For a circular drum we need to write the right-hand side in polar coordinates. It can be shown (extra credit if you can show this) that in polar coordinates the wave equation is
Our goal is to develop equations that govern the vibrations of a circular
drum. The wave equation in two spatial dimensions
is
For
a circular drum we need to write the right-hand side in polar coordinates. It
can be shown (extra credit if you can show this) that in polar coordinates the
wave equation
is
Here,
represents the vertical displacement of the drum head at a distance
from the center of the drum, and at angle
from the
,
and at time
This is where the Wikipedia article at
http://en.wikipedia.org/wiki/Vibrations_of_a_circular_drum
begins.
The eigenfunction solutions to this equation, both for the radially symmetric
case and the full case, are given in the article. In order to satisfy a
general initial condition, an infinite sum of the eigenfunction solutions
would have to be taken.
Assume that
and that the drum has radius
.
We will start with the radially symmetric case. This means that
does not depend on
,
and so we can write just
.
The boundary conditions are
and
.
After separation of variables into
the boundary conditions on
become
and
.
Fill in the details for the Wikipedia derivation of the solution to the
radially symmetric solution. One missing step is putting the differential
equation for
in the form of a Bessel equation, and then deriving the solution shown. Then
animate three modes
in Maple. You can use anything you want for the constants
and
.
You will need to be able to find the first three roots (zeros) of
.
The Lab 1 Maple worksheet has the necessary Maple commands for finding roots
and for creating animations.
In our text, the authors derive a series formula for Bessel functions by using
a series solution to the Bessel differential equation. In the next part of the
lab, we want to investigate how useful the series representation of Bessel
functions is. In particular, how many terms in the series do we need in order
to get useful results? We will try to replicate the results from the first
bullet without the built-in Maple Bessel function.
Use Maple to get a series solution to the ordinary differential equation for
given on the Wikipedia page. Use
and use the Maple dsolve command with type=series. Start with Order:=6. Then
apply the boundary conditions
and
.
To get the eigenvalues
you will need to find the roots (zeros) of the equation
using the Maple fsolve command as in the Lab 1 Maple worksheet. Determine how
many terms in the series solution (Order) it takes to get a reasonable
approximation to each of the three modes as in the first bullet. Note how long
it takes Maple to do certain calculations. How many modes (ie, how high can
you go) do you think you can approximate in a reasonable amount of time using
an infinite series in Maple? How accurate are the eigenvalues (roots of
)
when using the series method?
Write a paper incorporating the two bullets above. Is the infinite series
representation of a Bessel function practical for finding the higher modes of
vibration of a drum? Explain why it is hard, but important, to calculate
values of a Bessel function
for large
.