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
Read more about this topic: Vibrations Of A Circular Drum
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