Deriving the Wave Equation

A Wave Is Newton on a Chain

The convolutional model was a shortcut that skipped the physics of a travelling wave. Part 4 puts that physics back, and it starts with the most honest possible picture: a row of masses joined by springs. Displace one mass and its springs pull it back while pushing its neighbours, so the disturbance spreads. That is a wave, and its governing equation drops straight out of Newton's law.

Consider one mass. Its two springs pull it toward the average of its neighbours, so the net force is proportional to how much the chain curves at that point, the second spatial derivative $\partial^2 u/\partial x^2$. Newton's law sets mass times acceleration equal to that force, and acceleration is the second time derivative $\partial^2 u/\partial t^2$. Divide through and you have the wave equation:

$\dfrac{\partial^2 u}{\partial t^2} = c^2\, \dfrac{\partial^2 u}{\partial x^2}, \qquad c = \sqrt{\dfrac{k}{m}}\,h.$

The speed $c$ is set by the spring stiffness $k$ and the mass $m$: stiffer springs or lighter masses carry the wave faster. In rock, stiffness is the elastic modulus and mass is density, which is why hard, low-density rock is fast.

The Chain Is the Equation

Pluck the chain and watch the pulse do exactly what the equation demands: it splits into two halves that travel outward at speed $c$, keep their shape, and reflect off the fixed ends. Raise the stiffness and the pulse speeds up. There is nothing hidden here; the animation is the wave equation being solved, one mass at a time, by the very rule you just derived.

That rule, new displacement from the curvature of the current one, is the seed of the finite-difference method. The next section makes it precise as a numerical stencil on a grid, and from there the 2D engine that powers the rest of this course is a short step. Every diffraction, multiple, and bowtie the convolutional model could not make will come from running this same equation in two dimensions.