In our original post on drum waves, we looked at the vibrations of a circular drum. I chose that shape because just about every drum is circular, which I guess is because it is the easiest shape to manufacture.
However, the mathematics of the vibrations of a circular drum are not the simplest. We saw in the first drum waves post that the solution to the wave equation for a circular drum involes the Bessel functions, which are not the simplest functions to work with, and knowledge of the zeros of these functions is also required, and these zeros are not trivial to determine either.
The simplest shape, mathematically speaking is a rectangular drum. In this case, the solution to the wave equation is simply a product of trigonometric functions. The solution is similar to the circular drum, in the sense that it can be decomposed as a product of functions that depend on only one variable: radius, angle, or time, in the case of the circular drum, and length, width, or time, in the case of the rectangular drum.
The terms of a general solution to the wave equation for a rectangular drum are:
\[\cos (c \lambda_{mn} t) \sin\left(\frac{m \pi x}{a}\right) \sin\left(\frac{n \pi y}{b}\right),\]
where \(a\) and \(b\) are the length and width of the drum, and the oscillation frequencies are determined by \(\lambda_{mn} = \sqrt{\left(\frac{m\pi}{a}\right)^2 + \left(\frac{n\pi}{b}\right)^2}\). The constant \(c\) does not depend on the oscillation mode, and depends only on the material characteristics of the drum.
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