Mathematics
The Wave Equation and d'Alembert's Formula for Solutions
Quick fact
In 1746, Jean le Rond d'Alembert discovered that the solution to the one-dimensional wave equation is simply the sum of two arbitrary functions—one moving left and the other moving right—each preserving its initial shape perfectly.
Why this is interesting
If you pluck a guitar string, the bump splits into two waves that run away from each other. Could you predict exactly what the string looks like at any later time?
Read the full explanation
Understanding The Wave Equation and d'Alembert's Formula for Solutions
Imagine a long, taut string. The wave equation, ∂²u/∂t² = c² ∂²u/∂x², governs its displacement u(x,t). The constant c is the wave speed. d'Alembert's formula gives an explicit solution for initial displacement f(x) and initial velocity g(x): u(x,t) = ½[f(x−ct) + f(x+ct)] + (1/2c)∫{x−ct}^{x+ct} g(s) ds. This says that the initial shape splits into two copies—one traveling right (f(x−ct)) and one traveling left (f(x+ct))—each at speed c. The velocity term adds a cumulative effect that also propagates in both directions. This formula works for the infinite line, where no boundaries interfere.
A deeper explanation
The power of d'Alembert's formula comes from the method of characteristics. The wave equation can be factored into two first-order operators: (∂/∂t + c∂/∂x)(∂/∂t − c∂/∂x)u = 0. This reveals that the solution is constant along lines x ± ct = constant, called characteristics. The general solution is u = F(x−ct) + G(x+ct), where F and G are arbitrary twice-differentiable functions. Applying the initial conditions determines F and G, yielding the integral formula. The formula shows that the wave equation is non-dissipative: the initial shape propagates without change, which is why the solution is a superposition of traveling waves. This contrasts with the heat equation, where disturbances diffuse and smooth out. d'Alembert's formula is a cornerstone of mathematical physics, demonstrating how initial data on a boundary surface (the initial time) determines the entire future evolution, and it paves the way for more advanced solution techniques like Fourier series and transforms.