Vibrations of A Circular Drum - The General Case

The General Case

The general case, when can also depend on the angle is treated similarly. We assume a solution in separated variables,

Substituting this into the wave equation and separating the variables, gives

where is a constant. As before, from the equation for it follows that with and

From the equation

we obtain, by multiplying both sides by and separating variables, that

and

for some constant Since is periodic, with period being an angular variable, it follows that

where and and are some constants. This also implies

Going back to the equation for its solution is a linear combination of Bessel functions and With a similar argument as in the previous section, we arrive at

where with the -th positive root of

We showed that all solutions in separated variables of the vibrating drum head problem are of the form

for

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