mation, a single-valued relationship is established between pressure and density, and 

 there results a complete set of equations for the fluid motion in which the entropy 

 does not appear explicitly. 



The magnitude of the errors introduced both by an isentropic approximation 

 and by later use of a single-scattering approximation to solve the associated wave- 

 equation, may be estimated from the exact inhomogeneous wave equation for the pres- 

 sure field, the right side of which will involve density, pressure and velocity. One may 

 transform this equation to the corresponding integral equation and determine the con- 

 tributions of successive orders of an appropriate iteration procedure, using the other 

 equations of motion for relations between pressure, density, and velocity. 



For examination of a scattering process, the appropriate starting point for the 

 iteration could be zeroth order density, pressure, and velocity fields which are the 

 sums of the fields corresponding to an infinitesimal sound wave and very low Mach 

 number turbulence, together with the entropy field associated with these infinitesimal 

 disturbances. It is to be noted that the infinitesimal sound wave represents a solution 

 of the homogeneous wave-equation; because the integral equation for the pressure field 

 provides for explicit introduction of this homogeneous part of the solution (auto- 

 matically maintaining proper boundary conditions at infinity during iteration), it is a 

 more suitable instrument for investigating orders of magnitudes of correction terms 

 than the differential equation, for which boundary conditions at infinity must be taken 

 account of at every step. 



Now, if the amplitudes of sound wave and turbulence are sufficiently small, and 

 if the volume in which scattering takes place is sufficiently small, the first Born approxi- 

 mation — the solution resulting from a single iteration starting with the zeroth order 

 fields described — may be justified (R. H. Kraichnan, J. Acoust. Soc. Am. 25, 1096 

 (1953)). This implies that departures from isentropic conditions, and what is loosely 

 termed "multiple scattering" will both have small effect on the scattered wave, for 

 both these phenomena here show up in higher steps of the iteration procedure. 



The expression "sufficiently small" which I have used requires discussion. It 

 is notoriously difficult in physics to give compact and generally applicable sufficient 

 conditions for the validity of the first Born approximation. An obvious (but sometimes 

 ignored) necessary condition can be stated however: The difference between the exact 

 field and the zeroth order field must be small compared to both throughout the region 

 of scattering. When an applicable criterion of this sort is not satisfied, and the first 

 Born approximation breaks down, it is frequently meaningful to ask to what extent the 

 breakdown is due to nonisentropic conditions or to inappropriate procedure in solving 

 the isentropic equations. The two effects (each represented by terms in the iteration 

 expansion of the source function for the pressure field) are often fairly distinct 

 physically. 



For a simple example, consider a plane sound wave propagating through a 

 very large region of homogeneous turbulence. If the turbulence is very weak, the error 

 in the first Born approximation may be ascribable principally to a non-isentropic effect 

 — the attenuation of the sound wave by viscosity. If the turbulence is stronger, how- 

 ever, the attenuation of the plane sound wave may be due principally to energy scattered 

 out of the wave by the turbulence, and an accurate description of this could be 

 obtained by applying better scattering approximations to the isentropic equations. In 

 either of these simple cases, provided there is small attenuation within the correlation 

 length of the turbulence, a better description may be obtained by using a "doctored" 

 Born approximation in which the zeroth order sound field is exponentially damped. 

 (This may be considered the first Born approximation to an inhomogeneous wave 

 equation in which the homogeneous part contains a damping term.) 



On the other hand, in the interaction of intense sound waves with strong turbu- 

 lence, as in a high-speed turbulent jet, it is not very meaningful to speak separately of 

 non-isentropic effects and multiple scattering effects. In this case, in fact, the distinction 



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