Diffusion Equation in Coupled Rooms: Simulation, Eigenvalue Analysis, and Validation

The diffusion equation is an efficient method to model sound energy decay in rooms. It is particularly well suited to coupled volumes with different decay characteristics — for example, a small, highly absorbent room connected through an aperture to a larger, less absorbent space. Such configurations arise frequently in architectural acoustics, concert hall design, and spatial audio. A diffusion-based model can capture the dominant energy exchange between volumes at a fraction of the cost of full wave-equation or geometrical-acoustics simulations. However, the number of relevant decay components and their accuracy are not well understood for coupled systems.

The aim of this work is to develop a simulation framework for the diffusion equation in coupled rooms (using finite-difference time-domain discretisation), to apply eigenvalue analysis to the resulting state-space system de/dt = Ae in order to identify the dominant decay components and assess how many are needed for an accurate description, and to validate the results against measurement data or wave-equation simulations. This investigation should reveal the strengths and limitations of the diffusion approach for coupled spaces and provide guidelines on the number of decay components required for a given accuracy.

Prerequisites

Basic knowledge in numerical methods, Python (NumPy, SciPy), and affinity to mathematics.

References

• [1] Fichera, I., Hoorickx, C. V. & Hornikx, M. An empirical diffusion coefficient function for the acoustic diffusion equation model in long rooms. Appl. Acoust. 246, 111234 (2026).
• [2] Sequeira, M. E. & Cortínez, V. H. A simplified two-dimensional acoustic diffusion model for predicting sound levels in enclosures. Appl. Acoust. 73, 842–848 (2012).

Contact

Dr. Albert Prinn, International AudioLabs Erlangen (albert.prinn@audiolabs-erlangen.de), Prof. Sebastian J. Schlecht. Available immediately.