Elementary solutions

The homogeneous scalar wave equation (49) satisfies the plane wave solution:

p’ = F (n ■ x — c0t) (50)

with П as the unit vector in the direction of propagation. This can easily be verified for П = (1, 0,0), in which case the wave equation (45) reduces to:

1 d [1] [2]p’ d 2p’

c2 dt2 dx2

Using the chain rule we can verify that p’ = F(x — c0t) is a solution. The function F(x) is determined by initial and boundary conditions. Also p’ = G(x + c0t) is a solution, representing a wave propagating in the opposite direction П = ( —1, 0, 0). For harmonic waves with a frequency f we can write this solution with the complex notation as:

p = A exp
iw	A exp І	wt — к ■ x	(52)

where A is the complex amplitude, к = (w/c0)n the wave vector and w = 2nf. Substitution of the plane wave solution into the momentum equation (45)with f = 0 yields:

u’ = —— П. (53)

Po co

Another elementary solution is obtained by considering spherical symmetric waves emanating from a point at source y. The pressure field is then only a function of time and of distance r = x — y between the source position f and the observer’s position x. The mass conservation law and momentum equation reduce to:

dP + po d (r2 =0

dt r2 dr dr j

and

dVr + dp = 0 Pod + dr = 0

(55)

where v’ is the fluid velocity in the radial direction. Eliminating the velocity and the density p’ = p’/c^ yields:
which is satisfied by the one-dimensional d’Alembert solution for the prod­uct of pressure p’ and distance r:

p’ = – F(r — cot) . (57)

r

By using this equation, we actually assume “free field” conditions. We assume that there are only outgoing waves and no incoming (or reflected) waves converging towards the source. For harmonic waves equation (57) becomes in complex notation:

A

p’ = — exp [i(ut — kr)] (58)

r

with k = ш/co. The corresponding radial velocity is found by substitution in the momentum equation:

We observe that for large distances compared to the wave length kr >> -, the solution can locally be approximated by a plane wave with: p’ = p0c0v’r. In this so-called “far field” approximation we have:

dp’ – dp’

dr c0 dt

In the opposite limit of near field kr << – the velocity varies quadrati – cally with the distance r, which is typical for the incompressible flow from a point volume source. Whenever characteristic flow dimensions are small compared to the wave length we can neglect wave propagation. Such a flow is called a “compact” flow.

Using these results (58-59) we can now consider the sound radiated by a pulsating sphere of radius

a = ao + a exp(iwt)

where a/a0 << – and Fa/c0 << -. Substituting (6-) into (59) and using (58) we find:

. л A exp(—ikao)F – , .

iwa = ———————- – + —— (62)

Pocoao V ikao J

and

This result shows that in the limit kaQ << 1 for a given volume flux ampli­tude 4^a2 wa, the amplitude of the radiated sound wave increases linearly with the frequency. At low frequency the pulsating sphere is compact and is a very inefficient source of sound. In the opposite limit kaQ >> 1 the radiated amplitude is independent of the frequency.