> [!NOTE] > Any solution to the wave equation in one spatial dimension, $u_{tt}=c^2u_{x x}$ where $t\geq 0, x\in \mathbb{R}$ has the form $u(x,t)=g(x-ct)+f(x+ct)$for some functions $f,g:\mathbb{R}\to \mathbb{R}.$ **Proof (Coordinate Method, Strauss)**: Observe that [[1D Wave Operator is a Composite of Two Opposite Direction Linear Advection Operators|wave operator can be factored into transport operators]]. Hence, introduce the [[Method of Characteristics for Transport Equation|characteristic coordinates]] $\xi=x+ct,\quad\eta=x-ct.$By the chain rule, we have $\partial_x=\partial_\xi+\partial_\eta$ and $\partial_t=c\partial_\xi+c\partial_\eta.$ Assuming $u$ is sufficiently smooth, equality of mixed partial derivatives yields $(\partial_{t}-c\partial_{x})u=-2c\partial_{\eta}u$ and $(\partial_t+c\partial_{x})u=2c\partial_{\xi}u.$ So the wave equation becomes $(\partial_t-c\partial_x)(\partial_t+c\partial_x)u=(-2c\partial_\xi)(2c\partial_\eta)u=0,$ which means that $u_{\xi\eta}=0$ since $c\neq0.$ Integrating with respect to $\eta$ then $\xi$ yields $u=f(\xi)+g(\eta)$for some some functions $f,g:\mathbb{R}\to \mathbb{R}.$ # Sources 1. W Strauss.