发表机构
Department of Architecture and Living Design, Nihon University Junior College（日本大学短期大学部建筑与生活设计系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出扩散声场中广义混响理论的完整数学基础，通过宏观能量微分方程与微观卷积模型的统一，揭示Eyring低估源于时间方差扩展的结构性遗漏，并经射线追踪验证其尺度不变准确性。
AI 中文摘要
Sabine的基础理论奠定了现代建筑声学的基石。然而，它无法预测在完全吸声房间中混响时间为零的情况。Eyring随后通过提出一个新公式解决了这一问题，Knudsen后来将其扩展以纳入空气吸收。尽管如此，Eyring的基本方法包含根本性的理论矛盾。为解决这些缺陷，作者先前引入了用于修正混响理论的宏观和微观模型；然而，它们的数学统一、从微分方程的严格推导以及Eyring低估背后的机制仍未阐明。本文提出了该修正理论的全面推广和数学基础。宏观模型直接从基本能量微分方程推导而来，纳入了空气吸收和广义平均残余自由程。通过将微观模型重新表述为顺序卷积过程，从数学上证明了其与宏观模型的基本一致性。至关重要的是，研究表明Eyring的低估可以被解释为普遍源于多次反射固有的时间方差扩展（{\sigma}_{n}^{2}=n{\sigma}^{2}）的结构性遗漏，无论假定的概率分布如何。最后，射线追踪模拟验证了所提出理论的尺度不变准确性。最终，这一数学上一致的框架为未来非扩散声场中的广义理论确立了理论极限。
英文摘要
Sabine's foundational theory established the cornerstone of modern architectural acoustics. However, it fails to predict zero reverberation time in perfectly absorptive rooms. Eyring subsequently addressed this issue by proposing a new formula, which Knudsen later extended to incorporate air absorption. Nevertheless, Eyring's underlying approach contains fundamental theoretical contradictions. To resolve these flaws, the author previously introduced macroscopic and microscopic models for a revised reverberation theory; however, their mathematical unification, rigorous derivation from differential equations, and the mechanism behind Eyring's underestimation remained unclarified. This paper presents a comprehensive generalization and mathematical foundation of the revised theory. The macroscopic model is derived directly from fundamental energy differential equations, incorporating air absorption and a generalized mean residual free path. By reformulating the microscopic model as a sequential convolution process, its fundamental consistency with the macroscopic model is mathematically demonstrated. Crucially, it is shown that Eyring's underestimation can be interpreted as stemming universally from the structural omission of temporal variance expansion (σ_{n}^{2}=nσ^{2}) inherent to multiple reflections, regardless of the assumed probability distribution. Finally, ray-tracing simulations validate the scale-invariant accuracy of the proposed theory. Ultimately, this mathematically consistent framework establishes the theoretical limit for future generalized theories in non-diffuse sound fields.
Comments35 pages, 15figures, 1 Table
