Solution of the Three-Dimensional Helmholtz Equation with Nonlocal Boundary Conditions

The Helmholtz equation is solved within a three-dimensional rectangular duct with a sound source at the duct entrance plane, local admittance conditions on the side walls, and a new, nonlocal radiation boundary condition at the duct exit plane. The formulation employs a truncation of an infinite matrix, the generalized modal admittance tensor, that represents the transformation of the modal pressure coefficients to the modal axial velocity coefficients. This condition accurately models the acoustic admittance (a relationship between the velocity and pressure of an acoustic wave) at an arbitrary computational boundary plane. In particular, the proper physical admittance may correspond to a nonreflecting radiation condition on an arbitrarily located boundary plane that is used to limit the size of the computational domain. A linear system of equations is constructed with second-order central differences for the Helmholtz operator and second-order backward differences for both local admittance conditions and the gradient term in the nonlocal radiation boundary condition. The resulting matrix equation is large, sparse, and non-Hermitian. The size and structure of the matrix makes direct solution techniques impractical; as a result, a nonstationary iterative technique is used for its solution. The theory behind the nonstationary technique is reviewed, and numerical results are presented for radiation from both a point source and a planar acoustic source in both soft-walled and hard-walled ducts. The solutions with the nonlocal boundary conditions are invariant to the location of the computational boundary, and the same nonlocal conditions are valid for all solutions. The nonlocal conditions thus provide a means of minimizing the size of three-dimensional computational domains.