Wave Equation

3 min read#waves

The partial differential equation whose solutions are anything that travels left or right at a fixed speed without changing shape.

Contents

Wave Equation

The wave equation is the law every non-dispersive wave obeys:

\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}.

Read it as a statement about curvature: the acceleration of the medium at a point (\partial_{tt}u) is proportional to how sharply the medium is bent there (\partial_{xx}u). A taut string that is curved upward gets pulled downward by its own tension, and the constant of proportionality c^2 sets how fast the resulting motion races along. The speed c=\sqrt{T/\mu} for a string of tension T and linear density \mu.

From masses on springs to a continuum

The wave equation is not fundamental — it emerges. Take a row of Coupled Oscillators: beads of mass m on a string, each tied to its neighbors by springs. Newton's law for bead n is

m\,\ddot u_n = \kappa\,(u_{n+1}-2u_n+u_{n-1}).

The right-hand side is a discrete second difference — the lattice's version of a second derivative. Now let the beads shrink and crowd together, taking the spacing to zero. The second difference becomes \partial_{xx}u, and the discrete chain melts into the smooth wave equation. A wave is the continuum limit of infinitely many coupled oscillators, which is exactly why every point still moves like a Simple Harmonic Oscillator.

d'Alembert: left- and right-movers

The general solution, found by d'Alembert, is breathtakingly simple:

u(x,t)=f(x-ct)+g(x+ct).

Any function f of the combination x-ct is a shape that slides rightward at speed c without distorting; any g(x+ct) slides leftward. The full solution is just a right-mover plus a left-mover, set by the initial shape and velocity. Nothing about the profile matters — a smooth bump, a sharp kink, a sine — it simply translates.

Two pulses launched from opposite ends: a right-mover (accent) and a left-mover (violet). Each keeps its shape exactly as d'Alembert promises. Watch them overlap in the middle and emerge unchanged.

Where the two pulses overlap they simply add (the Superposition Principle) and then continue on, perfectly intact. If c instead depended on frequency, the shape would not survive — that is the story of Dispersion. And if the ends are tied down so the movers reflect and recombine forever, you get a Standing Wave.

See also

Related topics

  • Superposition PrincipleWhen two waves meet, the medium's displacement is simply the sum of what each wave would do alone — the linearity that underlies all of wave physics.
  • Standing WaveTrap a wave between two fixed ends and it stops traveling — freezing into a pattern of motionless nodes and violently swinging antinodes.
  • WavesFrom a single swinging oscillator to ripples that travel, interfere, and carry frequency through space — the bridge between Oscillations and Fourier Analysis.
  • DispersionWhen wave speed depends on frequency, a pulse made of many frequencies spreads and reshapes as it travels, and phase and group velocities part ways.
  • Simple Harmonic OscillatorA mass pulled back by a force proportional to its displacement — the linear ideal whose sinusoidal motion underlies every other oscillator.
  • Fourier AnalysisAny signal is a sum of pure sinusoids — and reading it as such turns the tangled language of time into the clean language of frequency.

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