On how to obtain diffuse field sound transmission loss from Cremer's thin plate transmission coefficient formula

The transmission coefficient for a single plane wave sent through a thin infinite homogeneous isotropic plate separating two fluid media has been derived by Cremer in 1942. Cremer's classical formula is converted into two distinct compact representations revealing the genuine dependencies on material parameters including damping. It is pointed out that the calculation of the diffuse sound field transmission loss is by no means trivial, and a safe and efficient method of calculation is presented. Furthermore, the asymptotical behaviour of the diffuse sound field transmission loss is worked out for both high and low frequencies, generalising Cremer's results with respect to arbitrary maximum incidence angle and general damping. With an adequate expansion a simple analytic estimate for the maximum of the transmission loss and its position has been found. An approximate formula valid for all frequencies is constructed on the basis of the low- and high-frequency expansions. Analytical approximations for all local extremal values and their positions have been generated by a numerical procedure. Finally generalisations for anisotropic plates and distinct fluid media are described. The combination of the suggested exact method with standard integrators enables safe calculation of the transmission loss of anisotropic plates including third-octave or octave frequency averages.