Conjugate Operators And Orthogonality Of Acoustic Waveguide Based On Radiation Boundary Condition

When an open waveguide(the transverse direction is open)is terminated by the radiation boundary condition(RBC),the wave propagation computation problem within the waveguide is approximately turned to the numerical solution of the Helmholtz equation within a bounded waveguide.The radiation boundary condition causes the wave to travel forward across the boundary with as little reflection or refraction as possible.However,there is no cross orthogonality between the linearly-independent eigenfunctions of Helmholtz operator with the introduction of radiation boundary conditions.For the fast computation of the coordinates under the basis of the eigenfunctions of Helmholtz operator,in this paper,for a flat interface,we derive the conjugate operator of the Helmholtz operator with the RBC and theoretically proof the cross orthogonality between the eigenfunctions of the original eigenoperators and the eigenfunctions of the conjugate eigenoperators.For the curved interface,we firstly use local orthogonal transformation to transform the coordinates.Then we construct its conjugate characteristic operator and proof the cross orthogonality between the eigenfunction of the original eigenoperator and the eigenfunction of the conjugate eigenoperator by the same theory.For the sake of convenience,we only simulate the case of flat interface.Since the radiation boundary conditions contain characteristic parameters,we cannot directly solve the characteristic modes.Therefore,in the case of flat interface,we transform the characteristic problem into a nonlinear equation to solve its characteristic mode.Because of the existence of analytic solution,we can use the adaptive gaussian integral to compute the eigenfunction integral directly.The experimental results verify the cross orthogonality between the eigenfunctions of the original eigenoperators and the eigenfunctions of the conjugate eigenoperators.