Application of singular eigenfunction expansions to the propagation of periodic disturbances in a radiating grey gas

Abstract

The effect of thermal radiation on the propagation of small disturbances generated by the harmonic oscillation of a planar wall, either in position or temperature or both, is considered. By separation of variables, it is shown that the governing linearized equations can be reduced to a homogeneous integral equation that admits both regular and singular solutions to form a complete set; thus, the problem can be solved by the application of singular eigenfunction expansions analogous to those used in neutron-transport theory. An exact closed form solution is obtained for disturbed quantities in the flow and radiation fields. The exact solution shows that the disturbances consist of a damped continuum mode with infinite wave speed in addition to two discrete modes with finite wave speed. One of the discrete modes represents the "modified classical" wave whereas the other discrete mode plus the continuum mode corresponds to the radiation-induced wave. For sufficiently small Bouguer numbers, the discrete mode of the radiation-induced wave disappears. The newly found continuum mode damps faster than the discrete mode. Consequently, only the discrete modes persist away from the boundary. The differential approximation is found to predict the modified classical wave accurately but predicts the radiation-induced wave only approximately. The implications as well as the accuracy of the differential approximation are discussed and compared.