Modal problems in non-rectangular rooms
Wave acoustics studies the low-frequency behavior of an enclosure, characterizing its own modes, among others. In a rectangular room, its determination is very simple using the Rayleigh formula:
fk,m,n = 172,5 . √((k/Lx)2+(m/Ly)2+(n/Lz)2)
where Lx, Ly i Lz represent the room in meters, and k,m,n are integer values (0, 1, 2, 3, …)
Knowing the low frequency behavior of a room is of vital importance in critical listening spaces such as the control rooms of a recording studio. Knowing where and how these standing waves appear allows you to design the appropriate materials for each case, as well as their optimal placement. In the same way, exhaustive knowledge of this frequency range allows defining optimal listening points and the position of the listening system.
When one of the surfaces of the room loses parallelism, the standing waves do not disappear, they simply move and vary their frequency, making it more difficult to make predictions. In these cases, mathematical tools for numerical approximation must be used, such as finite element techniques, which allow solving the Helmholtz integral of interior and exterior radiation.
In the figure above you can see an example of how the sound pressure varies in a rectangular room and a non-rectangular room, at 63Hz, both with the same surface and volume and with two ideal pressure sources of 0.1Pa.