# Wave Simulator - Superposition, Standing Waves, Beats & Interference

> Interactive wave simulator: superpose traveling waves, see standing waves on strings and pipes with nodes and harmonics, hear beats, and explore a ripple tank.

- URL: https://qilingtools.com/en/math/waves
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Four classic wave phenomena in one place: two traveling waves adding up, standing waves on strings and air columns, beats between two close frequencies, and the interference pattern of two coherent sources in a ripple tank. Every scene is animated and responds to the sliders instantly, with the formulas, the numbers plugged in and the key results shown underneath — and the standing-wave and beats panels can play the actual sound in your browser.

## How to use

1. Pick a scene at the top: superposition, standing waves, beats or two-source interference
2. Choose a preset (guitar string, clarinet, tuning forks, classic interference…) or drag the sliders for amplitude, wavelength, frequency, length, tension or source spacing
3. Watch the animation and read the resultant amplitude, harmonic frequencies, beat frequency or path difference with the worked formulas
4. Optionally press the sound button to hear the note or the beats, and press again to stop

## Features

- Two traveling waves shown separately or overlaid, with automatic detection of constructive/destructive, standing and wave-packet cases, plus resultant amplitude, SWR and group velocity
- Strings (fixed–fixed, fixed–free) and pipes (open–open, open–closed) with labelled displacement nodes N and antinodes A and an optional pressure pattern
- Wave speed from tension and linear density v = √(T/μ) or from air temperature, with optional end correction
- Beats with a zoomed carrier, the full envelope, beat period and cents, and live Web Audio playback
- Ripple tank with instantaneous and time-averaged views, hyperbolic antinodal and nodal lines, screen intensity and a draggable probe
- Bilingual and touch-friendly

## FAQ

### What is the difference between a standing wave and a traveling wave?

A traveling wave moves its shape — and its energy — along the medium. A standing wave forms when two waves of equal amplitude and frequency travel in opposite directions: y = 2A·sin(kx)·cos(ωt), so space and time separate and the shape doesn’t move. Nodes never move, antinodes swing with amplitude 2A, adjacent nodes are λ/2 apart, and everything between two nodes moves in step while neighbouring loops move in opposite phase.

### Why does a pipe closed at one end only have odd harmonics?

The air at a closed end can’t move, so it must be a displacement node, while an open end is close to a displacement antinode. The distance from a node to an antinode can only be an odd number of quarter wavelengths, so L = (2n−1)λ/4 and f = (2n−1)v/(4L): only 1, 3, 5… times the fundamental. That is why a clarinet sounds hollow and plays about an octave lower than a flute of the same length. Real pipes need an end correction of about 0.61r per open end.

### How do you calculate the beat frequency?

Adding two close tones gives cos(2πf₁t) + cos(2πf₂t) = 2cos(π(f₁−f₂)t)·cos(2πf̄t): you hear the average pitch f̄, but its loudness is modulated by a slow envelope that swells |f₁ − f₂| times per second — the beat frequency. Beats up to about 15 Hz are easy to count, which is how tuners match pitches (tune until the beats vanish); wider gaps sound rough and then become two separate notes.

### Why are the interference lines hyperbolas?

Points with a constant path difference Δ = r₁ − r₂ to two sources lie on a hyperbola with the sources as foci. For in-phase sources, Δ = mλ gives constructive interference (antinodal lines) and Δ = (m + ½)λ gives cancellation (nodal lines). Far away the hyperbolas become straight lines at sinθ = mλ/d, and since |Δ| can’t exceed the spacing d there are at most 2⌊d/λ⌋ + 1 antinodal lines — the same physics as Young’s double slit.

### What sets the pitch of a vibrating string?

The wave speed on a string is v = √(T/μ), where T is the tension and μ the mass per unit length. A string fixed at both ends has fundamental f₁ = v/(2L) = (1/2L)·√(T/μ): shorter, tighter and lighter strings sound higher. Quadrupling the tension raises the pitch an octave, and fretting shortens L. A 0.648 m guitar A string with μ ≈ 3.6 g/m needs about 73 N of tension to sound 110 Hz.

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