The wave equation and d'Alembert's formula

The wave equation \( u_{tt} = c^2 u_{xx} \) factors: \( (\partial_t - c\,\partial_x)(\partial_t + c\,\partial_x)u = 0 \). It is two transport equations stacked. In the characteristic coordinates \( \xi = x + ct \), \( \eta = x - ct \) the chain rule turns it into \( u_{\xi\eta} = 0 \), whose general solution is \[ u(x,t) = F(x + ct) + G(x - ct): \] a wave moving left plus a wave moving right, each at speed \( c \), each keeping its shape. The example checks \( \sin(x - 2t) \) with \( c = 2 \).

Fitting the initial position \( u(x,0) = f(x) \) and velocity \( u_t(x,0) = g(x) \) gives d'Alembert's formula \[ u(x,t) = \frac{f(x + ct) + f(x - ct)}{2} + \frac1{2c}\int_{x - ct}^{x + ct} g(s)\,ds. \] An initial bump with zero velocity splits into two half-height copies that separate; an initial kick \( g \) spreads out as a widening plateau.

The formula shows at a glance that \( u(x,t) \) depends only on the data in the domain of dependence \( [x - ct, x + ct] \). Conversely, data near \( x_0 \) affect only the cone \( |x - x_0| \le ct \): signals travel at speed at most \( c \). And nothing is smoothed: a corner in \( f \) produces corners in \( u \) that travel along the lines \( x \pm ct = \text{const} \) forever.

Picture it: in the \( (x, t) \) plane, from any point draw two lines back down to the axis with slopes \( \pm 1/c \). Everything the solution knows at that point lies in the triangle between them.

Think it: the heat and wave equations sit at opposite extremes. Heat has infinite speed and infinite smoothing; waves have finite speed and no smoothing, and so the wave equation is time-reversible while the heat equation is not. The Navier-Stokes equations mix both behaviours: the transport term carries features at the fluid velocity, and viscosity smooths them.