Vibrations of A Circular Drum - The Radially Symmetric Case

The Radially Symmetric Case

We will first study the possible modes of vibration of a circular drum head that are radially symmetric. Then, the function does not depend on the angle and the wave equation simplifies to

We will look for solutions in separated variables, Substituting this in the equation above and dividing both sides by yields

The left-hand side of this equality does not depend on and the right-hand side does not depend on it follows that both sides must equal to some constant We get separate equations for and :

The equation for has solutions which exponentially grow or decay for are linear or constant for and are periodic for Physically it is expected that a solution to the problem of a vibrating drum head will be oscillatory in time, and this leaves only the third case, when Then, is a linear combination of sine and cosine functions,

Turning to the equation for with the observation that all solutions of this second-order differential equation are a linear combination of Bessel functions of order 0,

The Bessel function is unbounded for which results in an unphysical solution to the vibrating drum head problem, so the constant must be null. We will also assume as otherwise this constant can be absorbed later into the constants and coming from It follows that

The requirement that height be zero on the boundary of the drum head results in the condition

The Bessel function has an infinite number of positive roots,

We get that for so

Therefore, the radially symmetric solutions of the vibrating drum head problem that can be represented in separated variables are

where