Find the characteristic frequencies for sound vibration in a...

Find the characteristic frequencies for sound vibration in a rectangular box (say a room) of sides $a, b, c$. Hint : Separate the wave equation in three dimensions in rectangular coordinates. This problem is like Problem 3 but for three dimensions instead of two. Discuss degeneracy (see Problem 3).

Key Concepts

Separation of Variables

A powerful mathematical technique used to reduce partial differential equations (PDEs) into simpler ordinary differential equations (ODEs) by assuming that the solution can be written as a product of functions, each depending on a single coordinate. This method is particularly useful in problems with separable geometries (like rectangular boxes) and allows one to obtain independent equations for each spatial dimension.

Eigenvalue Problem and Normal Modes

When separating the wave equation in a rectangular domain, the resulting ODEs typically become eigenvalue problems. The eigenfunctions, or normal modes, represent the standing wave patterns inside the box, and their associated eigenvalues determine the characteristic frequencies of the system. The quantization arises from the constraint imposed by the boundary conditions.

Boundary Conditions

Boundary conditions are essential in defining the allowed solutions for the vibration modes. In a rectangular room, the conditions at the walls (such as fixed or pressure-release boundaries) lead to discrete permitted values of the wave numbers in each dimension. These conditions ensure that the physical solutions are confined within the space and determine the specific form of the eigenfunctions.

Characteristic Frequencies

The frequencies of sound vibration are determined by the eigenvalues obtained during the separation process. They are typically expressed in terms of the dimensions of the room and the indices corresponding to the mode numbers along each coordinate axis. These frequencies are calculated as combinations (often via a square root of sums of squares) of contributions from each spatial dimension.

Degeneracy

Degeneracy occurs when distinct sets of mode numbers yield the same frequency, meaning different vibration patterns share the same eigenvalue. In a rectangular box, depending on the ratios of its dimensions, some characteristic frequencies may be degenerate, leading to multiple independent normal modes with identical frequencies. This concept is important in understanding the symmetry properties and the energy distribution among the modes.

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