〰️ Travelling wave

About

🎯 A traveling wave on a string modeled as a mass-spring system
🧠 Based on original code by Rhett Allain
👉 An elaborate explanation is given in his video
🐍 A VPython demo is available as well, see traveling_wave.py

Live experiment

Model and implementation

The traveling wave is modeled as a one-dimensional chain of identical point-like bodies connected by linear springs.

The chain consists of $N$ bodies with equal mass. The equilibrium positions are equally spaced along the $x$-axis. Neighboring bodies are connected by springs that exert a restoring force according to Hooke’s law,

$$\boxed{ \overrightarrow{F} =-k\left(r-r_0\right)\hat{\mathbf r} }$$

where $r$ is the current distance between two neighboring bodies, $r_0$ is the spring’s rest length, and $k$ is the spring constant.

In this simulation, the spring rest length is deliberately chosen to be smaller than the initial distance between neighboring bodies:

$$r_0 = 0.9\frac{L}{N-1}$$

while the bodies are initially separated by

$$\Delta x = \frac{L}{N-1}$$

The springs are therefore initially stretched. This creates a non-zero tension in the chain and makes the transverse disturbance propagate as a wave through a tensioned medium.

The chain is fixed at both ends. The first body is additionally subject to a time-dependent boundary condition. During one oscillation period it is displaced in the transverse $y$-direction according to

$$\boxed{ y(t)=A\sin(\omega t) }$$

where $A$ is the amplitude and $\omega$ is the angular frequency. The boundary condition is applied only for one complete period,

$$T=\frac{2\pi}{\omega}$$

After this initial excitation, the first body is no longer actively driven, allowing the generated disturbance to propagate through the chain.

The resulting disturbance is a transverse traveling wave. Although the individual bodies mainly oscillate perpendicular to the chain, the spring forces transmit the disturbance from one body to the next.

A damping coefficient is applied to the spring interactions. Damping reduces the energy of the wave as it propagates and therefore prevents the chain from oscillating indefinitely with constant amplitude.

The physical model is implemented using the Lattice class. The lattice stores the bodies, the neighboring body pairs, their spring forces, and the boundary conditions. The ChainTopology creates the one-dimensional chain and connects each body to its immediate neighbor.

At every integration step, the following sequence is used:

1. The boundary conditions are applied.
2. The spring forces are calculated for all neighboring pairs.
3. The forces are accumulated on the bodies.
4. The bodies are integrated forward in time.

Conceptually, the update therefore has the form

$$\text{boundary condition} \rightarrow \text{spring forces} \rightarrow \text{integration}.$$

The simulation uses a fixed integration timestep. Multiple integration steps can be performed during each simulation-clock interval through the substeps setting. This allows the spring dynamics to be integrated with smaller repeated steps while keeping the simulation clock and rendering loop separate.

The visualization is independent of the physical model. LatticeView represents the bodies as spheres and the spring connections as cylinders or springs. The simulation controls allow the spring constant, damping, driving frequency, and driving amplitude to be varied interactively.

The resulting model demonstrates how a localized transverse disturbance propagates through a mechanically coupled chain and illustrates the role of spring stiffness, tension, damping, amplitude, and driving frequency in wave propagation.