d'Alembert's solution
/ dah-lahm-BAIR /
What does a wave on an infinite string actually do? Before any series or transform, there is a strikingly clean answer for the one-dimensional wave equation, found by Jean le Rond d'Alembert in 1747: the most general solution is just two fixed shapes, one sliding to the right and one sliding to the left, both at the wave speed c, added together. Nothing oscillates in place; everything translates.
The wave equation u_tt = c^2 u_xx has general solution u(x, t) = F(x - c t) + G(x + c t), where F and G are arbitrary (twice-differentiable) functions. The combination x - c t is constant along a rightward-moving observer, so F(x - c t) is a shape rigidly travelling right at speed c; likewise G(x + c t) travels left. Given the initial shape u(x, 0) = phi(x) and initial velocity u_t(x, 0) = psi(x), the two unknown functions are pinned down explicitly, yielding d'Alembert's formula: u(x, t) = (1/2)[phi(x - c t) + phi(x + c t)] + (1/(2c)) times the integral from x - c t to x + c t of psi(s) ds. In words: the displacement at (x, t) is the average of the initial shape at the two points that exactly c t away to the left and right, plus a contribution from the initial velocity integrated over the interval between them.
This formula is the cleanest illustration of the hyperbolic structure laid bare. It shows the finite speed of propagation directly: the solution at (x, t) depends only on initial data in the interval [x - c t, x + c t], its domain of dependence, and on nothing outside. It explains why a plucked string splits into two counter-running pulses, and it is the exact, closed-form benchmark against which numerical wave solvers are checked. On a finite string the same idea works by reflecting the waves off the boundaries, which is the travelling-wave picture dual to the standing-wave Fourier series.
An infinite string released from rest (psi = 0) with an initial bump phi has solution u(x, t) = (1/2)[phi(x - c t) + phi(x + c t)]: the bump splits into two half-height copies, one running left and one running right, each rigid and unchanging.
Release from rest and the initial shape simply halves and runs both ways — the signature picture of d'Alembert's formula.
D'Alembert's formula applies cleanly to the one-dimensional wave equation; in even space dimensions (like waves on a 2D surface) the analogue fails to have a sharp trailing edge, so a flash of sound in a plane would not give a clean echo — the simple travelling-wave picture is special to odd dimensions.