Wave Equation: d'Alembert's Solution

∂²u/∂t² = c²∂²u/∂x². d'Alembert (1746): u(x,t) = f(x−ct) + g(x+ct). Any wave splits into right-traveling f and left-traveling g components. Boundary conditions create reflections.

Wave speed c: 1.0
Amplitude: 1.0
Width σ: 0.10

Total u(x,t)

Right-moving f(x−ct)

Left-moving g(x+ct)

d'Alembert: u(x,t) = ½[f(x−ct)+f(x+ct)] + 1/(2c)∫[x-ct,x+ct] g(ξ)dξ