# Physics: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

$\displaystyle{ W = K \frac{\rho_o}{c_o^5} L^2 U^8, }$

where

• $\displaystyle{ W }$ is the acoustic power in the far-field,
• $\displaystyle{ K }$ is the proportionality constant (or Lighthill's constant),
• $\displaystyle{ \rho_o }$ is the uniform fluid density,
• $\displaystyle{ c_o }$ is the speed of sound,
• $\displaystyle{ L }$ is the characteristic length scale of the turbulent source and
• $\displaystyle{ U }$ 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, $\displaystyle{ M\lt 1 }$. And also, the source has to be compact to apply this law.

## References

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).