Local Orthogonal Mappings and Operator Formulation for Varying Cross-sectional Ducts

A method is developed for solving the two dimensional Helmholtz equation in a duct with varying cross-section region bounded by a curved top and flat bottom, having one region inside. To compute the propagation of sound waves in a curved duct with a curved internal interface is difficult problem. One method is to transform the wave equation into a solvable form and making the curved interface plane. To this end a local orthogonal transformation is developed for the varying cross-sectional duct having one medium inside. This transformation is first used to make the curved top of the waveguide flat and to transform the Helmholtz equation into an initial value problem. Later on the local orthogonal transformation is developed for a waveguide having two media inside with flat top, a flat bottom and a curved interface. This local orthogonal transformation is used to flatten the interface and also to transform the Helmholtz equation into a simple, solvable ordinary differential equation. In this paper we present operator formulation for the part with flat bottom and curved top including a curved interface. In the ordinary differential equation with operators in coefficients, obtained after the transformation, all the operations related to the transverse variable are treated as operators while the derivative with respect to the range variable is kept. Key-words: Helmholtz equation; Local orthogonal transform; operator equation