3.2 The continuum limit and the 1-D wave equation

Start from the discrete-chain equation we just derived,

m\, \ddot y_n \;=\; \kappa\, (y_{n+1} - 2 y_n + y_{n-1}),

and take the limit $N \to \infty$ , $a \to 0$ with $L = Na$ fixed and $\mu \equiv m/a$ fixed. The displacement $y_n(t)$ becomes a continuous field $y(x, t)$ with $x = n a$ . Watch what happens.

Spatial second difference becomes second derivative

Taylor-expand $y_{n \pm 1} = y(x \pm a, t)$ around $x$ :

y(x \pm a, t) \;=\; y \pm a\, y_x + \tfrac12 a^2\, y_{xx} \pm \tfrac16 a^3\, y_{xxx} + \cdots

Adding the $+$ and $-$ versions:

y(x + a, t) + y(x - a, t) \;=\; 2 y + a^2\, y_{xx} + O(a^4).

So the second difference is

y_{n+1} - 2 y_n + y_{n-1} \;=\; a^2\, \frac{\partial^2 y}{\partial x^2} \;+\; O(a^4).

Substituting into the equation of motion and using $m = \mu a$ and $\kappa a = T$ (where $T$ is the tension — see derivation below):

\mu a\, \ddot y \;=\; \kappa\, a^2\, y_{xx} \;\;\Longrightarrow\;\; \mu\, \ddot y \;=\; (\kappa a)\, y_{xx} \;=\; T\, y_{xx},

or in canonical form,

\boxed{\;\;\frac{\partial^2 y}{\partial t^2} \;=\; c^2\, \frac{\partial^2 y}{\partial x^2}, \qquad c^2 \;=\; T/\mu.\;\;}

This is the one-dimensional wave equation. The propagation speed is $c = \sqrt{T/\mu}$ — the square root of tension over mass per unit length. It depends only on the properties of the medium, not on the amplitude or the shape of the disturbance.

▶ Why

\kappa a = T

in the continuum limit Derivation

The springs in the discrete chain have stiffness $\kappa$ . In the continuum limit, the same physical material is described by a tension $T$ — the force per unit cross-section needed to stretch it. To extract the right scaling, hold a small piece of string of length $\ell = N_\text{piece}\, a$ in tension $T$ . Extending the piece by $\delta\ell$ requires force $T \cdot \delta\ell / \ell$ (Hooke for a continuous rod). For the equivalent discrete chain to behave the same way, the total extension $\delta\ell$ is the sum of extensions of $N_\text{piece}$ springs in series, each with stiffness $\kappa$ . The force to extend the chain by $\delta\ell$ is $\kappa \delta\ell / N_\text{piece}$ (because springs in series sum compliances). Equating, $T / \ell = \kappa / N_\text{piece}$ , i.e. $T = \kappa a$ .

The shortcut: a discrete-chain stiffness $\kappa$ between masses spaced by $a$ corresponds to a continuum tension $T = \kappa a$ . Both have units of force (cross-section dropped because we’re 1-D).

A real wave equation

The 1-D wave equation has two spatial derivatives and two time derivatives. The two-ness on both sides is crucial: it is what makes the equation invariant under $t \to -t$ (waves travel either direction) and what allows the existence of two independent travelling-wave families (next lesson). A “wave equation” with one time derivative and two spatial derivatives is the heat equation — and behaves entirely differently (diffusion, no propagation).

For a string of mass density $\mu = 0.5\,$ g/m under tension $T = 50\,$ N — about a steel guitar string — the speed is

c \;=\; \sqrt{T/\mu} \;=\; \sqrt{50 / 5 \times 10^{-4}} \;\approx\; 320\,\text{m/s}.

Suggestively close to the speed of sound in air. This is also no accident: both are square roots of restoring-force-per-unit-extension over mass-per-unit-length, with the relevant moduli computed differently.

The two interpretations

The same equation can be read two ways:

Each point oscillates in time. Hold $x$ fixed; $y(x, t)$ as a function of $t$ obeys $\ddot y = c^2 y_{xx}$ , where the right side is set by what the spatial profile looks like at that instant.
A spatial pattern travels. Watch the whole profile $y(x, t)$ evolve in $x$ and $t$ ; it moves rigidly at speed $c$ in either direction.

The next lesson formalises the second interpretation through d’Alembert’s solution.