Wave Equation in 1D

PDEs · Axiom Academy

Understanding the fundamental equation governing wave propagation in one dimension The 1D wave equation is a second-order partial differential equation that describes how a wave u(x,t) evolves in space and time: Here, u(x,t) represents the displacement of the wave at position x and time t, while c is the wave speed (a positive constant). The equation states that the acceleration of the wave (second time derivative) is proportional to its curvature (second space derivative). Because the wave equation is second-order in time, we need two initial conditions to uniquely determine the solution: Initial displacement: u(x,0) = f(x) — the shape of the wave at t = 0 Initial velocity: u t (x,0) = g(x) — how fast each point is moving at t = 0 Think of a vibrating string: to predict its future motion, you need to know both its initial shape and how fast each point is moving initially. For waves on a finite domain [0, L], we need boundary conditions at both endpoints. Common examples include: Fixed (Dirichlet): u(0,t) = 0, u(L,t) = 0 — endpoints held fixed (like a guitar string) Free (Neumann): u x (0,t) = 0, u x (L,t) = 0 — zero slope at endpoints Periodic: u(0,t) = u(L,t), u x (0,t) = u x (L,t) — wave wraps around On an infinite domain (-∞, ∞), we typically require that u and its derivatives decay to zero as |x| → ∞. The wave equation initial value problem is well-posed in the sense of Hadamard, meaning: Existence: A solution exists for reasonable initial data

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