2.3 The Laplacian and harmonic functions

The previous two lessons developed the three first-order vector operators: gradient, divergence, and curl. This third lesson builds the second-order operator $\nabla^2$ — the Laplacian — that combines them. It is, in the operator algebra of physics, the central object: every wave, every diffusion, every steady-state field equation in this bookshelf is built on it.

The Laplacian

The Laplacian is the divergence of the gradient:

\boxed{\;\nabla^2 \phi \;=\; \nabla \cdot (\nabla \phi) \;=\; \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + \frac{\partial^2 \phi}{\partial z^2}.\;}

(Some textbooks write $\Delta\phi$ instead of $\nabla^2\phi$ ; both notations are universal.)

The Laplacian measures how much a scalar field at a point differs from the average of its immediate neighbours — sometimes called the “neighbourly-average” operator. The argument is short. The average of $\phi$ over a small sphere of radius $r$ centred at $\mathbf{r}_0$ is, by Taylor expansion,

\langle \phi \rangle_r \;\approx\; \phi(\mathbf{r}_0) + \frac{r^2}{6}\, \nabla^2 \phi(\mathbf{r}_0) + \mathcal{O}(r^4).

The difference $\langle\phi\rangle_r - \phi(\mathbf{r}_0)$ is proportional to $\nabla^2 \phi$ , with sign and magnitude depending on the local curvature:

If $\phi$ is locally a maximum (a hill peak), neighbours are all lower, so $\nabla^2 \phi < 0$ .
If $\phi$ is locally a minimum (a bowl bottom), neighbours are all higher, so $\nabla^2 \phi > 0$ .
Flat or linear regions give $\nabla^2 \phi = 0$ .

The Laplacian generalises the second derivative from one dimension to many: in 1-D, $\nabla^2 \phi = \partial^2 \phi / \partial x^2$ is exactly the second derivative, which measures concavity. In 2-D and 3-D, the same idea applies to all directions simultaneously.

field:

Negative at the centre (the field is locally maximal), positive in a ring outside (the field is locally minimal along that ring after passing the inflection).

The interactive shows a scalar field $\phi$ on the left and its Laplacian $\nabla^2 \phi$ on the right. The colour code: red where the Laplacian is positive (the field is a local minimum), blue where it’s negative (local maximum), cream where it’s zero (no net curvature). Notice the saddle preset: the Laplacian is identically zero — the rise in one direction exactly cancels the fall in the perpendicular direction. Such functions are called harmonic.

Harmonic functions

A function satisfying $\nabla^2 \phi = 0$ everywhere is called harmonic. Equivalently, it is a solution of Laplace’s equation:

\nabla^2 \phi \;=\; 0.

Harmonic functions have several remarkable properties:

Mean-value property. The value of a harmonic function at any point equals the average of its values on any sphere centred at that point. (This is the operator interpretation, made global.) The interactive’s saddle preset is a harmonic function: at every point, the value equals the average over the neighbourhood.
Maximum principle. A harmonic function on a bounded region attains its maximum and minimum on the boundary, not in the interior. (Unless it is constant.) This is a direct consequence of the mean-value property — at an interior max, the value would exceed its neighbourhood average, contradicting the mean-value property.
Uniqueness. Two harmonic functions on a region with the same boundary values are identical. (Their difference is harmonic and zero on the boundary, hence zero everywhere by the maximum principle.) This is why specifying $\phi$ on the boundary uniquely determines the interior.

These properties make Laplace’s equation a boundary-value problem rather than an initial-value problem. There is no time evolution; the boundary data determines everything. Develop this further in Foundations 6.6.

Examples of harmonic functions:

Electrostatic potential in a region with no charges. Solving Laplace’s equation with grounded-conductor boundary conditions is the standard electrostatic problem.
Gravitational potential in vacuum.
Steady-state temperature in a thermally-equilibrated body with fixed boundary temperatures.
Velocity potential of an incompressible irrotational flow — used implicitly in Sound Ch 4 when we work with the linearised acoustic potential.
The real and imaginary parts of any holomorphic complex function $f(z) = u(x, y) + i v(x, y)$ , with $z = x + iy$ . The Cauchy–Riemann equations force $u$ and $v$ to be harmonic. This is the bridge between complex analysis and 2-D potential theory.

▶ Verifying that a function is harmonic Worked Example

Let $\phi(x, y) = e^{kx}\cos(ky)$ , a solution that arises in 2-D acoustic duct problems. Verify $\nabla^2\phi = 0$ .

\frac{\partial^2 \phi}{\partial x^2} = k^2 e^{kx}\cos(ky), \qquad \frac{\partial^2 \phi}{\partial y^2} = -k^2 e^{kx}\cos(ky).

Sum:

\nabla^2\phi = k^2 e^{kx}\cos(ky) - k^2 e^{kx}\cos(ky) = 0. \;\checkmark

The positive curvature in $x$ (exponential growth) is exactly cancelled by the negative curvature in $y$ (cosine oscillation). By the mean-value property, $\phi$ at any point equals its average over any surrounding circle — no local maxima or minima exist in the interior.

The Laplacian in wave physics

The Laplacian is the most important operator in wave physics. The 3-D acoustic wave equation reads

\frac{\partial^2 \phi}{\partial t^2} \;=\; c^2\, \nabla^2 \phi.

Every other derivation in the Sound book either produces this equation, manipulates it, or unpacks one of its consequences. The five canonical linear PDEs of Foundations 6:

Wave equation: $\partial_t^2 u = c^2 \nabla^2 u$ .
Heat equation: $\partial_t u = D \nabla^2 u$ .
Laplace’s equation: $\nabla^2 u = 0$ .
Helmholtz equation: $\nabla^2 u + k^2 u = 0$ .
Schrödinger equation: $i\hbar \partial_t \Psi = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi$ .

All five are different ways of relating $\nabla^2$ to either time evolution, a source term, or both. The Laplacian is the universal “spatial” operator of linear physics; choosing what’s on the other side of the equality gives you the four canonical equations of mathematical physics.

A plane-wave field $\phi(\mathbf{r}, t) = e^{i(\mathbf{k} \cdot \mathbf{r} - \omega t)}$ is an eigenfunction of the Laplacian: $\nabla^2 \phi = -|\mathbf{k}|^2 \phi$ . The Laplacian acts on plane waves by multiplying by $-k^2$ . This is what makes the wave equation reduce to the dispersion relation $\omega^2 = c^2 |\mathbf{k}|^2$ , and what makes the Helmholtz equation an eigenvalue problem with eigenvalues $-k^2$ — see Foundations 6.7 for the full development.

What we use it for

The Laplacian is invoked across the bookshelf:

The continuity equation uses $\nabla \cdot$ (Sound 4.2).
Euler’s equation for a fluid uses $\nabla$ on pressure (Sound 4.3).
The wave equation uses $\nabla^2$ on the velocity potential or pressure (Sound 4.5 onwards).
Plane-wave solutions use $\nabla$ acting on $e^{i \mathbf{k} \cdot \mathbf{r}}$ , which is just multiplication by $i \mathbf{k}$ . See Sound 5.1 — Plane harmonic waves for a 2-D animation of the pressure field varying with wavevector and wavelength.
Energy flux and intensity use $\mathbf{v} \cdot \nabla p$ and friends.
The heat equation uses $\nabla^2$ on the temperature field — see Foundations 6.6 and Sound 10.1.
Helmholtz cavity modes are eigenfunctions of the Laplacian — see Foundations 6.7, Sound 7.7.

That closes the chapter. Vector calculus is the algebraic toolkit you’ll be using for the rest of the bookshelf: gradient for force fields, divergence for conservation laws, curl for rotational structure, Laplacian for everything wavelike, harmonic, or diffusive.

Drill

Rote recall of the Laplacian and the canonical PDEs it generates, as a spaced-repetition deck. Reveal each card, then grade yourself — Again / Hard / Good / Easy — and SM-2 schedules when it returns. Progress is shared with the Foundations study deck.