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On the energy scattered from the interaction of turbulence with sound or shock waves

Published online by Cambridge University Press:  24 October 2008

M. J. Lighthill
M. J. Lighthill
Affiliation:
Department of MathematicsThe University Manchester
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Abstract

The energy scattered when a sound wave passes through turbulent fluid flow is studied by means of the author's general theory of sound generated aerodynamically. The energy scattered per unit time from unit volume of turbulence is estimated (§3) as

where I is the intensity and ∧ the wave-length of the incident sound, and is the mean square velocity and L1 the macro-scale of the turbulence in the direction of the incident sound. This formula does not assume any particular kind of turbulence, but does assume that ∧/L1 is less than about 1. For turbulence which is isotropic and homogeneous, the energy scattered, and its directional distribution, are obtained for arbitrary values of ∧/L1. It is predicted that components of the turbulence with wave-number κ will scatter sound of wave-number K at an angle 2 sin−1 (k/2k). The statistics of multiple successive scatterings is considered (§4), and it is predicted that sound of wave-length less than the micro-scale λ of the turbulence will become uniform (i.e. quite random) in its directional distribution in a distance approximately .

The theory is extended (§5) to the case of an incident acoustic pulse. However, this extended theory cannot be applied directly to the case of a shock wave, for which it would predict infinite scattered energy. This is due to the perfect resonance between successive rays emitted forwards which would occur if the shock wave were propagated at the speed of sound. By taking into account (§6) the true speed of the shock wave (subsonic relative to the fluid behind it), the theory is improved to give a finite value, 0·8s tunes the kinetic energy of the turbulence traversed by a weak shock of strength s, for the total energy scattered. However, the greater part of this energy catches up with the shock wave, and probably is mostly reabsorbed by it, and only the remainder (tabulated as a function of s in Table 1) is freely scattered, behind the shock wave, as sound. The energy thus freely scattered when turbulence is convected through the stationary shock-wave pattern in a supersonic jet may form an important part of the sound field of the jet.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 3 , July 1953 , pp. 531 - 551
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

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