Evaluation of acoustic Green's function in rectangular rooms with general surface impedance walls

Acoustic room modes and the Green's function mode expansion are well-known for rectangular rooms with perfectly reflecting walls. First-order approximations also exist for nearly rigid boundaries; however, current analytical methods fail to accommodate more general boundary conditions, e.g., when wall absorption is significant. In this work, we present a comprehensive analysis that extends previous studies by including additional first-order asymptotics that account for soft-wall boundaries. In addition, we introduce a semi-analytical, efficient, and reliable method for computing the Green's function in rectangular rooms, which is described and validated through numerical tests. With a sufficiently large truncation order, the resulting error becomes negligible, making the method suitable as a benchmark for numerical simulations. Additional aspects regarding the spectral basis orthogonality and completeness are also addressed, providing a general framework for the validity of the proposed approach.