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Standing Wave on a String (2D)

2D standing-wave lab: y(x,t) = A·sin(nπx/L)·cos(ωt) driven live, with a second superposable harmonic, node markers and a real-time frequency/wavelength/period readout.

Mathematics2DModerate60 FPS📱 Mobile-adapted⇄ 3D version

This 2D companion drives the same fixed-end standing-wave physics as the 3D version through a plain canvas view: a side panel exposes the primary harmonic number, its amplitude, an optional second superposed harmonic, and the wave speed, while a live readout tracks frequency, period, wavelength and node count as you change them — making the relationship between harmonic number, node spacing and pitch directly readable instead of only visible.

⚙ Under the hood

2D standing-wave lab: y(x,t) = A·sin(nπx/L)·cos(ωt) driven live, with a second superposable harmonic, node markers and a real-time frequency/wavelength/period readout.

standing wavessuperpositionresonanceharmonicsnodes and antinodes

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install

Why do only certain frequencies form a standing wave?

A string fixed at both ends forces y=0 at x=0 and x=L at all times. Only wavelengths where an integer number of half-cycles fits exactly between the ends satisfy that boundary condition, giving the discrete set λₙ = 2L/n and fₙ = nv/2L used here — any other frequency destructively interferes with its own reflection instead of building a stable pattern.

What happens when the second harmonic is turned on?

The displayed curve becomes y₁+y₂, the sum of two independent standing-wave modes (the superposition principle). Because n₁ and n₂ generally have different frequencies, the combined shape is not itself a standing wave — it changes over time in a way governed by the beat between the two frequencies, though the string always returns to a repeating pattern since both modes are periodic.