Foundations — Wave equation (hyperbolic) 1D — derivation

This page builds every piece of notation the parent derivation leans on, starting from things a curious 12-year-old already knows and adding exactly one new idea at a time. Nothing here is assumed. If the main note used a symbol, it is defined below before you meet it there.

1 · The string and its shape: what is $u(x,t)$?

Picture a guitar string stretched flat between two pegs. Lay a ruler along its resting line — call the direction along the ruler the $x$-direction (horizontal, left–right).

Now pluck it. Each point of the string lifts a little off the resting line. How far a point has moved up or down is its displacement.

Why we need two inputs. A single photo of the string (one frozen instant) is a function of $x$ alone — a curve. But the string moves, so its whole curve is different a moment later. To describe the motion we need a height that depends on both where ($x$) and when ($t$). That is exactly what a two-variable function $u(x,t)$ is.

2 · The Greek and symbol dictionary

Before any calculus, here are the plain-word meanings of every letter the parent note prints. Keep this table beside you.

We now earn the two ideas that turn these letters into physics: slope/tangent and derivatives.

3 · Angle, slope, and why $\theta$ shows up

The string is not flat once plucked; at each point it tilts. Two words describe the same tilt.

These connect through a right triangle. Step forward a tiny run and the string rises by some amount; the tangent is the hypotenuse. On that triangle: $\tan \theta =\frac{\text{rise}}{\text{run}}=\text{slope}.$

Why $\tan$ and not $\sin$ or $\cos$? $\tan \theta$ is opposite over adjacent — it is literally rise-over-run, so it is the slope. That is why the parent note writes $\tan \theta =u_x$: the slope of the string and the tangent of its tilt-angle are the same number.

4 · The derivative: turning "steepness" into a symbol

The single most important tool below is the derivative. Here it is from zero.

Why "partial" and the curly $\partial$? Because $u$ has two inputs, $x$ and $t$. When we measure the slope in the $x$-direction we hold $t$ frozen — we only wiggle one variable. The curly $\partial$ is a flag saying "there are other variables; I'm changing just this one."

Why we need the limit. Real slope is defined at a point, but a point has no width — you cannot compute rise-over-run at a single spot. The trick is to compute it across a small chunk and then let the chunk vanish. The later change of variables and the whole derivation depend on this shrinking idea; it is the engine of calculus.

5 · The second derivative = curvature

Now apply the same slope-machine again, this time to the slope itself.

Why the topic lives or dies on this. The whole wave equation says $u_{tt}=c^2{u}_{xx}$: acceleration is proportional to curvature. A point sitting in a dip ($\cup$, $u_{xx}>0$) gets pushed up; a point on a hump ($\cap$, $u_{xx}<0$) gets pushed down. Curvature is the messenger that tells each point which way its neighbours are yanking it.

Time gets the same treatment: $u_{tt}=\frac{{\partial }^{2}u}{\partial t^2}$ is acceleration — how fast the up/down velocity $u_t$ of a fixed point is changing. It is the "$a$" in $F=ma$.

6 · Tension $T$, density $\rho$, and the speed $c$

Two physical properties of the string finish the cast.

7 · How the foundations feed the topic

Read it top to bottom: geometry (slope, curvature) plus physics (tension, mass) meet inside $F=ma$, and out drops the wave equation. This same skeleton — a balance law feeding a PDE — underlies the heat equation and Laplace's equation too, which is why the classification of second-order PDEs groups them together.

8 · One combined coordinate preview: $\xi$ and $\eta$

The parent's d'Alembert step swaps $(x,t)$ for $\xi =x-ct$ and $\eta =x+ct$. You do not need to master it here, only to recognise the idea:

Equipment checklist

Self-test: cover the right side and answer each before revealing.