Recipe

Find the natural frequencies of a rectangular membrane

Use when: you have $u_{tt}=c^2(u_{xx}+u_{yy})$ on a rectangle with $u=0$ on all four edges

Goal: every natural frequency $\omega_{mn}$ , and (if needed) the coefficients for the full series solution

Taught in The vibrating rectangular membrane

Quick recipe

Solve each 1-D Dirichlet eigenvalue problem separately
Add the two eigenvalues to get $\lambda_{mn}$
Take $\omega_{mn}=c\sqrt{\lambda_{mn}}$ ; don't stop at $\lambda_{mn}$
If you need the full solution, expand $f$ and $g$ in the double basis

1
Treat the $x$ -direction and $y$ -direction as two independent 1-D Dirichlet problems, each already solved in earlier chapters: $\mu_m=(m\pi/a)^2$ with $X_m=\sin(m\pi x/a)$ , and $\nu_n=(n\pi/b)^2$ with $Y_n=\sin(n\pi y/b)$ .

Example

For $a=4,b=3$ , $m=n=1$ : $\mu_1=(\pi/4)^2$ and $\nu_1=(\pi/3)^2$ .

Why: each edge condition only involves one direction's own function. So the two 1-D eigenvalue problems never interact with each other.
2
The combined spatial eigenvalue is simply the sum, $\lambda_{mn}=\mu_m+\nu_n$ , with no other combination rule involved.

Example

$\lambda_{11}=\pi^2(\tfrac1{16}+\tfrac19)=\pi^2(\tfrac{25}{144})=(\tfrac{5\pi}{12})^2$ .

Why: the 2-D Laplacian splits into $X''/X+Y''/Y$ . So its eigenvalue is a plain sum of the two 1-D eigenvalues.
3
Multiply the square root of $\lambda_{mn}$ by the wave speed $c$ to get the actual natural frequency, since the time equation here is second order, just like the 1-D string's.

Example

With $c=2$ : $\omega_{11}=2\cdot\tfrac{5\pi}{12}=\tfrac{5\pi}6\approx2.618$ .

Why: an eigenvalue only ever feeds the spatial equation. A second-order time equation always converts it into a frequency, through this same $c\sqrt\lambda$ pattern.
4
Check whether the initial shape $f(x,y)$ is already one pure product mode, or a general function needing the double coefficient integral.

Example

For $f(x,y)=x(4-x)y(3-y)$ on the same rectangle, the double integral factors: $\int_0^4x(4-x)\sin(\pi x/4)\,dx=\tfrac{256}{\pi^3}$ and $\int_0^3y(3-y)\sin(\pi y/3)\,dy=\tfrac{108}{\pi^3}$ , giving $A_{11}=\tfrac4{12}\cdot\tfrac{256}{\pi^3}\cdot\tfrac{108}{\pi^3}=\tfrac{9216}{\pi^6}\approx9.586$ .
- $f$ is already a single product $\sin(m\pi x/a)\sin(n\pi y/b)$ — read that mode's coefficient directly off $f$ 's own scale factor, with every other coefficient zero.
- $f$ is a general or multi-mode function — compute $A_{mn}=\frac4{ab}\int_0^a\int_0^bf\sin(m\pi x/a)\sin(n\pi y/b)\,dy\,dx$ , factoring into two 1-D integrals whenever $f$ itself is a product.
Why: this is exactly ch.05 §8's own double coefficient formula. It is now paired with a second-order time equation, instead of a first-order one.

Common mistakes

Reporting $\lambda_{mn}$ itself as the frequency instead of $\omega_{mn}=c\sqrt{\lambda_{mn}}$ .
Swapping $m$ and $n$ between the two directions; $\omega_{mn}\neq\omega_{nm}$ whenever $a\neq b$ .
Extracting only one direction's factor and forgetting the other direction still needs its own integral.
Using the 1-D natural-frequency formula $cn\pi/L$ directly on one direction instead of combining both eigenvalues first.