D'Alembert's Solution

PDEs · Axiom Academy

A complete derivation of the general solution to the one-dimensional wave equation We begin with the one-dimensional wave equation: D'Alembert's key insight was to introduce new coordinates that move with the wave. We define: These are called characteristic coordinates . The variable ξ moves to the left (as t increases), while η moves to the right. 2. Transforming the Wave Equation Using the chain rule, we compute the derivatives. First, the time derivatives: Substituting into the wave equation u tt = c²u xx : Remarkably, the terms cancel and we get: This is much simpler than the original equation! It means that the mixed partial derivative equals zero. If u ξη = 0, then integrating with respect to η gives: where F is an arbitrary function of ξ alone. Integrating again with respect to ξ: Substituting back ξ = x - ct and η = x + ct: This is the general solution to the wave equation! It represents the sum of two waves: one traveling right F(x-ct) and one traveling left G(x+ct). 5. Applying Initial Conditions Consider the initial value problem with initial position f(x) and initial velocity g(x): At t = 0, our general solution gives: Differentiating with respect to time: Solving the system of equations from the initial conditions, we obtain: This is D'Alembert's formula , the explicit solution to the initial value problem for the wave equation. Characteristic coordinates: Variables ξ and η that move with the wave

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