3.1 A discrete chain of masses

A horizontal row of $N$ identical small masses, each of mass $m$ , is connected to its neighbours by identical springs of stiffness $\kappa$ , with the two ends pinned to fixed walls. The equilibrium spacing is $a$ . The dynamical variable is the transverse displacement $y_n(t)$ of the $n$ -th mass — perpendicular to the chain.

A row of N identical masses, each of mass m, joined by springs of stiffness κ at equilibrium spacing a, with fixed walls at either end. The dynamical variable y_n is the transverse displacement of the n-th mass — perpendicular to the chain.

The force on mass $n$ from the spring on its left is $\kappa\, (y_{n-1} - y_n)$ (pulling toward $y_{n-1}$ ); from the spring on its right, $\kappa\, (y_{n+1} - y_n)$ . Newton’s second law gives

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

This is a coupled system of $N$ second-order ODEs. Without the coupling (the term in parentheses), each mass would just oscillate independently at $\omega_0 = \sqrt{\kappa/m}$ . With the coupling, the masses talk to each other: a kick on one propagates to its neighbours.

A travelling pulse

Set the leftmost mass moving and leave the rest at rest. The first mass pulls on the second; the second responds, and starts pulling on the third; the third responds in turn. After some time, the disturbance has reached the far end of the chain. Energy that was originally on the left has been transported, mass by mass, to the right.

Number of masses N = 32

A pulse on a chain of masses on springs propagates as a wave. For small N the chain has dispersion — short-wavelength components travel slower than long ones — and the pulse spreads and ripples. As N grows, the chain's spectrum fills out smoothly and approaches the continuum wave equation u_tt = c²u_xx: the dashed reference. This is how Newton's particle-mechanics becomes continuum field theory.

The interactive shows a Gaussian pulse evolving on the chain (solid blue), compared with the solution of the continuous wave equation for the same initial condition (dashed green). For small $N$ the chain is dispersive — short-wavelength components travel slower than long ones, and the pulse spreads and ripples. As $N$ grows the two curves converge.

The speed of that transport depends on $\kappa$ , $m$ , and $a$ . By dimensional analysis,

c \;\sim\; a \sqrt{\kappa / m}.

(The units of $\kappa/m$ are $1/\text{time}^2$ ; $a$ has units of length; so $a\sqrt{\kappa/m}$ has units of length/time. There is no other dimensionally consistent combination.) The next lesson derives the exact relation — same form, prefactor $1$ .

Normal modes

Because the equations are linear and the chain is uniform, the system has $N$ normal modes, each oscillating sinusoidally in time at its own frequency. For a chain with fixed ends, the $k$ -th mode has spatial shape

y_n^{(k)} \;\propto\; \sin\!\left( \frac{k\, n\, \pi}{N+1} \right),

with $k = 1, 2, \ldots, N$ . The mode frequencies are

\omega_k \;=\; 2\sqrt{\kappa/m}\, \sin\!\left( \frac{k\pi}{2(N+1)} \right).

The interactive below shows the $k$ -th mode of an $N$ -mass chain (red dots) alongside a continuous sine $\sin(k\pi x / L)$ at the same wavenumber (green dashed). Slide $k$ from $1$ to $N$ to walk through every mode; slide $N$ to watch the same $k$ approach the continuum.

mode k

masses N

At the smallest k, the dots fall on the continuum sine wave exactly — the chain's low modes are the modes of a continuous string. At the largest k, neighbouring dots oscillate in opposite directions: the wavelength is now comparable to the spacing a, the chain disperses, and the chain frequency falls below the continuum prediction. Increase N and a fixed k moves further from this dispersive edge, recovering the continuum result. The ratio ω_chain / ω_cont approaches 1 as N → ∞ at fixed k.

For small $k$ relative to $N$ , $\sin(k\pi / 2(N+1)) \approx k\pi/(2(N+1))$ , so

\omega_k \;\approx\; \frac{k\pi}{N+1}\, \sqrt{\kappa/m} \;=\; c\, k_\text{wavenumber},

with $k_\text{wavenumber} = k\pi / L$ for total length $L = (N+1) a$ . The low modes of the discrete chain look exactly like the low modes of a continuous string. The next lesson shows that this is no accident.

The continuum limit

Now take $N \to \infty$ and $a \to 0$ while keeping the total length $L = N a$ fixed and the linear mass density $\mu = m/a$ fixed. The chain becomes a continuous string. The discrete second difference $y_{n+1} - 2 y_n + y_{n-1}$ approaches $a^2 \partial^2 y / \partial x^2$ . The discrete equation of motion becomes the wave equation — the topic of the next lesson.