Linear partial differential equations

Hyperbolic, parabolic, elliptic; characteristics, modes, boundary conditions.

Partial differential equations are the language of fields — quantities that vary across space and time, related to themselves through their partial derivatives. The pressure in air, the displacement of a vibrating membrane, the temperature in a metal bar, the electrostatic potential in vacuum — each obeys a PDE, and almost every interesting equation in this bookshelf is one. This chapter is the working refresher.

Three canonical second-order linear PDEs cover most of physics: the wave equation (propagation at finite speed), the heat equation (smoothing and diffusion), and Laplace’s equation (steady-state, no time). Each is the prototype of a class — hyperbolic, parabolic, elliptic — with its own causal structure, its own data requirements, and its own characteristic solution techniques. The chapter walks all three, plus the two workhorse methods (d’Alembert’s traveling-wave formula and separation of variables) and the boundary-condition language that ties them together.

If you have not thought about PDEs in a while, that is the audience this chapter is written for. Each lesson reintroduces its idea from the picture down before any algebra is required.

• 6.1 What is a PDE? — the three canonical types, side-by-side comparison of wave vs heat, why classification matters.
• 6.2 The 1-D wave equation: d’Alembert and characteristics — the explicit general solution, domain of dependence, region of influence, the geometric picture for hyperbolic PDEs.
• 6.3 Separation of variables — the workhorse technique, worked end-to-end on the clamped string.
• 6.4 Boundary conditions — Dirichlet, Neumann, Robin, periodic; physics-first treatment with a cautionary worked example.
• 6.5 Modes and mode sums — eigenfunctions, orthogonality, Fourier projection, modal density, the bank-of-resonators picture.
• 6.6 The heat equation and Laplace’s equation — parabolic and elliptic prototypes; the spatial machinery transferred, the time evolution replaced.
• 6.7 The Helmholtz equation — the time-harmonic reduction of the wave equation that frequency-domain acoustics actually solves, plus a worked 2-D cavity-modes derivation.
• 6.8 The Schrödinger equation — a brief detour outside acoustics, demonstrating that the same separation-of-variables machinery underwrites quantum mechanics.