Solution of the Ordinary Differential EquationsSolution of the Ordinary Differential Equations
Mathematica can solve the ordinary differential equation in r.
The Differential Equation in r
vSolution = DSolve[radiusDiffEqn2, v[r], r]This differential equation is called Bessel's equation and the solutions are the Bessel function of the first kind of order 0, J0, and the Bessel function of the second kind of order 0, Y0. These solutions involve the square root of k^2, which verifies that we needed to choose our separation constant to be a negative number, namely -k^2. Mathematica does not know that we can assume k is positive, so it will not simplify Sqrt[k^2]. Let's give it a little help.
vSolution2 = vSolution /.{Sqrt[k^2] -> k}
Let's take a look at the graphs of the Bessel functions. The graph of Y0
is
Plot[BesselY[0,r], {r,0,1}];
It appears that Y0 has a vertical asymptote at r = 0. In fact, Y0 acts like
Log[r] at r -> 0+.
coefficient =Thus, near r = 0, Y0 is approximatelyLimit[BesselY[0,r]/Log[r], r -> 0, Direction -> -1]
coefficient Log[r]Drumheads cannot have infinite displacements at their centers, so we must discard this solution; that is, we let C[1] = 0. (This gives us our second boundary condition in r -- v[r] must be bounded at r = 0.)
vSolution3 = vSolution2 /. {C[1] -> 0}
Our other boundary condition in r states that v[1] = 0; that is, the drumhead is stationary at the outer rim, r = 1. Substituting this condition, we get
vBoundaryCondition = ((v[r] /. vSolution3) /. {r -> 1}) == 0
Therefore, we need k to be a zero of J0. Let's plot J0 get an idea of the
locations of its zeroes.
Plot[BesselJ[0,k], {k,0,15}];
We can use Newton's method, FindRoot in Mathematica, to find the first
few zeroes of J0. (To find the exact solution to our drumhead problem, we
would have to find all of J0's zeroes.) First we construct a list of initial
guesses for the first four zeroes.
guesses = {2.5, 5.5, 9, 12}
Now apply Newton's method to these initial guesses.
j0zeroes = Map[FindRoot[BesselJ[0,k] == 0, {k,#}]&,
guesses]
Using these values of k in J0, we have satisfied the differential equation
in r and the boundary conditions. We now turn to the differential equation in
t and the initial conditions.