Roland BadeauProfessor

Statistical wave field theory

The statistical wave field theory establishes the statistical laws of the solutions to the wave equation in a connected and bounded domain, which hold at high frequency and after many reflections on the boundary surface. It thus provides the mathematical solution to a long-standing problem in room acoustics, which has been the subject of extensive research since the pioneering work of Wallace Clement Sabine in the late 19th century: the study of reverberation.

Compared to existing approaches, the statistical wave field theory offers various advantages:

• it provides a unified framework that encompasses all the previously known statistical properties of late reverberation, including the reverberation time in ergodic rooms;
• it also provides the closed-form expression of the power distribution and correlations of the wave field over time, frequency, and space, in terms of the domain’s geometry and boundary condition;
• it is not restricted to ergodic rooms: it also applies to geometric shapes that generate a homogeneous but anisotropic wave field;
• it allows for greater accuracy compared to classical approaches based on the statistics of reflections in billiards, by taking advantage of the “semi-classical” approximation of quantum physics;
• it reveals the existence of “black holes”, which behave like those of general relativity: a part of the energy of the wave field can be trapped in the vicinity of the domain’s boundary.

In the particular case of an isotropic wave field, the formula of the reverberation time predicted by theory has been verified experimentally using several numerical methods and in various room geometries. Its accuracy has proven to be remarkable. This formula has also been used to estimate the impedances of the different surfaces in a room with a high degree of reliability.

Current publication roadmap

Roland Badeau. On the spectral decomposition of the complex Robin Laplacian. Journal of the Acoustical Society of America, 2025, 158 (1), pp.  838-848.
Copyright (2025) Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.
Download updated preprint, including minor corrections posterior to publication (dated December 8, 2025)

Roland Badeau. Statistical wave field theory. Journal of the Acoustical Society of America, 2024, 156 (1), pp. 573-599.
Copyright (2024) Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.
Download updated preprint, including minor corrections posterior to publication (dated December 8, 2025)

Albert G. Prinn and Roland Badeau. Verification of statistical wave field theory reverberation time predictions, 2025, under review.
Download preprint

Roland Badeau. Statistical wave field theory: Curvature term. Journal of the Acoustical Society of America, 2025, 157 (3), pp.  1650-1664.
Copyright (2025) Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.
Download updated preprint, including minor corrections posterior to publication (dated December 8, 2025)

Roland Badeau. Statistical wave field theory: Special polyhedra. Journal of the Acoustical Society of America, 2025, 157 (3), pp.  2263-2278.
Copyright (2025) Acoustical Society of America. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the Acoustical Society of America.
Download updated preprint, including minor corrections posterior to publication (dated December 8, 2025)

Other research themes

• Room acoustics: statistical modeling of reverberation
• Data representation: dimensionality reduction, time-frequency analysis (high resolution)
• Modeling: probabilistic latent variable models, source models (positive matrix factorizations, sinusoidal models, sparse models, etc.), propagation models (convolutional, diffuse)
• Algorithms: Bayesian estimation, optimization methods, fast adaptive algorithms, performance analysis, convergence speed, numerical stability, algorithmic complexity
• Applications to audio signals: source separation/localization, audio coding, restoration, denoising, dereverberation, sound scene analysis, music information retrieval
• Other applications: biomedical data analysis, digital communications, image processing

Teaching

• Engineering training at Télécom Paris, 1st year and Master cycle (2nd and 3rd years, M1&M2 levels): Applied Mathematics and Signal Processing
• Master 2 ATIAM, Sorbonne Université: Music signal processing
• Master 2 MVA, ENS Paris-Saclay: Audio-frequency signal analysis
• Master 2 CIMES, ESPCI and Sorbonne Université: Signal and image processing, statistics

Responsibilities

• Supervision of doctoral and master’s theses
• Supervision of teaching units at Télécom Paris and in the ATIAM and MVA M2 programs
• Supervision of the TSIA study track (Signal Processing for Artificial Intelligence) at Télécom Paris
• Correspondent of the Master ATIAM at Télécom Paris