March 4, 2005 13:25 WSPC/130-JCA inhom˙DTBC Journal of Computational Acoustics c  IMACS DISCRETE TRANSPARENT BOUNDARY CONDITIONS FOR WIDE ANGLE PARABOLIC EQUATIONS FOR NON–VANISHING STARTING FIELDS MATTHIAS EHRHARDT Institut f¨ ur Mathematik, Technische Universit¨at Berlin, Straße des 17. Juni 136, D–10623 Berlin, Germany, ehrhardt@math.tu-berlin.de http://www.math.tu-berlin.de/˜ehrhardt/ Received (Day Month Year) Revised (Day Month Year) Transparent boundary conditions (TBCs) are an important tool for the truncation of the compu- tational domain in order to compute solutions on an unbounded domain. In this work we want to show how the standard assumption of ‘compactly supported data’ could be relaxed and derive TBCs for the wide angle parabolic equation directly for the numerical scheme on the discrete level. With this inhomogeneous TBCs it is not necessary that the starting field lies completely inside the computational region. However, an increased computational effort must be accepted. Keywords : wide angle parabolic equation; transparent boundary condition; non-compactly sup- ported initial data. 1. Introduction Transparent boundary conditions (TBCs) are an important tool for the truncation of the computational domain in order to compute solutions on an unbounded domain. In this work we want to show how the standard assumption of ‘compactly supported data’ could be relaxed and derive inhomogeneous TBCs for a finite difference discretization of standard and wide angle “parabolic” equations 1 . These models appear as one–way approximations to the Helmholtz equation in cylindrical coordinates with azimuthal symmetry and include as a special case the Schr¨ odinger equation. With this TBCs it is not necessary that the starting field is completely inside the computational region. This is the case in underwater acoustics if the source is close to the bottom or in radiowave propagation problems when computing coverage diagrams of airborne antennas 2 . In the past two decades “parabolic” equation (PE) models have been widely used for wave propagation problems in various application areas, e.g. seismology 3 , optics and plasma physics (cf. the references in 4 ). Further applications to wave propagation problems can be found in radio frequency technology 5 . Here we will be mainly interested in their application to underwater acoustics, where PEs have been introduced by Tappert 6 . An account on the vast recent literature is given in the survey article 7 and in the book 8 . In oceanography one wants to calculate the underwater acoustic pressure p(z,r) emerg- 1