and Sanders.

( = displacement from equilibrium, u= velocity = ((/(t,

s = 'condensation' = fractional change in density = ((-(o)/ (o (s taken to be = < po exp (i k(r - i(t) uo exp (i k(r - i(t) >

The understanding with complex notation is that that at the end, we will take the real part.

Re(p) = po cos (k(r - (t) [po understood to be real] and likewise for u. There is in general a phase difference between p and u. In the equations above the phase difference is (. To shorten the notation we will take (1 = k(r and (2 = k(r + (.

The time average is

< I > = pouo (1/T) 0( T dt [( cos ((1-(t) cos((2-(t) )]

< I > = pouo (1/T) 0( T dt [( cos (1 cos (t + sin (1 sin (t) (cos (2 cos (t + sin (2 sin (t) )]

The time average of sin (t cos (t vanishes, and the time averge is 1/2 of both cos2(t and sin2(t, so

= 1/2 pouo ( cos (1 cos (2 + sin (1 sin (2 ) = 1/2 pouo cos((2-(1) = 1/2 pouo cos(()

We have been aiming to show that the time-averaged intensity can just as well be written as

= 1/2 Re (pu*),

where u* is the complex conjugate of u. In this form both p and u are written in complex notation:

= 1/2 Re (po exp(i(k(r-(t)) uo* exp(-i(k(r-(t + ()) .

This reduces to the previous equation. Finally we can substitute for u in terms of p:

= 1/2 |po|2/((c) . time-average intensity, plane wave

Energy density (energy/volume) {Section 13.10, p. 420}.

In travelling waves, sound waves, or light waves or

others, we can think of energy being transported through

space at the speed of propagation of the wave. We could

imagine plane waves moving into a region without waves. A

Then a cylindrical region of volume dV = A (c dt) would

move a distance cdt into the 'unoccupied region', carrying

energy across a boundary at a rate P = energy/time. cdt

We'll call ( the energy per unit volume so the energy within the volume is dE = ( A cdt,. The energy per unit area per unit time is the intensity so we find I = 1/A dE/dt = (c . This tells us that the energy density is

( = I/c Energy density = Intensity/velocity.

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