Digital waveguide mesh structures have proved useful in a wide variety of modelling applications. When modelling the acoustics of an enclosed space the accurate simulation of specific boundary conditions is paramount but waveguide mesh related techniques have as yet to provide an appropriate and reliable solution. This paper gives an overview of boundary types that have been implemented by researchers to date and describes the application of a higher order approximation for the particular case of an anechoic boundary, that may prove useful as part of a more general solution.