6.1 What is a PDE?

In Foundations 5 the unknown was a function of one variable — a curve $x(t)$ . The equation related the curve’s value to its time derivatives, and the solution was a curve threading through a slope field. That was an ordinary differential equation.

A partial differential equation (PDE) is what we write when the unknown depends on more than one variable. The wave equation for sound pressure,

\frac{\partial^2 p}{\partial t^2} \;=\; c^2 \left( \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2} + \frac{\partial^2 p}{\partial z^2} \right),

has $p(x, y, z, t)$ as its unknown — a field of pressure values varying through space and time. Diffusion of heat through a metal bar has temperature $T(x, t)$ as its unknown. The shape of a soap film stretched on a wire loop is governed by the Laplace equation for height $h(x, y)$ .

The defining feature of PDEs is that the unknown is a field — a function of several variables — and the equation relates it to its partial derivatives with respect to each variable independently. Where an ODE is a relation in time alone, a PDE is a relation across all the dimensions the unknown lives in.

Three canonical PDEs run almost everything

Most of physics lives downstream of three second-order linear PDEs. They are the workhorses of Sound, Hearing, and (eventually) Cavitation, and almost every more complicated PDE you meet is either one of these in disguise or a small modification of one.

The wave equation. Pressure in air, displacement of a stretched string, voltage on a transmission line, the electromagnetic field in vacuum — each obeys (in one form or another)

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

A disturbance launched at one point travels outward at speed $c$ , preserving its shape. This is the equation of propagation. Almost everything in the Sound book is a special case of it.

The heat (or diffusion) equation.

\frac{\partial u}{\partial t} \;=\; D\, \nabla^2 u.

Heat flowing through a solid, a drug diffusing in tissue, the concentration of a chemical in solution — anything that smooths and spreads rather than propagates. This is the equation of relaxation.

Laplace’s equation.

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

The steady-state limit of the heat equation (set $\partial_t u = 0$ ), and the form taken by the gravitational and electrostatic potentials in empty regions. The solution at any interior point is the average of its neighbours; the boundary values alone determine the entire field. This is the equation of equilibrium.

Hyperbolic, parabolic, elliptic

The three equations above are the prototypes of the three classes into which second-order linear PDEs in two variables fall, named by analogy with the conic sections. The classification depends on the sign of a discriminant constructed from the second-derivative coefficients, but the physical content is what matters here:

Class	Prototype	Time derivatives	Behaviour	Where it shows up
Hyperbolic	wave equation $u_{tt} = c^2 u_{xx}$	second order	propagates at finite speed; preserves shape	acoustics, optics, EM, gravitational waves
Parabolic	heat equation $u_t = D u_{xx}$	first order	diffuses; smooths out instantly; loses information	thermal conduction, viscous diffusion, financial Black–Scholes
Elliptic	Laplace $u_{xx} + u_{yy} = 0$	none	steady-state; boundary data determines interior globally	electrostatics, gravity in vacuum, incompressible irrotational flow

The classification is not trivia. It controls three things that matter to anyone trying to set up and solve a PDE:

What data is needed to specify a unique solution. Hyperbolic equations need initial conditions (and possibly boundary conditions). Parabolic equations need initial conditions plus boundary conditions. Elliptic equations need boundary conditions only — there is no time, so there is no initial moment.
How fast information travels. Hyperbolic: at finite speed (the characteristic speed $c$ ). Parabolic and elliptic: instantaneously — a change anywhere is felt everywhere immediately, although with exponentially small amplitude far away.
What numerical methods are stable. Each class wants different schemes; mismatching scheme to class is the most common reason a simulation explodes.

The three classes, made visible

The cleanest way to feel the difference between hyperbolic and parabolic is to put them on the same interval, give them the same initial profile, and let them run side by side.

initial:

wave speed c = 1.00 diffusivity D = 0.050

Both panels start from the same initial profile, sit on the same interval, and run for the same elapsed time. The wave panel preserves shape and propagates the disturbance at finite speed c, reflecting off the clamped ends. The heat panel begins smoothing the profile everywhere at once — the moment t is positive, the field at every point has felt the initial data. Hyperbolic and parabolic PDEs answer the same kind of question with completely different causal structures.

Both panels start with the same Gaussian pulse, sit on the same interval $[0, L]$ , and obey the same homogeneous Dirichlet boundary conditions $u = 0$ at both ends. The clock at the bottom is shared. (The wave panel uses the leapfrog scheme; the heat panel uses forward-time, centred-space. Both are developed in Foundations 10.3, including the stability bounds that constrain the time-step.) Watch what each equation does with that pulse:

The wave panel splits the pulse into two halves, each travelling outward at speed $c$ , reflecting off the clamped ends, returning, interfering, and re-forming the pulse periodically. Sharp features stay sharp. The total motion is causally constrained: at time $t$ , the disturbance has only reached points within a distance $ct$ of where it started.
The heat panel does not split the pulse — it begins smoothing it everywhere at once. The peak shrinks, the tails fill in, and the profile relaxes towards zero. Every point of the field has felt the initial data the moment $t > 0$ . There is no light cone.

Elliptic equations are not in this comparison because they have no time. We meet Laplace’s equation properly in 6.6, where the absence of a time derivative becomes the equation’s defining feature.

Order, again

Just as for ODEs, the order of a PDE is the highest order of any partial derivative that appears. All three of our canonical PDEs are second order in space. The wave equation is second order in time; the heat equation is first order in time; Laplace’s equation has no time at all. That hierarchy in the time variable is exactly what produces the three different qualitative behaviours.

For linear PDEs the order of the time derivative also controls how much initial data you need. A second-order time derivative needs two initial functions — typically $u(x, 0)$ and $\partial_t u(x, 0)$ . A first-order time derivative needs one: $u(x, 0)$ . No time derivative needs none. Boundary data fills in the rest.

What this chapter does

The remaining five lessons walk the two solution techniques the rest of the bookshelf actually uses, the boundary-condition language they rely on, and the parabolic / elliptic cases that round out the canonical three.

6.2 — The 1-D wave equation: d’Alembert and characteristics. The explicit general solution of the 1-D wave equation, the characteristic curves along which information travels, and the geometric domain-of-dependence picture that hyperbolic PDEs are built around.
6.3 — Separation of variables. The workhorse technique for bounded domains: turn one PDE into two ODEs by factoring the unknown as $u(x, t) = X(x)\, T(t)$ . Worked end-to-end on the clamped string.
6.4 — Boundary conditions. The four families (Dirichlet, Neumann, Robin/impedance, periodic), the physics that selects each, and a cautionary worked example of what goes wrong when you state the boundary condition on the wrong field.
6.5 — Modes and mode sums. What the mode catalogue actually is — eigenfunctions of a self-adjoint spatial operator, orthogonal and complete — plus Fourier projection, modal density, and the forced-mode picture.
6.6 — The heat equation and Laplace’s equation. The parabolic and elliptic prototypes, what kind of data they ask for, and why their solution behaviour is so different from the wave equation.
6.7 — The Helmholtz equation. The time-harmonic reduction of the wave equation: the elliptic eigenvalue problem that frequency-domain acoustics actually solves, plus 2-D cavity modes worked end-to-end.
6.8 — The Schrödinger equation. A small detour outside acoustics, showing that the same separation-of-variables machinery underwrites quantum mechanics — the particle in a box is the clamped string, mathematically.

If your PDE intuition has rusted, this chapter is the audience it was written for. Each lesson reintroduces its idea from the picture down before any algebra is required.