D'Alembert's Solution

D'Alembert's solution is simply a formula saying that a wave on a long string is two copies of a shape, one sliding left and one sliding right. It's named after Jean le Rond d'Alembert, the French mathematician who found it in 1747. It's handy because it gives the answer directly, with no infinite series, and it makes the finite speed of waves obvious. The formula has three pieces: $f(x - ct)$ (the starting shape moving right), $f(x + ct)$ (the same shape moving left), each at half height, which is where the first $\frac12$ comes from, and the integral of $g$ (the effect of the starting push, spreading out both ways). It looks crowded, but it's two sliding shapes and one integral.

Derivation. In the variables $\xi = x - ct$ and $\eta = x + ct$, the wave equation becomes $u_{\xi\eta} = 0$, so $u = F(\xi) + G(\eta)$: a right-moving and a left-moving wave. Fitting $u(x, 0) = f$ and $u_t(x, 0) = g$ determines $F$ and $G$. Notice that $F(x - ct)$ is exactly the transport-equation solution from the first unit. The wave equation is really two transport equations stacked together, one for each direction.

Finite speed. The value $u(x, t)$ depends only on the initial data in $[x - ct, x + ct]$, the domain of dependence. That makes sense: nothing moves faster than $c$, so in time $t$ only the data within $ct$ of $x$ can have reached it. Signals never outrun $c$, unlike heat, which spreads instantly.

Standing waves are traveling waves. $\sin kx\cos ckt = \frac12\left[\sin k(x - ct) + \sin k(x + ct)\right]$: a standing wave is two equal traveling waves moving through each other. That ties back to the last lesson: each mode of a guitar string is really two traveling waves bouncing back and forth between the ends.

Sound reaching you from a source, the delay of seismic waves between stations, and signals on transmission lines all show the finite speed and splitting of d'Alembert's solution.