. Analogy of Curle

The analogy of Curle (1955) is the integral formulation (88) applied to Lighthill’s analogy (42) in terms of density fluctuations:

The observer is placed within the control volume V over which we carry out the integration. This equation is based on the assumption that at the listener’s position p’ = c2p’. We will further ignore the contribution from the external force field (f = 0). By means of partial integration we move

the space derivatives from the source terms towards the Green function:

Using the definition of the viscous stress tensor (26) and the momentum equation (9) we can write (102) in the form:

n

dG

dyi nidSy

Furthermore we neglect entropy fluctuations on the surface S.

By means of partial integration we move the time derivative in the sec­ond integral from the momentum flux to the Green’s function. Using the symmetry relations of the free field derivative with respect to space (85) and time derivatives (84), we find in the far field approximation (60):

In (104) we recognize the monopole sound production due to the volume flux leaving the surface (first integral), the dipole field generated by the force acting on the surfaces and the quadrupole field generated by fluctuations of the Reynolds stress tensor in the volume.