Appendix 4 Derivation of the Time-dependent Fourth-order Covariance in a Homogeneous Flow. Z Z Z Dr @ 4 @t 2 a @t 2 B

xx ? < T 0 xx >) > < T xx > 2 ; (A4:1) where P is the noise source correlation function, and we have made the assumption that < T xx > is equal to < T 0 xx >. Here T xx is the aligned component of T ij in the direction between the source and the distant observer, and the constant density term has been removed. P involves the fourth order covariance between the ow properties at two separated points in space and time. Now the acoustic power output from isotropic turbulence per unit volume in near incompressible ow at low Mach numbers is, p s (x; t : t) = 1 At low Mach numbers the frequency, , of noise at the observer, is equal to the frequency, !, of noise at the source, and then for the case of stationary ow: