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
Famous quotes containing the words general and/or case:
“In former times and in less complex societies, children could find their way into the adult world by watching workers and perhaps giving them a hand; by lingering at the general store long enough to chat with, and overhear conversations of, adults...; by sharing and participating in the tasks of family and community that were necessary to survival. They were in, and of, the adult world while yet sensing themselves apart as children.”
—Dorothy H. Cohen (20th century)
“Never in any case say I have lost such a thing, but I have returned it. Is your child dead? It is a return. Is your wife dead? It is a return. Are you deprived of your estate? Is not this also a return?”
—Epictetus (c. 55–135)