The physics
A wave on a stretched string is a disturbance that travels without the string itself moving along. With a small amount of damping (for example, air drag), the displacement y(x, t) obeys the damped wave equation:
A sinusoidal wave sent in from one end of the string is an exact solution:
Here s is the distance travelled from the source end, and the sign gives the direction of travel. Because equation (1) is linear, the sum of two solutions is also a solution. This is the principle of superposition, and the bold curve in the simulation is exactly the sum of the two dashed waves.
| Symbol | Meaning | SI unit |
|---|---|---|
| A | Amplitude at the source end | m |
| f, ω = 2πf | Frequency and angular frequency | Hz, rad/s |
| k = 2π/λ | Wavenumber (λ = wavelength) | rad/m |
| φ | Phase constant | rad |
| c | Wave speed without damping, √(T/μ) | m/s |
| β | Damping rate | 1/s |
| α | Attenuation coefficient | 1/m |
Damping: attenuation and dispersion
Substituting (2) into (1) gives the dispersion relation for the complex wavenumber κ = k + iα:
Without damping, k = ω/c and every frequency travels at the same speed. With damping, the wave fades over the attenuation length 1/α. It also travels slightly slower than c, and by an amount that depends on frequency, so the medium becomes dispersive. Both effects appear in the live values.
What to look for
- Standing wave: two equal waves travelling in opposite directions give 2A sin(kx) cos(ωt). Nodes never move, and they are λ/2 apart. Put the probe on a node and the displacement graph stays flat.
- Beats: two waves travelling the same way with close frequencies make the amplitude swell and fade at the beat frequency |f1 − f2|. Watch the probe graph.
- Interference: equal waves in phase add (constructive); waves half a cycle apart (φ = π) cancel (destructive).
Method and validation
The simulation evaluates the exact solution (2) for each wave at every frame. No numerical integration is involved, so the result doesn't depend on the frame rate or playback speed. Automated tests check that the formula satisfies the damped wave equation to finite-difference precision. They also check that standing-wave nodes are exactly stationary, that beats have the period 1/|f1 − f2|, and that antiphase waves cancel.
Assumptions and limitations
- Small, transverse displacements of a uniform, perfectly flexible string.
- Linear viscous damping (proportional to velocity).
- Steady sinusoidal waves fed in continuously from the ends. Start-up transients and reflections from the ends are not modelled.