Acoustic momentum and energy theorems for piecewise uniform ducts

Abstract

The problem of sound propagation in a constant-area duct with compliant walls is studied, including a uniform mean flow. The wall characteristics are assumed to be piecewise constant. Energy conservation leads to a unitary scattering matrix. A second relation of similar type is derived from momentum conservation which is, however, violated at the wall discontinuities, thereby introducing additional dyadic terms into this relation. The rank of the commutator of the scattering matrix and a wavenumber matrix is shown to be related to the number and type of the wall discontinuities. Explicit results are obtained if this rank is one. As an example, sound propagation in a duct with a semi-infinite splitter plate is discussed.