6.1 The Wave Equation

Vibrating bodies, regions of turbulent flow, the mixing of flows of different temperatures, and other unsteady phenomena produce fluctuations in pressure that propagate through a real fluid as sound, that is, as a succession of compressions and rarefactions. In applications we are usually concerned with wave amplitudes that are small relative to the mean undisturbed pressure. Therefore, in a first approximation the equations of motion may be linearised and combined to form the wave equation for sound propagation. Sound waves are nondispersive in a homogeneous fluid over a broad range of frequencies for which viscous and thermal diffusion are small; in particular, this is true at audible frequencies, and it ensures that pressure fluctuations representing speech remain coherent over large distances.

The intensity of sound in air is usually measured on a decibel scale by the quantity

where the p is a measure of the characteristic acoustic pressure and p _ref = 2 10 ^?5 N/m ² is a standardised reference pressure. Thus p = p _o ? 1 atm (= 10 ⁵ N/m ²) is equivalent to 194 dB. A very loud sound, ~120 dB, corresponds to

Nonlinear effects begin to be important for deafening sounds of intensities ~160 dB, for which p/p _o ~ 0.02 and p ~ 0.3 lbs/in ².

We shall see that sound waves are longitudinal disturbances , such that the passage of a wave in the form of a pressure fluctuation...