Method of Characteristics — Wave Equation

Method of Characteristics: The wave equation uₜₜ = c²uₓₓ factors into (∂ₜ+c∂ₓ)(∂ₜ−c∂ₓ)u = 0. Information travels along characteristic lines x±ct = const. d'Alembert's solution: u(x,t) = ½[f(x−ct)+f(x+ct)] + 1/(2c)∫ g(s)ds where f=u₀ (initial displacement) and g=v₀ (initial velocity). The characteristic diagram (x-t plane) shows how initial data at x₀ influences the solution at (x,t) only within the domain of dependence |x−x₀| ≤ ct. This is causality built into the mathematics.