Laplace's Equation on a Rectangle

Main idea

Wait long enough and a plate's temperature stops changing. Then $u_t = 0$ , and the heat equation leaves $u_{xx} + u_{yy} = 0$ : the bending in $x$ has to cancel the bending in $y$ . On a rectangle with one heated edge, that gives sines along the edge and hyperbolic sines that fade across the plate.

{\htmlData{sym=0}{\nabla^2u}} = u_{xx} + u_{yy} = 0

What the symbols mean

$\nabla^2u$
Laplacian
Measures how u compares with its average nearby; zero means perfectly balanced.

Hot top edge, other edges at 0

u(x, y) = \sum_{n=1}^{\infty} {\htmlData{sym=0}{B_n}}\sin\frac{n\pi x}{a}\,{\htmlData{sym=1}{\frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)}}}

What the symbols mean

$B_n$
Edge coefficients
The sine coefficients of the top edge's temperature f(x).
$\frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)}$
Growth across the plate
0 at the bottom edge, 1 at the top; high modes stay near the top edge.

Explanation

Laplace's equation is simply the rule a temperature follows once it has stopped changing: every point sits at the average of the points around it. It's named after Pierre-Simon Laplace, who used it to study gravity, and it's good for any "settled down" situation, like the final temperature of a plate, the voltage around charged objects, or groundwater seeping through soil. Here's where it comes from: take the 2D heat equation $u_t = k(u_{xx} + u_{yy})$ , hold the edges at fixed temperatures, and wait until $u_t = 0$ (no more change in time). What's left is $u_{xx}$ (how much a point bends compared with its neighbors left and right) plus $u_{yy}$ (the same thing up and down) adding to zero, so any bending one way in $x$ is cancelled by bending the other way in $y$ . It's just a balance condition, nothing scarier; bear with the hyperbolic sines below, they're simpler than they look.

Functions that solve Laplace's equation are called harmonic functions. The same equation shows up for steady temperatures, electrostatic potentials, and the slow flow of ideal fluids, so everything here carries over to all three.

Solving on a rectangle. Take $0 \le x \le a$ , $0 \le y \le b$ with $u = f(x)$ on the top and $0$ on the other three sides. Separation works just like it did for the rod: $u = X(x)Y(y)$ gives $X'' + \lambda X = 0$ with zero ends (so sines) and $Y'' - \lambda Y = 0$ with $Y(0) = 0$ (so $\sinh$ ). Notice the sign flip. Since $X''/X + Y''/Y = 0$ , if $X$ oscillates then $Y$ has to grow, which is why the second factor is a hyperbolic sine instead of a decaying exponential in time. Then match the top edge with a sine series, the same way you matched an initial temperature before.

With heat on several edges, solve one edge at a time and add. Laplace's equation is linear, so the pieces don't interfere.

A common mistake. After the rod, it's tempting to put sines in both directions, $\sin\frac{n\pi x}{a}\sin\frac{m\pi y}{b}$ , because every edge looks like a "fixed end." It isn't that simple: a product of two sines is zero on all four edges, so it can never match a hot top edge. Only the direction with two zero edges gets sines. The other direction gets $\sinh$ , chosen to vanish on the one cold edge and to carry the boundary data on the hot one.

The mean value property. The value of a harmonic function at any point equals its average around any circle centered there. That's the "perfectly balanced" idea made exact, and we'll see why it's true in the next lesson, on a disk.

The maximum principle. It follows straight from the mean value property. A point can't be hotter than every point around it if it's the average of them. So a harmonic function has no maximum or minimum inside the region: the hottest and coldest points of a steady plate are on its boundary.

See it

A square plate with its far edge held at sin πx + a·sin 3πx and the other three edges at 0. As the wiggle a grows, the hot edge gets bumpy, but notice how fast the bumps flatten out once you step into the plate. A mode with n bumps fades like e^(−nπ × distance), so the 3-bump part is nearly gone by the middle: it moves the center temperature by under 1% of its size at the edge. Steady heat smooths things out. Drag to turn the picture.

The steady temperature of a unit plate with its top edge at sin πx and the other edges at 0. The hottest point is right on the hot edge, at height 1, and the temperature slopes down toward the cold edges with no bumps or dips inside, just as the maximum principle says. The center sits at about 0.2.

Worked examples

Example 1

Solve $u_{xx} + u_{yy} = 0$ on the unit square with $u = 0$ on three edges and $u(x, 1) = \sin\pi x$ .

1
The top edge is already a single sine, so only the $n = 1$ mode is needed: $u = B_1\sin\pi x\,\sinh\pi y$ . This is zero on the other three edges automatically.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Which function is harmonic?

Problem 2+5 XP

Solve on the unit square: $u = 0$ on three edges, $u(x, 1) = \sin\pi x$ .

Problem 3+5 XP

Solve on the unit square: $u = 0$ on three edges, $u(x, 1) = 3\sin 2\pi x$ .

Problem 4+5 XP

For $u = \frac{\sin\pi x\,\sinh\pi y}{\sinh\pi}$ , find the temperature at the center $\left(\frac12, \frac12\right)$ .

Problem 5+5 XP

Where does a harmonic function on a plate reach its maximum?

Problem 6+5 XP

A 1 m square metal plate has its top edge held at $100\sin\pi x$ °C and its other edges at 0 °C, so the steady temperature is $u = \frac{100\sin\pi x\,\sinh\pi y}{\sinh\pi}$ . What's the temperature 10 cm below the middle of the hot edge, at $(0.5, 0.9)$ ?

Problem 7+5 XP

Is $u = \ln(x^2 + y^2)$ harmonic away from the origin?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A unit square plate has its top edge held at $\sin\pi x$ , its right edge held at $\sin\pi y$ , and the other two edges at 0. Use superposition to find the steady temperature at the center.

Where it's used

Steady heat in walls and circuit boards, voltages around electrodes, groundwater flow and the shape of a soap film stretched over a nearly flat wire all satisfy Laplace's equation.

Electric Fields and EquipotentialsCalc III · Physics

D'Alembert's Solution Laplace's Equation on a Disk

Laplace's Equation on a Rectangle

Main idea

{\htmlData{sym=0}{\nabla^2u}} = u_{xx} + u_{yy} = 0

What the symbols mean

$\nabla^2u$
Laplacian
Measures how u compares with its average nearby; zero means perfectly balanced.

Hot top edge, other edges at 0

u(x, y) = \sum_{n=1}^{\infty} {\htmlData{sym=0}{B_n}}\sin\frac{n\pi x}{a}\,{\htmlData{sym=1}{\frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)}}}

What the symbols mean

$B_n$
Edge coefficients
The sine coefficients of the top edge's temperature f(x).
$\frac{\sinh(n\pi y/a)}{\sinh(n\pi b/a)}$
Growth across the plate
0 at the bottom edge, 1 at the top; high modes stay near the top edge.

Explanation

With heat on several edges, solve one edge at a time and add. Laplace's equation is linear, so the pieces don't interfere.

See it

Worked examples

Example 1

Solve $u_{xx} + u_{yy} = 0$ on the unit square with $u = 0$ on three edges and $u(x, 1) = \sin\pi x$ .

1
The top edge is already a single sine, so only the $n = 1$ mode is needed: $u = B_1\sin\pi x\,\sinh\pi y$ . This is zero on the other three edges automatically.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

Which function is harmonic?

Problem 2+5 XP

Solve on the unit square: $u = 0$ on three edges, $u(x, 1) = \sin\pi x$ .

Problem 3+5 XP

Solve on the unit square: $u = 0$ on three edges, $u(x, 1) = 3\sin 2\pi x$ .

Problem 4+5 XP

For $u = \frac{\sin\pi x\,\sinh\pi y}{\sinh\pi}$ , find the temperature at the center $\left(\frac12, \frac12\right)$ .

Problem 5+5 XP

Where does a harmonic function on a plate reach its maximum?

Problem 6+5 XP

Problem 7+5 XP

Is $u = \ln(x^2 + y^2)$ harmonic away from the origin?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A unit square plate has its top edge held at $\sin\pi x$ , its right edge held at $\sin\pi y$ , and the other two edges at 0. Use superposition to find the steady temperature at the center.

Where it's used

Steady heat in walls and circuit boards, voltages around electrodes, groundwater flow and the shape of a soap film stretched over a nearly flat wire all satisfy Laplace's equation.

Electric Fields and EquipotentialsCalc III · Physics