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\parindent=0in

\title{MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations\\
Lecture 5: Diffusions on and Vibrations of a Membrane in 2D/3D--II 2D Disk}
\author{Lecturer: Naoki Saito\\ Scribe: Matthew Herman}
\date{April 12 \& 17, 2007}

\begin{document}
\maketitle
%\setcounter{section}{-1}
\section{Vibrations of a 2D Drumhead}
The basic references for this lecture are the texts by Strauss~\cite[Sec.~10.2]{STRAUSS},
and Courant and Hilbert~\cite[Sec.~V.5.5]{COURANT-HILBERT}.

\subsection{Dirichlet Boundary Conditions}

Given a disk domain $\Omega=\{(x,y) \cond x^2+y^2<a^2\}\subset\Rf^2$, consider the following Dirichlet boundary condition (BC) problem\\
\begin{equation}
\label{2D_DBC_Prob}
\begin{array}{lll}
\text{(DE):} & u_{tt} = c^2\varDelta u = c^2\gr{u_{xx}+u_{yy}} & \text{in $\Omega$}\\
\text{(BC):} & u = 0 & \text{on $\partial\Omega$}\\
\text{(IC):} & u(\bx,0) = f(\bx), \;u_t(\bx,0) = g(\bx)\\
\end{array}
\end{equation}

\ \\Recall in polar coordinates that $\gr{x,y} =
\gr{r\cos\theta,r\sin\theta}$ for $0\leq r<a$ and
$-\pi\leq\theta<\pi$. In 2D the Laplace operator can be expressed as
$$\varDelta = \frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} = \frac{\partial^2}{\partial r^2} +
\frac{1}{r}\frac{\partial}{\partial r} +
\frac{1}{r^2}\frac{\partial^2}{\partial \theta^2}.$$

\ \\As before, assume our solution admits a separation of variables
$$u(r,\theta,t) = R(r)\Theta(\theta)T(t).$$

\ \\This results in $$\frac{T''}{c^2T} = \frac{R''}{R} + \frac{R'}{rR}
 + \frac{\Theta''}{r^2\Theta} = -\lambda.$$

\ \\Let $\frac{\Theta''}{\Theta} = -\gamma$, where $\gamma$ is a
constant. We obtain the three ODEs
\begin{enumerate}
\item[(i)] $T'' + c^2\lambda T = 0$
\item[(ii)] $\Theta'' + \gamma\Theta = 0$
\item[(iii)] $R'' +  \frac{1}{r}R' + \gr{\lambda - \frac{\gamma}{r^2}}R = 0.$
\end{enumerate}

\ \\We begin by solving (ii) first (we will save (i) for last
because it involves the IC). Note that we have the periodic BC
$\Theta(\theta+~\pi)=\Theta(\theta), \,\forall \theta$. So the
characteristic equation for (ii), which is $p^2 + \gamma = 0$ or $p
= \pm \sqrt{-\gamma}$. This will generate a feasible solution only
when $\gamma>0$. Thus,
$$\Theta(\theta) = A\cos\sqrt{\gamma}\theta +
B\sin\sqrt{\gamma}\theta$$
which implies that $\sqrt{\gamma}=n\in\N$ due to the $2\pi$-periodicity.
Finally, we arrive at
$$\Theta(\theta) = \left\{
  \begin{array}{ll}
    \frac{1}{2}A_0, & n = 0;\\
    A_n\cos n\theta + B_n\sin n\theta, & n\in\N.
  \end{array}
\right.
$$
where $A_0$, $A_n$ and $B_n$ are appropriate constants.

\ \\Next, we solve (iii) for $0<r<a$. We realistically impose that
at the origin $R(0)$ be finite. Also, the Dirichlet BC requires
$R(a) = 0$. From the last lecture we know that the
Dirichlet-Laplacian eigenvalues are positive, i.e., $\lambda\geq 0$.
Therefore, there are two cases of $\lambda$ to consider in (iii).

\ \\\textbf{Case 1:} $\lambda = 0$.

We have $r^2R'' + rR' - n^2R = 0$, which is an ODE of the Euler type
\cite[Sec.~6.3]{INCE}. We seek solutions of the form $R(r) =
r^\alpha$ which results in
\begin{center}
$r^2\alpha\gr{\alpha-1}r^{\alpha-2} + r\alpha r^{\alpha-1} -
n^2r^\alpha = 0$\\
$\alpha\gr{\alpha-1} + \alpha - n^2 = 0$\\
$\alpha^2 - n^2 = 0.$\\
\end{center}

Therefore, $\alpha = \pm n$ which results in a solution of the form
$R(r) = Ar^n + Br^{-n}$. Since $R(0)$ is bounded, this implies that $B = 0.$
Thus, $R(r) = Ar^n$. Now using the Dirichlet BC we see that $R(a) = Aa^n = 0$ also
implies that $A = 0$. We conclude that for $\lambda = 0$, the only solution for $R(r)$
is the zero solution.

\ \\\textbf{Case 2:} $\lambda > 0$.

First use the change of variable: $\rho = \sqrt{\lambda}\,r$, which
results in
\[R_r=R_{\rho}\dif{\rho}{r}=\sqrt{\lambda}R_{\rho},\qquad R_{rr}=\lambda R_{\rho\rho}.\]

Now, (iii) can be rewritten as
\begin{equation} \label{eq:Eqn3}
R_{\rho\rho} + \frac{1}{\rho}R_\rho + \gr{1-\frac{n^2}{\rho^2}}R = 0
\end{equation}
This is \textbf{Bessel's differential equation of order $\bn$}. For
more about Bessel functions see \cite{WATSON},  \cite{BOWMAN},
\cite[Ch.~5]{FOLLAND-FOURIER}, \cite[Ch.~8]{HOCHSTADT},
\cite[Ch.~5,6]{LEBEDEV}.

\subsection{Solutions of Bessel's Differential Equation}
At $\rho=0$ the coefficients of the $R_\rho$ and $R$ terms in
\eqref{eq:Eqn3} blow up. So instead, multiply first by $\rho^2$:
$$\rho^2R_{\rho\rho} + \rho R_\rho + \gr{\rho-n^2}R = 0.$$
Now at $\rho=0$ the coefficients of the $R_\rho$ and $R$ do not blow up.

\ \\This $\rho=0$ is called a \emph{singular point}. However, it isn't
really a ``bad'' singular point since the denominators of the
coefficients of $R_\rho$ and $R$ in \eqref{eq:Eqn3} approach zero
\emph{polynomially} as $\rho\rightarrow0$. In other words, the pole does not
have \emph{exponential} growth.

\ \\More precisely, if
$$
\lim_{\rho\rightarrow0}\;
\rho\cdot\frac{\textrm{coef}(R_\rho)}{\textrm{coef}(R_{\rho\rho})}<\infty
$$
and
$$ \lim_{\rho\rightarrow0}\;
\rho^2\cdot\frac{\textrm{coef}(R)}{\textrm{coef}(R_{\rho\rho})}<\infty,
$$
then such a singular point is called a \emph{regular singular
point}. In this case we seek a solution in the form of the following
series
(see \cite[Part~I, Sec.~5.2]{MORSE-FESHBACH}, \cite[Ch.~VII]{INCE}
for more about the method of series solution for ODEs, as well as regular
singular points)
$$R(\rho) = \rho^\alpha \sum_{k=0}^\infty a_k \rho^k, \qquad a_0\neq0$$
for some $\alpha$ and $\{a_k\}_{k=0}^{\infty}$ to be determined. By
plugging this into \eqref{eq:Eqn3} we get
$$\rho^\alpha \sum_{k=0}^\infty \Vect{\gr{\alpha+k}\gr{\alpha+k-1}a_k\rho^{k-2}
+ \gr{\alpha+k}a_k\rho^{k-2} + a_k\rho^k - n^2a_k\rho^{k-2}} = 0$$

\ \\Now rewriting $\sum_{k=0}^\infty a_k \rho^k = \sum_{k=2}^\infty
a_{k-2} \rho^{k-2}$ we have
$$
%\left\{
  \begin{array}{ll}
    \vect{\alpha\gr{\alpha-1} + \alpha - n^2}a_0 = 0, & \hbox{when }k=0; \\
    \vect{\gr{\alpha+1}\alpha + \alpha + 1 - n^2}a_1 = 0, & \hbox{when }k=1; \\
    \vect{\gr{\alpha+k}\gr{\alpha+k-1} + \alpha + k - n^2}a_k + a_{k-2} = 0,
     & \hbox{when }k\geq2.
  \end{array}
%\right.
$$

So for $a_0\neq0$, from the Case $k=0$, we have that $\alpha^2 - n^2
= 0$, i.e., $\alpha = \pm n$.

\ \\Consider $\alpha=n$. Substituting in the Case $k=1$ above we get
$$\vect{n^2 + n + n + 1 - n^2}a_1 = 0,$$
$$\vect{2n +1}a_1 = 0.$$
Thus, $a_1 = 0$.
Note that when $k\geq2$ above we see that
the square bracket factor simplifies to
$\vect{\gr{\alpha+k}^2-n^2}$. Therefore,
$$a_k = \frac{-a_{k-2}}{\gr{n+k}^2-n^2}, \qquad k = 2,3,\ldots.$$
Since $a_1=0$ we know that $a_k=0$ whenever $k$ is odd.
So choose $a_0 = 1/\gr{2^nn!}$. Then we get
\begin{eqnarray*}
R(\rho) &=& \frac{\rho^n}{2^nn!}\Vect{1 - \frac{\rho^2}{2^2(n+1)} +
\frac{\rho^4}{2!2^4(n+1)(n+2)} - \cdots}\\
& = & \sum_{k=0}^\infty \frac{(-1)^k}{k!\gr{n+k}!}
\Gr{\frac{\rho}{2}}^{n+2k}\\
& \equiv & J_n(\rho)
\end{eqnarray*}
where $J_n(\cdot)$ is defined as Bessel function of the first kind
of order $n$, $n=0,1,2,\cdots$.

\ \\Asymptotically, $J_n(\rho) \sim \sqrt{\frac{2}{\pi\rho}}\cos\gr{\rho
- \frac{\pi}{4} - \frac{n\pi}{2}} +O(\rho^{-3/2})$ as
$\rho\rightarrow\infty.$ Here, the $\sqrt{\frac{2}{\pi\rho}}$ factor
represents a decay.

\subsection{The Eigenfunction Expansion}
Now we know $R(\rho)=cJ_n(\rho)$, where $c$ is an arbitrary
constant. With $\rho=\sqrt{\lambda}r$, we have the solution of the
form (here we just consider the stationary part of the solution)

$$\underbrace{J_n(\sqrt{\lambda}r)}_{radial}
\underbrace{\gr{A_n\cos n\theta + B_n\sin n\theta}}_{angular}$$

which separates the radial and angular components.

\ \\The Dirichlet BC dictates at $r=a$ that the radial component
$J_n(\sqrt{\lambda}a)=0$. Thus $\sqrt{\lambda}a$ is a zero of
$J_n(x)$.

\ \\Note that each $J_n(x)$ has an infinite number of positive
zeros, and that they are \emph{not} regularly or periodically
spaced. This can be seen more clearly in
Figure~\ref{fig:Bessel_Plot}. For a table of the zeros of Bessel
functions, see \cite{WATSON} and \cite[Ch.~9]{ABROMOWITZ-STEGUN}.
\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=5.5in, trim=0.5in 0 0 0]{besselplot.epsc} %[trim=l b r t]
 \caption{\label{fig:Bessel_Plot}$\Omega\subset\Rf^d$, with $\nu$ is the normal vector.}
\end{center}
\end{figure}

\ \\In general, let $j_{n,m}$ be the $m^{\mbox{th}}$ positive zero
of $J_n(x)$ for $n=0,1,2,\ldots$ and $m=1,2,3,\ldots$. Then the
zeros obey the following
$$0 < j_{n,1} < j_{n,2} < \cdots \qquad \mbox{and} \qquad
j_{n,m} < j_{n+1,m} < j_{n,m+1},$$ where in our scenario
$j_{n,m}=\sqrt{\lambda_{n,m}}\,a$, i.e.,
$\lambda_{n,m}=\left(\fr{j_{n,m}}{a}\right)^2$.

\ \\Finally, we have the solution modulo IC as
\begin{eqnarray} \label{eq:Bessel_soln} \nonumber
u\gr{r,\theta,t} &=& \sum_{m=1}^\infty J_0(\sqrt{\lambda_{0,m}}\,r)
\Gr{C_{0,m}\cos
\gr{\sqrt{\lambda_{0,m}}ct} + D_{0,m}\sin \gr{\sqrt{\lambda_{0,m}}ct}} \\
\nonumber &&\; + \; \sum_{n=1}^\infty \sum_{m=1}^\infty
J_n(\sqrt{\lambda_{n,m}}\,r) \Gr{A_{n,m}\cos \gr{n\theta} +
B_{n,m}\sin \gr{n\theta}} \\
&& \qquad\qquad\qquad \cdot\Gr{C_{n,m}\cos
\gr{\sqrt{\lambda_{n,m}}ct} + D_{n,m}\sin
\gr{\sqrt{\lambda_{n,m}}ct}}.
\end{eqnarray}

\ \\Note that the factors $J_0(\sqrt{\lambda_{0,m}}\,r)$ and
$J_n(\sqrt{\lambda_{n,m}}\,r) \Gr{A_{n,m}\cos \gr{n\theta} +
B_{n,m}\sin \gr{n\theta}}$ represent eigenfunctions of the Laplacian
operator.

\ \\In order to determine the coefficients $A_{n,m}, B_{n,m}$ for $n,m=1,2,\ldots\,$,
and $C_{n,m}, D_{n,m}$ for $n=0,1,\ldots,\,m=1,2,\ldots\,$ we use the ICs
$u(\bx,0)= f(\bx), u_t(\bx,0)= g(\bx)$.

\ \\Write $\sqrt{\lambda_{n,m}}=\beta_{n,m}=j_{n,m}/a$. Then at
$t=0$, \eqref{eq:Bessel_soln} reduces to
\begin{eqnarray} \nonumber
u\gr{r,\theta,0} &=& f(r,\theta)\\\nonumber
&=& \sum_{m=1}^\infty C_{0,m}J_0(\beta_{0,m}\,r) + \sum_{n=1}^\infty \sum_{m=1}^\infty
C_{n,m}J_n(\beta_{n,m}\,r) \\\nonumber
&& \hspace{2in}\cdot\Gr{A_{n,m}\cos \gr{n\theta} + B_{n,m}\sin \gr{n\theta}}
\end{eqnarray}
and
\begin{eqnarray} \nonumber
u_t\gr{r,\theta,0} &=& g(r,\theta)\\\nonumber
&=& \sum_{m=1}^\infty c\beta_{0,m}D_{0,m}J_0(\beta_{0,m}\,r) + \sum_{n=1}^\infty \sum_{m=1}^\infty
c\beta_{n,m}D_{n,m}J_n(\beta_{n,m}\,r) \\\nonumber
&& \hspace{2in}\cdot\Gr{A_{n,m}\cos \gr{n\theta} + B_{n,m}\sin \gr{n\theta}}.
\end{eqnarray}

\ \\
Let us exploit the orthogonality of
$\varphi_{n,m,l}(r,\theta)=J_n(\beta_{n,m}r){\cos\choose\sin}(m\theta)$,
$l=0$ for $\cos(m,\theta)$ and $l=1$ for $\sin(m,\theta)$.

\ \\Define the inner product for any $p,q\in L^2(\Omega)$ as
\begin{eqnarray*}
\inner{p}{q} &:=& \int_\Omega  p(\bx)\overline{q(\bx)} \,\mathrm{d}\bx \mathrm{d}\by \\
&=& \int_{-\pi}^\pi \int_0^a p(r,\theta)\overline{q(r,\theta)}\, r\!\dd{r}\!\dd{\theta} \\
\end{eqnarray*}

where the functional forms of $p$ and $q$ are adjusted with respect to the respective
coordinate system (i.e.,  Cartesian, polar, etc.).

\ \\For fixed $m$ it is easy to see that
$$\int_0^a J_n(\beta_{n,m}\,r) J_{n'}(\beta_{n',m}\,r)\, r\!\dd{r} = c\delta_{n,n'}$$
for some normalization constant $c$. Here, $\delta$ is the Kronecker
delta function, and different $n$'s correspond to different
``frequencies.'' Also, for fixed $n$ (i.e., for a fixed
``frequency'') we have that
$$\int_0^a J_n(\beta_{n,m}\,r) J_n(\beta_{n,m'}\,r)\, r\!\dd{r}  = \tilde{c}\delta_{m,m'}$$
for some normalization constant $\tilde{c}$. Furthermore, $\cos
n\theta$ and $\sin n\theta$ are mutually perpendicular for
$-\pi\leq\theta<\pi$.
% (Also $\cos n\theta$ is perpendicular to $\cos n'\theta$,
% and $\sin n\theta$ is perpendicular to $\sin n'\theta$?)
In summary, we have
$$\inner{\varphi_{n,m,l}}{\varphi_{n,m,l'}} = \mbox{const}\cdot\delta_{n,n'}\,\delta_{m,m'}\,
\delta_{l,l'}.$$

\ \\So, we can do the Fourier series expansion of $f(r,\theta)$ and
$g(r,\theta)$, and find the matching coefficients
$\{C_{o,m},C_{n,m}A_{n,m},C_{n,m}B_{n,m},D_{0,m},D_{n,m}A_{n,m},D_{n,m}B_{n,m}\}$
as
\begin{eqnarray*}
C_{0,m} &=& \frac{1}{2\pi a_{0,m}}
\int_{-\pi}^\pi \int_0^a f(r,\theta)J_0(\beta_{0,m}r)\, r\!\dd{r}\!\dd{\theta} \\
C_{n,m}A_{n,m} &=& \frac{1}{\pi a_{n,m}}
\int_{-\pi}^\pi \int_0^a f(r,\theta)J_n(\beta_{n,m}r)\cos n\theta\, r\!\dd{r}\!\dd{\theta} \\
C_{n,m}B_{n,m} &=& \frac{1}{\pi a_{n,m}}
\int_{-\pi}^\pi \int_0^a f(r,\theta)J_n(\beta_{n,m}r)\sin n\theta\, r\!\dd{r}\!\dd{\theta} \\
\end{eqnarray*}
and
\begin{eqnarray*}
c\beta_{0,m}D_{0,m} &=& \frac{1}{2\pi a_{0,m}}
\int_{-\pi}^\pi \int_0^a g(r,\theta)J_0(\beta_{0,m}r)\, r\!\dd{r}\!\dd{\theta}\\
c\beta_{n,m}D_{n,m}A_{n,m} &=& \frac{1}{\pi a_{n,m}}
\int_{-\pi}^\pi \int_0^a g(r,\theta)J_n(\beta_{n,m}r)\cos n\theta\, r\!\dd{r}\!\dd{\theta} \\
c\beta_{n,m}D_{n,m}B_{n,m} &=& \frac{1}{\pi a_{n,m}}
\int_{-\pi}^\pi \int_0^a g(r,\theta)J_n(\beta_{n,m}r)\sin n\theta\, r\!\dd{r}\!\dd{\theta}, \\
\end{eqnarray*}
where the normalization constant $a_{n,m}$ is defined as
\begin{eqnarray*}
a_{n,m} &\define& \int_0^a \Vect{J_n(\beta_{n,m}r)}^2 \, r\!\dd{r}\!\dd{\theta} \\
&=& \frac{1}{2} \, a^2\Vect{J_n'(\beta_{n,m}a)}^2 \\
&=& \frac{1}{2} \, a^2\Vect{J_{n\pm1}(\beta_{n,m}a)}^2. \\
\end{eqnarray*}

\textbf{Example:} Beat this drum with a stick in the middle (i.e.,
at $r=0$) at $t=0$, with ICs $u\gr{r,\theta,0} = f(r,\theta) = 0$,
and $u_t\gr{r,\theta,0} = g(r,\theta) = g(r)$. Since $f(r,\theta) =
0$, it must be that $C_{n,m}=0, \forall n,m$. Moreover, in examining
$g(r)$ we see that $n=0$ is the only remaining term, i.e.,
$D_{n,m}A_{n,m}=D_{n,m}B_{n,m}=0$, for $n\geq 1$. So we have
$$u\gr{r,\theta,t} = u\gr{r,t} = \sum_{m=1}^\infty D_{0,m}
J_0(\beta_{0,m}\,r)\sin \gr{\beta_{0,m}ct}$$ where
\begin{eqnarray*}
D_{0,m} &=& \frac{1}{c\beta_{0,m}a_{0,m}}\int_0^a g(r)J_0(\beta_{0,m}r)\,
r\!\dd{r} \\
&=& \frac{\int_0^a g(r)J_0(\beta_{0,m}r)\, r\!\dd{r}} {\frac{1}{2}
\, a^2c\beta_{0,m}\Vect{J_{1}(\beta_{0,m}a)}^2}.
\end{eqnarray*}

Therefore, the lowest ``note'' (or ``frequency'') we hear is the
fundamental frequency corresponding to $\beta_{0,1}c =
\frac{1}{a}j_{0,1}c \approx \frac{2.40483c}{a}$. Compare this with a
1-D string of length $a$ (with density $\rho$ and under
tension~$\mathcal{T}$). There, the lowest note would be $\frac{\pi
c}{a} \approx \frac{3.141592c}{a}$, where $c =
\sqrt{\frac{\mathcal{T}}{\rho}}$. Thus, if the $c$ parameter is the
same in both of the 1-D and 2-D cases, we see that the 2-D drum can
support a lower note. In the next lecture we will see, curiously,
under the same conditions that the lowest note of a 3-D ball is
actually equal to that of the 1-D string! For the science of real
drums and percussion, see \cite{ROSSING}.


\section{Neumann BC}
Denote $\lambda^{(N)}$ as Neumann-Laplacian eigenvalues, and
$\lambda^{(D)}$ as Dirichlet-Laplacian eigenvalues. Let
$$\psi(r,\theta) = \psi_n(r,\theta) = J_n(\sqrt{\lambda^{(N)}}r)
\Gr{A_n\cos \gr{n\theta} + B_n\sin \gr{n\theta}}.$$

\ \\The Neumann BC specifies
$$\left.\pdif{\psi_n}{r}\right|_{\partial\Omega} =\;
\pdif{\psi_n}{r}(a,0) \;=\; 0$$ which implies
$$\sqrt{\lambda^{(N)}}J_n'(\sqrt{\lambda^{(N)}}a)
\Gr{A_n\cos \gr{n\theta} + B_n\sin \gr{n\theta}} = 0.$$

\ \\Therefore, either $J_n'(\sqrt{\lambda^{(N)}}a)=0,
n=0,1,2,\ldots$, or $\lambda^{(N)} = 0$.

\subsection{Recurrence relations of Bessel Functions}
The Bessel functions obey the following relationships:
\begin{eqnarray*}
J_{\nu-1}(x) + J_{\nu+1}(x) &=& \frac{2\nu}{x} J_\nu(x)\\
J_{\nu-1}(x) - J_{\nu+1}(x) &=& 2J'_\nu(x)\\
J'_\nu(x) &=& J_{\nu-1}(x) - \frac{\nu}{x} J_\nu(x)\\
J'_\nu(x) &=& -J_{\nu+1}(x) + \frac{\nu}{x} J_\nu(x).
\end{eqnarray*}
In particular,
\begin{eqnarray*}
J'_0(x) &=& -J_1(x) \\
J'_1(x) &=& J_0(x) - \frac{J_1(x)}{x}\\
        &=& \frac{1}{2}\Gr{J_0(x) - J_2(x)} \\
J'_2(x) &=& \frac{1}{2}\Gr{J_1(x) - J_3(x)}.
\end{eqnarray*}

\ \\Therefore, $\lambda_{0,m}^{(N)} = \lambda_{1,m}^{(D)}, m = 1, 2,
\ldots\,$, and from Figure~\ref{fig:Bessel_Plot}, we have
$$0=\lambda_{0,1}^{(N)} \;<\; \lambda_{1,1}^{(N)} \;<\; \lambda_{2,1}^{(N)}
\;<\; \lambda_{0,2}^{(N)} \;<\; \lambda_{3,1}^{(N)} \;<\; \cdots$$
Also, the Dirichlet BC Laplacian eigenvalue $\lambda_{0,1}^{(D)}$
lies in the interval
$$\lambda_{0,1}^{(D)}\in \gr{\lambda_{1,1}^{(N)}, \lambda_{2,1}^{(N)}}$$
and
$$\lambda_{1,2}^{(D)} = \lambda_{0,2}^{(N)}.$$ Thus, Neumann BCs have the potential
in general to give lower fundamental frequencies.

\ \\In general, we can show that
$$\lambda_{n+1}^{(N)} \;\leq\; \lambda_n^{(D)}, \quad n = 1,2,\ldots\,$$
which is a special case of the Friedlander Theorem discussed in Lecture 3.

\begin{thebibliography}{10}
\bibitem{STRAUSS}
{\sc W.~A. Strauss}, {\em Partial Differential Equations: An Introduction},
Brooks/Cole Publishing Company, 1992.

\bibitem{COURANT-HILBERT}
{\sc R. Courant, D. Hilbert}, {\em Methods of Mathematical Physics},
Vol.~I, Wiley-Interscience, 1953.

\bibitem{INCE}
{\sc E. L. Ince}, {\em Ordinary Differential Equations}, Dover, 1956.

\bibitem{WATSON}
{\sc G. N. Watson}, {\em A Treatise on the Theory of Bessel Functions},
2nd ed., Cambridge Mathematical Library, Cambridge Univ. Press, 1995
(Originally published in 1948).

\bibitem{BOWMAN}
{\sc F. Bowman}, {\em Introduction to Bessel Functions}, Dover, 1958.

\bibitem{FOLLAND-FOURIER}
{\sc G.~B. Folland}, {\em Fourier Analysis and Its Applications},
Brooks/Cole Publishing Company, 1992.

\bibitem{HOCHSTADT}
{\sc H. Hochstadt}, {\em The Functions of Mathematical Physics},
Dover, 1986 (Originally published by Wiley-Interscience in 1971).

\bibitem{LEBEDEV}
{\sc N.~N. Lebedev}, {\em Special Functions and Their Applications},
Dover, 1972 (Originally published by Prentice-Hall in 1965).

\bibitem{MORSE-FESHBACH}
{\sc P. M. Morse, H. Feshbach}, {\em Methods of Theoretical Physics},
McGraw-Hill, 1953 (Republished by Feshbach Publishing in 2005).

\bibitem{ABROMOWITZ-STEGUN}
{\sc M. Abramowitz, I.~A. Stegun}, {\em Handbook of Mathematical Functions},
Dover, 1972, http://www.math.sfu.ca/~cbm/aands/.

\bibitem{ROSSING}
{\sc T. D. Rossing}, {\em Science of Percussion Instruments},
Series in Popular Science, Vol.3, World Scientific, 2000.

\end{thebibliography}
\end{document}

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