An orthogonality relation for a class of problems with high-order boundaryconditions; Applications in sound-structure interaction

There are numerous interesting physical problems, in the fields of elastici ty, acoustics and electromagnetism etc., involving the propagation of waves in ducts or pipes. Often the problems consist of pipes or ducts with abrup t changes of material properties or geometry. For example, in car silencer design, where there is a sudden change in cross-sectional area, or when the bounding wall is lagged. As the wavenumber spectrum in such problems is us ually discrete, the wavefield is representable by a superposition of travel ling or evanescent wave modes in each region of constant duct properties. T he solution to the reflection or transmission of waves in ducts is therefor e most frequently obtained by mode-matching across the interface at the dis continuities in duct properties. This is easy to do if the eigenfunctions i n each region form a complete orthogonal set of basis functions; therefore, orthogonality relations allow the eigenfunction coefficients to be determi ned by solving a simple system of linear algebraic equations. The objective of this paper is to examine a class of problems in which the boundary conditions at the duct walls are not of Dirichlet, Neumann or of i mpedance type, but involve second or higher derivatives of the dependent va riable. Such wall conditions are found in models of fluid-structural intera ction, for example, membrane or plate boundaries, and in electromagnetic wa ve propagation. In these models the eigenfunctions are not orthogonal, and also extra edge conditions, imposed at the points of discontinuity, must be included when mode matching. This article presents a new orthogonality rel ation, involving eigenfunctions and their derivatives, for the general clas s of problems involving a scalar wave equation and high-order boundary cond itions. It also discusses the procedure for incorporating the necessary edg e conditions. Via two specific examples from structural acoustics, both of which have exact solutions obtainable by other techniques, it is shown that the orthogonality relation allows mode matching to follow through in the s ame manner as for simpler boundary conditions. That is, it yields coupled a lgebraic systems for the eigenfunction expansions which are easily solvable , and by which means more complicated cases, such as that illustrated in Fi g. 1, are tractable.