between sound and turbulence is unclear, although one may distinguish a transverse 

 velocity and a longitudinal velocity. What is most important to keep in mind is that 

 generalizations about the validity of a particular scattering approximation are risky, 

 and the self-consistency of a given approximation should be investigated for the appli- 

 cation at hand. For example, in certain applications the scattering due to turbulent 

 density fluctuations may be negligible, in others not. 



Dr. Batchelor noted some similarities between scattering by random scalar 

 inhomogeneities (scattering of radio and light waves by turbulent refractive index 

 fluctuations) and the scattering of sound by turbulence. I should like to stress that there 

 are essential differences in the range of validity of the Born approximation in the two 

 cases. In the scalar case, given a size of the scattering volume, the first Born approxi- 

 mation can remain valid at indefinitely high frequency if the scalar index fluctuations 

 are small. In the sound-scattering case, the interaction between sound and turbulence 

 involves quadratically the gradients of the velocities, thereby leading to a strong diverg- 

 ence of the first Born approximation cross-section toward short sound wave-lengths. 

 By the criterion mentioned previously, this approximation is therefore not valid for 

 indefinitely short wave-lengths; for example, it cannot properly be applied to the scatter- 

 ing of energy out of a shock-wave traversing turbulence. In some cases, the sound- 

 scattering at very short wavelengths is satisfactorily described by a "physical optics" 

 approximation in which the corrugation of coherent wavefronts is computed. Here 

 again, safe criteria of validity are not easy to state. 



Now I should like to turn briefly to the effects of time-variation of the 

 turbulence. Although in most cases where electromagnetic or sound waves are scattered 

 by turbulence the characteristic frequencies of the turbulence are much smaller than 

 the wave frequency, it does not follow that the effect of the time variation is unimpor- 

 tant. Slow time dependence will not alter the time-average scattering cross-sections, 

 but it will alter the time-structure and spectrum of the scattered radiation. In general, 

 the scattered radiation will be smeared over a frequency bandwidth range the order of 

 the characteristic frequencies of the turbulence, and it will exhibit fading times the 

 order of the characteristic period of the turbulence. In the scattering of sound by 

 turbulence the spreading in frequency of the first Born approximation scattered wave 

 has an interesting angular dependence; the energy scattered in nearly forward directions 

 is very little spread in frequency, compared to radiation at larger scattering angles. 



Further effects of the time-structure of the turbulence include phenomena of 

 energy exchange between wave field and turbulence. For electromagnetic waves, these 

 effects are not of much interest in nonionized media, but they may be significant for 

 the propagation of sound through turbulence, especially in large regions. The energy 

 exchange has two aspects: radiation of broad-band acoustic noise by the turbulence, 

 and transformation of energy from the incident sound field into turbulent kinetic 

 energy (that is to say, the scattering is not exactly "elastic," or conservative) . The 

 relative magnitude of the two effects depends on the ratio of turbulence and sound 

 particle velocity Mach numbers, and on the spectral structure of the fields. 



Both of these effects tend to degrade an initially monochromatic sound field 

 by increasing the ratio of energy in diffuse spectrum to that in the line spectrum. For 

 strong enough turbulence, the noise radiation may actually contain much more energy 

 than the scattered sound wave, although the latter may still be distinguishable because 

 of its sharper frequency structure. When Mach numbers of both sound and turbulence 

 are high, the distinction between scattered and radiated energy is not very meaningful. 



Finally I should like to comment on Dr. Mintzer's proposal for investigation of 

 turbulence structure by scattering experiments. It is of interest in this connection that 

 different information would be obtainable in principle by the scattering of electro- 

 magnetic or acoustic waves in the same wavelength range, because the coupling is 

 scalar in one case and tensor in the other. 



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