Lighthill's eighth power law

In aeroacoustics, Lighthill's eighth power law states that power of the sound created by a turbulent motion, far from the turbulence, is proportional to eighth power of the characteristic turbulent velocity, derived by Sir James Lighthill in 1952.^[1]^[2] This is used to calculate the total acoustic power of the jet noise. The law reads as

W=K{\frac {\rho _{o}}{c_{o}^{5}}}L^{2}U^{8},

where

$W$ is the acoustic power in the far-field,
$K$ is the proportionality constant (or Lighthill's constant),
$\rho _{o}$ is the uniform fluid density,
$c_{o}$ is the speed of sound,
$L$ is the characteristic length scale of the turbulent source and
$U$ is the characteristic velocity scale of the turbulent source.

The eighth power is experimentally verified and found to be accurate for low speed flows, i.e., Mach number is small, $M<1$ . And also, the source has to be compact to apply this law.

References

^ Lighthill, M. J. (1952, March). On sound generated aerodynamically I. General theory. In Proc. R. Soc. Lond. A (Vol. 211, No. 1107, pp. 564–587). The Royal Society.
^ Lighthill, M. J. (1954, February). On sound generated aerodynamically. II. Turbulence as a source of sound. In Proc. R. Soc. Lond. A (Vol. 222, No. 1148, pp. 1–32).

[1] Lighthill, M. J. (1952, March). On sound generated aerodynamically I. General theory. In Proc. R. Soc. Lond. A (Vol. 211, No. 1107, pp. 564–587). The Royal Society.

[2] Lighthill, M. J. (1954, February). On sound generated aerodynamically. II. Turbulence as a source of sound. In Proc. R. Soc. Lond. A (Vol. 222, No. 1148, pp. 1–32).

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