Wave Phenomena and d'Alembert's Solution

The wave equation models vibrations, electromagnetic waves, and gravitational waves. Its solutions exhibit characteristic features -- propagation at finite speed, superposition, and dispersion -- that distinguish hyperbolic from elliptic and parabolic physics.

The Wave Equation

Definition6.3d'Alembert's Solution

The one-dimensional wave equation $u_{tt} = c^2 u_{xx}$ on $\mathbb{R}$ with initial conditions $u(x,0) = f(x)$ , $u_t(x,0) = g(x)$ has the d'Alembert solution: $u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$ . This decomposes into left-traveling and right-traveling waves, with the solution at $(x,t)$ depending only on the initial data in the domain of dependence $[x-ct, x+ct]$ .

ExampleKirchhoff's Formula in 3D

In three dimensions, the wave equation $u_{tt} = c^2\nabla^2 u$ with $u(x,0) = f$ , $u_t(x,0) = g$ has the Kirchhoff solution: $u(\mathbf{x},t) = \frac{\partial}{\partial t}\left(\frac{1}{4\pi c^2 t}\iint_{|\mathbf{y}-\mathbf{x}|=ct} f(\mathbf{y})\,dS\right) + \frac{1}{4\pi c^2 t}\iint_{|\mathbf{y}-\mathbf{x}|=ct} g(\mathbf{y})\,dS$ . This is Huygens' principle: in odd dimensions $\geq 3$ , disturbances propagate on the light cone (sharp signals), unlike even dimensions where residual effects persist (no sharp Huygens principle in 2D).

Dispersion and Wave Packets

Definition6.4Dispersion Relation

For a linear wave equation with plane wave solutions $u = Ae^{i(kx-\omega t)}$ , the dispersion relation $\omega = \omega(k)$ relates frequency to wavenumber. Non-dispersive: $\omega = ck$ (standard wave equation, all frequencies travel at speed $c$ ). Dispersive: $\omega(k)$ nonlinear (e.g., Schrodinger: $\omega = \hbar k^2/(2m)$ ). The group velocity $v_g = d\omega/dk$ governs wave packet propagation; the phase velocity $v_p = \omega/k$ governs individual crests.

RemarkStationary Phase and Asymptotics

For dispersive equations, long-time asymptotics of wave packets are governed by the method of stationary phase: $\int e^{i\phi(k)t}\hat{u}_0(k)\,dk \sim \sqrt{2\pi/(t|\phi''(k_0)|)}e^{i\phi(k_0)t \pm i\pi/4}\hat{u}_0(k_0)$ for large $t$ , where $k_0$ satisfies $\phi'(k_0) = 0$ . This gives the $t^{-1/2}$ decay rate for the Schrodinger equation and explains why wave packets spread over time.