Reciprocity in the wave reflection and transmission problem

Abstract

Matrices of reflection and transmission coefficients of plane waves in media or normal waves in waveguides are, in general case, not symmetric. When all the waves are of the propagating type, the matrix can be symmetrized by normalizing the wave amplitudes with the wave power flow. For evanescent (inhomogeneous) waves this is not valid because of their zero power flow. This difficulty is overcome in the present paper. The main result is that a reflection and transmission matrix becomes symmetric if the wave amplitudes are normalized with the certain energy‐like quantity that coincides with the power flow in the case of propagating waves. The result is valid for all types of waves (including those with complex wave numbers) and for all media and waveguides where the classical reciprocity theorem is valid. All symmetry relations, known in the literature for reflection and transmission coefficients follow from the result as particular cases. The result is useful in analysis of multimodal sound fields in composite media. It is illustrated in examples with inhomogeneous waves in fluids and solid waveguides.