Heat in a Rectangular Plate

Heat in a rectangular plate is simply the heat equation from the rod, spread out over a flat sheet whose edges are held cold. Each "mode" is a basic temperature pattern that keeps its shape and just fades, and the full solution is a mix of them, a double Fourier series because it needs sines in two directions. It's good for anything flat that's cooling: computer chips, griddles, walls. All we're really saying is that the plate cools the way a rod does, with patterns that wiggle in two directions instead of one. The formula looks heavy, but it's a few simple parts: $B_{mn}$ (how much of each pattern you start with), $\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b}$ (a pattern with $m$ bumps across and $n$ bumps down) and $e^{-k\lambda_{mn}t}$ (a fade-out timer that runs faster the bumpier the pattern is). Bear with the double sums; they're just the rod twice.

From a rod to a plate. On a rod with zero ends, the heat equation gave modes $e^{-k(n\pi/L)^2t}\sin\frac{n\pi x}{L}$. A rectangular plate $0 \le x \le a$, $0 \le y \le b$ with its edges held at 0 is the same problem with one more direction, $u_t = k(u_{xx} + u_{yy})$. So it makes sense to try the same trick with one more factor.

Three-way separation. $u = X(x)Y(y)T(t)$ gives $X'' + \mu X = 0$, $Y'' + \nu Y = 0$ (each with zero ends) and $T' = -k(\mu + \nu)T$. The $x$ and $y$ problems are each just the rod problem again, so $X$ and $Y$ are sines. The only new thing is that the decay rate adds the two: $\lambda_{mn} = \mu_m + \nu_n$. Matching the initial temperature takes a double sine series, with one integral in each direction.

The slowest mode is $(1, 1)$: it has the fewest wiggles, so its curvature is smallest and heat leaks out of it slowest. In the long run the plate cools like $e^{-k\lambda_{11}t}$, with a single hump in the middle. Everything else dies first.

Degenerate modes. On a square, $(1, 2)$ and $(2, 1)$ have the same eigenvalue: different shapes decay at the same rate. That couldn't happen on a rod, where each eigenvalue had just one sine. It's a sign of the square's symmetry, since rotating the square turns one mode into the other.

The same separation handles drums ($u_{tt}$ instead of $u_t$) and three-dimensional boxes (a triple series). We'll use it on a round drum in the next lesson.

A common mistake. It's tempting to think the mode with the biggest coefficient $B_{mn}$ is the one you'll see later on, since it starts out largest. It isn't. The exponentials win: a mode that starts ten times bigger but decays five times faster is still gone long before $(1, 1)$. There's an animation further down showing a ripple with five times the hump's decay rate vanishing while the hump is still mostly there.

Cooling of computer chips and cooking surfaces, and the thermal design of buildings, use 2D and 3D heat models; the same double series describe rectangular membranes.