
# 4.5.2: Oscillation of a Membrane


Let $$\Omega\subset\mathbb{R}^2$$ be a bounded domain. We consider the initial-boundary value problem

\begin{eqnarray}
\label{mem1}\tag{4.5.2.1}
u_{tt}(x,t)&=&\triangle_xu\ \ \mbox{in}\ \Omega\times\mathbb{R}^1,\\
\label{mem2} \tag{4.5.2.2}
u(x,0)&=&f(x),\ \ x\in\overline{\Omega},\\
\label{mem3} \tag{4.5.2.3}
u_t(x,0)&=&g(x),\ \ x\in\overline{\Omega},\\
\label{mem4} \tag{4.5.2.4}
u(x,t)&=&0\ \ \mbox{on}\ \partial\Omega\times\mathbb{R}^1.
\end{eqnarray}

As in the previous subsection for the string, we make the ansatz (separation of variables)

$$u(x,t)=w(t)v(x)$$

which leads to the eigenvalue problem

\begin{eqnarray}
\label{evpmem1} \tag{4.5.2.5}
-\triangle v&=&\lambda v\ \ \mbox{in}\ \Omega,\\
\label{evpmem2} \tag{4.5.2.6}
v&=&0\ \ \mbox{on}\ \partial\Omega.
\end{eqnarray}

Let $$\lambda_n$$ are the eigenvalues of $$(\ref{evpmem1})$$, $$(\ref{evpmem2})$$ and $$v_n$$ a complete associated orthonormal system of eigenfunctions. We assume $$\Omega$$ is sufficiently regular such that the eigenvalues are countable, which is satisfied in the following examples. Then the formal solution of the above initial-boundary value problem is

$$u(x,t)=\sum_{n=1}^\infty\left(\alpha_n\cos(\sqrt{\lambda_n}t)+\beta_n\sin(\sqrt{\lambda_n}t)\right)v_n(x),$$

where

\begin{eqnarray*}
\alpha_n&=&\int_\Omega\ f(x)v_n(x)\ dx\\
\beta_n&=&\frac{1}{\sqrt{\lambda_n}}\int_\Omega\ g(x)v_n(x)\ dx.
\end{eqnarray*}

Note

In general, eigenvalues of (\ref{evpmem1}), (\ref{evpmem1}) are not known explicitly. There are numerical methods to calculate these values. In some special cases, see next examples, these values are known.

## Examples

Example 4.5.2.1: Rectangle membrane

Let
$$\Omega=(0,a)\times (0,b).$$
Using the method of separation of variables, we find all eigenvalues of (\ref{evpmem1}), (\ref{evpmem2}) which are given by
$$\lambda_{kl}=\sqrt{\frac{k^2}{a^2}+\frac{l^2}{b^2}},\ \ k,l=1,2,\ldots$$ and associated eigenfunctions, not normalized, are
$$u_{kl}(x)=\sin\left(\frac{\pi k}{a}x_1\right)\sin\left(\frac{\pi l}{b}x_2\right).$$

Example 4.5.2.2: Disk membrane

Set
$$\Omega=\{x\in\mathbb{R}^2:\ x_1^2+x_2^2<R^2\}.$$
In polar coordinates, the eigenvalue problem (\ref{evpmem1}), (\ref{evpmem2}) is given by
\begin{eqnarray}
\label{evppol1} \tag{4.5.2.6}
-\frac{1}{r}\left((ru_r)_r+\frac{1}{r}u_{\theta\theta}\right)&=&\lambda u\\
\label{evppol2} \tag{4.5.2.7}
u(R,\theta)&=&0,
\end{eqnarray}
here is $$u=u(r,\theta):=v(r\cos\theta,r\sin\theta)$$. We will find eigenvalues and eigenfunctions by separation of variables
$$u(r,\theta)=v(r)q(\theta),$$
where $$v(R)=0$$ and $$q(\theta)$$ is periodic with period $$2\pi$$ since $$u(r,\theta)$$ is single valued.
$$-\frac{1}{r}\left((rv')'q+\frac{1}{r}vq''\right)=\lambda v q.$$
Dividing by $$vq$$, provided $$vq\not=0$$, we obtain

\label{disk1} \tag{4.5.2.8}
-\frac{1}{r}\left(\frac{(rv'(r))'}{v(r)}+\frac{1}{r}\frac{q''(\theta)}{q(\theta)}\right)=\lambda,

which implies
$$\frac{q''(\theta)}{q(\theta)}=const.=:-\mu.$$
Thus, we arrive at the eigenvalue problem
\begin{eqnarray*}
-q''(\theta)&=&\mu q(\theta) \\
q(\theta)&=&q(\theta+2\pi).
\end{eqnarray*}
It follows that eigenvalues $$\mu$$ are real and nonnegative. All solutions of the differential equation are given by
$$q(\theta)=A\sin(\sqrt{\mu}\theta)+B\cos(\sqrt{\mu}\theta),$$
where $$A$$, $$B$$ are arbitrary real constants. From the periodicity requirement
$$A\sin(\sqrt{\mu}\theta)+B\cos(\sqrt{\mu}\theta)=A\sin(\sqrt{\mu}(\theta+2\pi))+B\cos(\sqrt{\mu}(\theta+2\pi))$$
it follows\begin{eqnarray*}
\sin x-\sin y&=&2\cos\frac{x+y}{2}\sin\frac{x-y}{2}\\
\cos x-\cos y&=&-2\sin\frac{x+y}{2}\sin\frac{x-y}{2}\end{eqnarray*}
$$\sin(\sqrt{\mu}\pi)\left(A\cos(\sqrt{\mu}\theta+\sqrt{\mu}\pi)-B\sin(\sqrt{\mu}\theta+\sqrt{\mu}\pi)\right)=0,$$
which implies, since $$A$$, $$B$$ are not zero simultaneously, because we are looking for $$q$$ not identically zero,
$$\sin(\sqrt{\mu}\pi)\sin(\sqrt{\mu}\theta+\delta)=0$$
for all $$\theta$$ and a $$\delta=\delta(A,B,\mu)$$. Consequently the eigenvalues are
$$\mu_n=n^2,\ \ n=0,1,\ldots\ .$$
Inserting $$q''(\theta)/q(\theta)=-n^2$$ into (\ref{disk1}), we obtain the boundary value problem
\begin{eqnarray}
\label{disk2} \tag{4.5.2.9}
r^2v''(r)+rv'(r)+(\lambda r^2-n^2)v&=&0\ \ \mbox{on}\ (0,R)\\
\label{disk3} \tag{4.5.2.10}
v(R)&=&0\\
\label{disk4} \tag{4.5.2.11}
\sup_{r\in(0,R)}|v(r)|&<&\infty.
\end{eqnarray}
Set $$z=\sqrt{\lambda}r$$ and $$v(r)=v(z/\sqrt{\lambda})=:y(z)$$, then, see (\ref{disk2}),
$$z^2y''(z)+zy'(z)+(z^2-n^2)y(z)=0,$$
where $$z>0$$. Solutions of this differential equations which are bounded at zero are Bessel functions of first kind and $$n$$-th order $$J_n(z)$$. The eigenvalues follows from boundary condition (\ref{disk3}), i. e., from $$J_n(\sqrt{\lambda}R)=0$$. Denote by $$\tau_{nk}$$ the zeros of $$J_n(z)$$, then the eigenvalues of (\ref{evppol1})-(\ref{evppol1}) are
$$\lambda_{nk}=\left(\frac{\tau_{nk}}{R}\right)^2$$
and the associated eigenfunctions are
\begin{eqnarray*}
J_n(\sqrt{\lambda_{nk}}r)\sin(n\theta), &&\ n=1,2,\ldots\\
J_n(\sqrt{\lambda_{nk}}r)\cos(n\theta), &&\ n=0,1,2,\ldots.
\end{eqnarray*}
Thus the eigenvalues $$\lambda_{0k}$$ are simple and $$\lambda_{nk},\ n\ge1$$, are double eigenvalues.

Remark. For tables with zeros of $$J_n(x)$$ and for much more properties of Bessel functions see \cite{Watson}. One has, in particular, the asymptotic formula
$$J_n(x)=\left(\frac{2}{\pi x}\right)^{1/2}\left(\cos(x-n\pi/2-\pi/5)+O\left(\frac{1}{x}\right)\right)$$
as $$x\to\infty$$. It follows from this formula that there are infinitely many zeros of $$J_n(x)$$.

## Contributors

• Integrated by Justin Marshall.