The Wave Equation

The wave equation is simply the rule for how a stretched string moves once it's disturbed: every bit of it gets pulled back toward straight, and the more it's bent, the harder the pull. The same rule governs guitar strings, sound in air, earthquake waves and light, which is why it's worth knowing well. The equation $u_{tt} = c^2u_{xx}$ has three small pieces: $u_{tt}$ (how fast a bit of string is speeding up), $u_{xx}$ (how sharply the string is bent at that spot), and $c^2$ (a constant set by how tight and how heavy the string is). That's Newton's "force equals mass times acceleration," one tiny bit of string at a time. It looks complex, but it's really just a spring law spread out along a line.

Same curvature, different response. Put the heat equation $u_t = k\,u_{xx}$ next to the wave equation. The right side is the same, curvature, but on the left the heat equation has a velocity and the wave equation has an acceleration. That one change makes all the difference. A hot spot responds to curvature by cooling directly, so bumps just flatten. A bit of string responds by accelerating, so it picks up speed, overshoots, swings back, and keeps going, like a mass on a spring.

Derivation. A piece of string from $x$ to $x + \Delta x$ is pulled by tension at both ends. For small slopes, the vertical components are $T\,u_x$, so the net vertical force is $T\,u_x(x + \Delta x) - T\,u_x(x) \approx T\,u_{xx}\Delta x$. Newton: $\rho\,\Delta x\,u_{tt} = T\,u_{xx}\,\Delta x$. Divide by $\rho\,\Delta x$ and you get $u_{tt} = c^2u_{xx}$ with $c^2 = T/\rho$. It makes sense that $c$ grows with tension and shrinks with density: a tighter string snaps back harder, and a heavier one is slower to get moving.

Two initial conditions. The equation is second order in time, so the starting shape isn't enough. A string sitting still in some shape and one passing through that same shape at full speed will do different things next. So we need the initial shape $u(x, 0) = f(x)$ and the initial velocity $u_t(x, 0) = g(x)$, just like the position and velocity of a mass on a spring.

Boundary conditions: fixed ends $u = 0$, or free ends $u_x = 0$.

Heat vs waves. Heat is irreversible: it smooths and forgets where it started. Waves keep their energy and their shape; sharp features persist and travel at speed $c$. You'll see both behaviors come straight out of the formulas once we solve these equations in the next unit.

Guitar and piano strings, sound in air, seismic waves, light and radio waves (Maxwell's equations) and water waves all obey wave equations.