analogous situation in water, since the mechanisms involved are fundamentally the 

 same. However, because of the greater density and lower speeds encountered in water, 

 the relative importance of various phenomena can be different. The present brief 

 review will be limited to those aspects of the problem of greatest interest in connection 

 with flowing water. 



Recognition of the association of certain sounds with unsteady flow goes back 

 to the nineteenth century. Rayleigh, for example, attributed the "aeolian tones" made 

 by wind streaming past a wire to the vortex wake behind the wire; and the "jet-edge 

 tones" of whistles were also believed to result from fluctuations in the jet. Richard- 

 son [39] has described the early investigations of these sounds. These investigations 

 were concerned primarily with the frequency of the sound. No serious attempt seems 

 to have been made to understand the factors controlling the intensity of the sound. 



A theoretical basis for investigating the magnitude of the sound pressure asso- 

 ciated with unsteady motion was provided by Lighthill [40] in 1950. He discarded 

 the classical form of the acoustic wave equation, which is admittedly only a small- 

 amplitude approximation, and rederived an exact wave equation from the exact equa- 

 tions of fluid motion. This exact wave equation is inhomogeneous and contains terms 

 which represent sources of sound associated with fluctuating velocities. For a liquid, 

 the essential terms in Lighthill's equation are 



1 d 2 p d 2 (UiUj) 

 V 2 p =- p f (21) 



c 2 dt 2 dxidxj 



where p is the pressure, c the velocity of sound, u i the i-th component of particle 

 velocity, x h the i-th space coordinate, V 2 = d 2 /dx i dx i and a repeated index indicates 

 summation in accordance with tensor notation. If the velocity components are suffi- 

 ciently small, i.e., if pMjMj <^.p s , then the right side of the equation is negligible and 

 the classical homogeneous wave equation results. If, however, the velocities are not 

 negligible, the right side represents sources of sound. 



The fluctuating pressure in an unbounded fluid can be related to the fluctuating 

 velocities by a volume integral over all space of the right side of Eq. (21). Thus 



p s (x,t) = - G&r) / (l/r)q(x',t') dV, (22) 



where p s (x, t) is the instantaneous sound pressure at time t and position x in space 

 (jc without subscript indicates the three coordinates x v x 2 , x 3 ). In the integral, r is 

 the distance between the point x and the volume element dV at position x', and q(x f , t') 

 is the value of the right side of Eq. (21) at position x' and earlier time t' — t-r/c. 

 In the application of this solution, the fluctuating velocities constituting q(x, t) are 

 considered to be independent variables whose values are given as part of the descrip- 

 tion of the flow. 



In the usual unsteady flow, the fluctuating component of the velocity differs 

 significantly from zero in only a limited portion of the unbounded space. The asso- 

 ciated pressure fluctuations are then very much larger directly within the unsteady 

 region than outside it. 



At a point within the unsteady region, the fluctuating pressure depends primarily 

 on the values of q at points in the immediate vicinity, corresponding to small values 

 of r. In this vicinity, the difference r/c between the time t and earlier time f is negli- 

 gible. Since the sound velocity c does not appear in Eq. (22) in any other way, the 

 pressure fluctuations are independent of the sound velocity. This implies that the 

 fluctuations inside the unsteady region do not depend on the compressibility of the 

 medium. 



Far from the unsteady region, at distances large compared with the dimensions 

 of the region, the situation is different. If the compressibility of the fluid is neglected 

 in the calculation for these distances also, the magnitude of the pressure fluctuations is 

 found to decrease very rapidly with increasing distance, as r~ 3 . If the difference between 



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