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

Matteo Calafà, Yuanxin Xia, Jonas Brunskog, Cheol-Ho Jeong

TL;DR

This work develops a semi-analytical framework to compute the Green's function for Helmholtz problems in rectangular rooms with general surface impedance. By deriving first-order asymptotics for eigenvalues across four impedance regimes and establishing a robust eigenfunction basis, the authors enable an efficient eigenfunction expansion-based Green's function that remains accurate for a wide range of boundary conditions. They prove orthogonality in a generalized bilinear form and show completeness for almost all impedances, then present a Newton–Raphson-based algorithm that leverages the asymptotic estimates for fast convergence. Numerical tests, including 2D simulations and real-room measurements, demonstrate strong agreement with finite-element results and experimental data, while offering substantial computational savings and a reliable benchmark for simulations. The approach provides practical tools for accurate room acoustics modeling and resonance analysis in realistic, lossy environments.

Abstract

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.

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

TL;DR

This work develops a semi-analytical framework to compute the Green's function for Helmholtz problems in rectangular rooms with general surface impedance. By deriving first-order asymptotics for eigenvalues across four impedance regimes and establishing a robust eigenfunction basis, the authors enable an efficient eigenfunction expansion-based Green's function that remains accurate for a wide range of boundary conditions. They prove orthogonality in a generalized bilinear form and show completeness for almost all impedances, then present a Newton–Raphson-based algorithm that leverages the asymptotic estimates for fast convergence. Numerical tests, including 2D simulations and real-room measurements, demonstrate strong agreement with finite-element results and experimental data, while offering substantial computational savings and a reliable benchmark for simulations. The approach provides practical tools for accurate room acoustics modeling and resonance analysis in realistic, lossy environments.

Abstract

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.
Paper Structure (18 sections, 3 theorems, 22 equations, 6 figures, 1 table, 1 algorithm)

This paper contains 18 sections, 3 theorems, 22 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Mode and eigenvalue problems
  4. Problem formulation in rectangular rooms
  5. Analysis of solutions
  6. Analysis of eigenvalues
  7. First-order asymptotics of eigenvalues
  8. Analysis of modes
  9. Construction of Green's function
  10. Orthogonality of eigenfunctions
  11. Completeness of the spectral basis
  12. Algorithm for calculation of Green's function
  13. Numerical tests
  14. Approximation of eigenvalues
  15. Calculation of Green's function
  16. ...and 3 more sections

Key Result

Theorem 1

There exists $m_0\in\mathbb{N}$ such that for any $m\ge m_0$:

Figures (6)

  • Figure 1: Evaluation of eigenvalue first-order asymptotics from Section \ref{['sec:firstorderasymptotics']} for different admittance values. (A) Hard walls. (B) Soft walls. (C) Highly asymmetric walls. Admittance and cutoff transition values are provided.
  • Figure 2: Comparison of the EE (Algorithm \ref{['alg:1']}) and FEM simulation. The same color map is applied on each row. The two solutions appear considerably similar with a slightly larger error at the source singular location.
  • Figure 3: Convergence of the EE (Algorithm \ref{['alg:1']}) as the truncation order increases. The error is shown relative to both the lower and high-fidelity FEM solutions. The value of $q$ is marked as a vertical dotted line.
  • Figure 4: Comparison of Green's function with the analytical resonance modes from Equation \ref{['eq:mode_real']}.
  • Figure 5: Sketch of the acoustic measurement room. The sound source is placed at coordinates [0.4 m, 0.396 m, 1.241 m] relative to the room corner, while the receiver is positioned at [0.567 m, 3.085 m, 1.033 m]. The absorbing panel is placed on the decorated wall face.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 1: Number of solutions
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof