fix(ch13): add worked example for 2D wave equation

chapters/ch13_wave_and_laplace.tex

@@ -635,6 +635,42 @@ and the angular frequencies are
   where $\omega_{mn} = c\pi\sqrt{m^2/a^2 + n^2/b^2}$ and the coefficients are determined by the initial displacement and velocity.
 \end{keyresult}
 
+\begin{workedexample}
+  A rectangular membrane of size $a \times b$ is fixed on all edges. The initial displacement is $u(x,y,0) = \sin(\pi x/a)\sin(\pi y/b)$ and the initial velocity is $u_t(x,y,0) = 0$. Find the solution $u(x,y,t)$.
+
+  \textbf{Solution.} The general solution for a rectangular membrane with fixed edges is
+  \[
+    u(x,y,t) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}
+               \sin\!\left(\frac{m\pi x}{a}\right)\sin\!\left(\frac{n\pi y}{b}\right)
+               \Bigl[A_{mn}\cos(\omega_{mn}t) + B_{mn}\sin(\omega_{mn}t)\Bigr],
+  \]
+  where $\omega_{mn} = c\pi\sqrt{m^2/a^2 + n^2/b^2}$.
+
+  The initial velocity condition gives
+  \[
+    u_t(x,y,0) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}
+                 \omega_{mn}\,B_{mn}\,
+                 \sin\!\left(\frac{m\pi x}{a}\right)\sin\!\left(\frac{n\pi y}{b}\right) = 0,
+  \]
+  so by the orthogonality of sine products, $B_{mn} = 0$ for all $m, n$.
+
+  The initial displacement condition gives
+  \[
+    u(x,y,0) = \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}
+               A_{mn}\,\sin\!\left(\frac{m\pi x}{a}\right)\sin\!\left(\frac{n\pi y}{b}\right)
+               = \sin\!\left(\frac{\pi x}{a}\right)\sin\!\left(\frac{\pi y}{b}\right).
+  \]
+  Again by orthogonality, $A_{11} = 1$ and $A_{mn} = 0$ for all $(m,n) \neq (1,1)$.
+
+  The solution is a single mode:
+  \[
+    u(x,y,t) = \sin\!\left(\frac{\pi x}{a}\right)\sin\!\left(\frac{\pi y}{b}\right)
+               \cos\!\left(c\pi\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}\,t\right).
+  \]
+
+  \textbf{Interpretation.} The initial displacement already has the shape of the fundamental mode $(m,n) = (1,1)$, so only the fundamental mode is excited. The membrane oscillates at its lowest eigenfrequency $\omega_{11} = c\pi\sqrt{1/a^2 + 1/b^2}$ with unit amplitude. If the initial displacement had contained higher modes (e.g., a sum of several products of sines), each mode would oscillate independently at its own frequency.
+\end{workedexample}
+
 \paragraph{Discussion.} Unlike the one-dimensional string, the frequencies of a rectangular membrane are \emph{not} integer multiples of a fundamental. The ratio $\omega_{mn}/\omega_{11} = \sqrt{m^2 a^2 + n^2 b^2}\,/\,\sqrt{a^2 + b^2}$ is generally irrational. This is why a drum produces a sound with no clear fundamental pitch --- a \textbf{non-harmonic} spectrum.
 
 \subsection{Laplace's Equation in Rectangles}