The general solution to the one-dimensional wave equation $u_{tt} = c^2 u_{xx}$ is: $u(x,t) = F(x - ct) + G(x + ct)$ where $F$ and $G$ are arbitrary twice-differentiable functions representing right-moving and left-moving waves, respectively.

Given initial conditions $u(x,0) = f(x)$ and $u_t(x,0) = g(x)$, the solution is: $u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds$

This is D'Alembert's formula, showing how initial displacement and velocity evolve into traveling waves.

For a string fixed at both ends ($x = 0$ and $x = L$), separation of variables $u(x,t) = X(x)T(t)$ yields: $u_n(x,t) = \sin\left(\frac{n\pi x}{L}\right)\left[A_n\cos\left(\frac{n\pi ct}{L}\right) + B_n\sin\left(\frac{n\pi ct}{L}\right)\right]$

These are normal modes with frequencies $\omega_n = \frac{n\pi c}{L}$. The general solution is a superposition: $u(x,t) = \sum_{n=1}^\infty u_n(x,t)$

In three dimensions with spherical symmetry, setting $v = ru$ transforms the wave equation to: $v_{tt} = c^2 v_{rr}$

D'Alembert's solution gives: $u(r,t) = \frac{F(r - ct)}{r} + \frac{G(r + ct)}{r}$

The first term represents an outgoing spherical wave, the second an incoming wave. The $1/r$ decay reflects energy spreading over expanding spherical surfaces.

The plane wave $u(x,t) = A\cos(\mathbf{k} \cdot \mathbf{x} - \omega t + \phi)$ is a solution if the dispersion relation $\omega = c|\mathbf{k}|$ holds, where $\mathbf{k}$ is the wave vector. These solutions are fundamental in Fourier analysis of wave phenomena.