Not taking any classes in the math department, leaves me only to have fun with math in my free time. So today, I thought I would look into the math around the vibrations of a drum. This is an example of a Boundary Value Problem, which we can utilize Fourier series to solve. We can model a drum as a vibrating circle, with a radius b, where y(b, t) = 0, where u is a function of r and t, namely y(r, t) = R(r)*T(t).
Which is defined from t≥0, 0≤r≤b. Let y(r, t) = R(r)T(t), then:
Which we can set equal to a constant since it is a function of t equal to a function of r, therefore the two functions must equal a constant for it to hold true for all values of r and t.
We can separate this equation into one depending on t, and the other on r.
We can then multiply by r2, yielding:
We can derive the boundary condition from y(b,t)=0, since y(b,t)=R(b)T(t)=0. We don’t want T(t)=0 otherwise the problem is trivial, thus we let R(b)=0.
If we let λ=α2:
If we let x=αr, then:
This is a zeroth order Bessel Equation governed by the general form of:
However, Y0 blows up at the origin, therefore B=0, or R will blow up as well.
If we let xi be the zeros of the zeroth order Bessel Function of the first kind then:
Now we can view the drum problem two different ways, the first is pushing the drum to some position y(r,0)= f(r). The second is hitting the drum with some velocity: y'(r,0)= f(r).
We will consider the first case for now.
Therefore B must equal 0 and:
Therefore, we can set:
As for the second case:
If we let λ=0 then:
But when r = 0, the natural log blows up.
So we are left with:
But R(b)=0, therefore A = 0, and we are left with the trivial solution of Y = 0.
If we let λ=-β2 then:
Which is a modified zeroth order Bessel Equation, which we can write in its general form of:
This has the solutions of zeroth order Bessel Functions:
However, I0 blows up at the origin, and therefore B = 0.
The only solution exists when B = 0, which is not the case since we need to have some non-zero radius. Therefore λ=-β2 yields no solution.
If we take all of the solutions for , we end up with
And the initial condition, f(r) is defined as:
Since Bessel’s Equation satisfies Sturm-Liouville Theory, we can determine the coefficients:
Similarly, in the case where , we end up with:
I hope to get around to simulating these results and numerically showing the results to a given input, but for today I am out of time.