Energy conservation via coupled modes in waveguides with an impedance boundary condition

A statement of energy conservation for a coupled mode formulation with real mode functions and eigenvalues has been demonstrated to be consistent with the statement of conservation derivable from the Helmholtz equation. The restriction to real mode functions and eigenvalues precludes coupled mode descriptions with waveguide absorption or untrapped modes. The demonstration, along with the derivation of the coupled mode range equation, relies on orthonormality in terms of a product of two modal depth functions integrated to infinite depth. This paper shows that energy conservation and the derivation of the coupled mode range equation can be extended to complex mode functions and eigenvalues, and that energy is conserved for ocean waveguides with a penetrable bottom boundary at a finite depth beneath any range dependence. For this, the penetrable bottom boundary is specified by an impedance condition for the mode functions. The new derivations rely on completeness and a modified orthonormality statement. Mode coupling is driven solely by waveguide range dependence. Thus, the form of the range equation and the values of the coupling coefficients are unaffected by a finite depth waveguide. Applications of energy conservation to examine the accuracy of a numerical coupled mode calculation are presented.