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.

💡 Research Summary

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The paper addresses the long‑standing problem of evaluating the acoustic Green’s function in rectangular rooms when the walls exhibit general, possibly highly absorptive, surface impedances. Classical solutions exist only for perfectly rigid (Neumann) boundaries, and first‑order approximations have been limited to nearly rigid walls. This work extends the theory to arbitrary complex admittances by deriving a set of first‑order asymptotic approximations that cover hard, soft, highly asymmetric, and negative‑reactance wall conditions, and by providing a rigorous proof of eigenfunction orthogonality and completeness for the non‑self‑adjoint operator that arises when β≠0.

The authors start from the Helmholtz equation with a normalized surface admittance β defined on each wall. They formulate three related problems: (i) the boundary‑value problem for the Green’s function, (ii) the modal problem where the wavenumber is unknown, and (iii) the eigenvalue problem where the physical wavenumber k is prescribed and the eigenvalue (\hat{k}) is sought. For a rectangular geometry, separation of variables reduces the three‑dimensional problem to a one‑dimensional transcendental equation (Eq. 6) for a nondimensional quantity (\hat{q}) that depends on the wall admittances through the parameters (\gamma_{\pm}=k,l,\beta_{\pm}).

A key contribution is Theorem 1, which uses Rouche’s theorem to count the number of solutions of Eq. 6 inside a disc of radius m+½, thereby guaranteeing that all eigenvalues are captured when the truncation order exceeds a modest threshold. This counting result underpins the completeness claim.

The paper then classifies the solutions of Eq. 6 into four asymptotic groups:

• Group 1 (Hard walls) – applicable when the impedance is small compared with the modal index n; the solution reduces to the classical rigid‑wall result with a small imaginary correction proportional to ((\gamma_{-}+\gamma_{+})/n).
• Group 2 (Soft walls) – relevant when the impedance dominates; the eigenvalue acquires a large imaginary part, reflecting strong damping.
• Group 3 (Negative reactance) – arises when the imaginary parts of the admittances are large and negative; two additional eigenvalues appear, approximately (\gamma_{\pm}/\pi).
• Group 1P (Highly asymmetric walls) – covers the case where one wall is very soft and the opposite wall is very rigid; the eigenvalue is shifted by a term proportional to ((\gamma_{-}+\gamma_{+})) around half‑integer multiples of the modal index.

Each group is derived by expanding the tan function around its zeros, poles, or at infinity, yielding first‑order closed‑form expressions for (\hat{q}). The authors also provide practical transition criteria (A₁₋₂ and A₁₋₁P) that allow a user to select the appropriate asymptotic regime based on the magnitude of the admittance parameters.

Having obtained the eigenvalues (\hat{k}_n) and eigenfunctions (\phi_n(\mathbf{x})) (including the normalization constant Λ from Eq. 5), the Green’s function is expressed as an eigenfunction expansion: \