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.

翻译：对于具有完美反射壁的矩形房间，声学房间模态和格林函数模态展开是众所周知的。对于近似刚性边界也存在一阶近似；然而，当前的分析方法无法适应更一般的边界条件，例如当壁面吸声显著时。在本工作中，我们提出了一种综合分析，通过纳入考虑软壁边界的附加一阶渐近项扩展了先前的研究。此外，我们引入了一种半解析、高效且可靠的方法来计算矩形房间中的格林函数，该方法通过数值测试进行了描述和验证。在截断阶数足够大的情况下，所得误差可忽略不计，使得该方法适合作为数值模拟的基准。关于谱基正交性和完备性的其他方面也进行了讨论，为所提方法的有效性提供了一个通用框架。