The problem of accurately computing low Mach number flows, with the specific intent of studying the interaction of sound waves with incompressible flow structures, such as concentrations of vorticity is considered. This is a multiple time (and/or space) scales problem, leading to various difficulties in the design of numerical methods. Concentration is on one of these difficulties - the development of boundary conditions at artificial boundaries which allow sound waves and vortices to radiate to the far field. Nonlinear model equations are derived based on assumptions about the scaling of the variables. Then these are linearized about a uniform flow and exact boundary conditions are systematically derived using transform methods. Finally, useful approximations to the exact conditions which are valid for small Mach number and small viscosity are computed. Hagstrom, Thomas and Lorenz, Jens Glenn Research Center NASA-TM-106374, ICOMP-93-38, E-8184, NAS 1.15:106374 DE-FG03-92ER-25128; NSF DMS-91-08072; NSF DMS-91-07612; NCC3-233...