# Wave Pulse Simulator - Reflection, Transmission, Superposition, Standing Waves

> Pulses on a string: inverted or upright reflection at a junction, measured r and t vs theory, fixed, free and absorbing ends, colliding pulses, standing waves.

- URL: https://qilingtools.com/en/physics/wave-pulse
- 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: 2026-10-29
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Send a Gaussian, square or sine pulse from the left along a string made of a light and a heavy section, and watch it split at the junction into a reflected and a transmitted pulse: light-to-heavy reflections come back inverted, heavy-to-light ones upright. The tool solves the wave equation with finite differences, measures the reflected and transmitted amplitudes and energies once the pulses separate, and compares them with r = (Z₁−Z₂)/(Z₁+Z₂) and t = 2Z₁/(Z₁+Z₂), usually agreeing within 0.5%. You can also switch between fixed, free and absorbing ends, collide two pulses to see superposition, or drive the string continuously to build a standing wave.

## How to use

1. Pick a scenario, or switch between Send from the left and Colliding pulses
2. Set the waveform, amplitude, pulse width or frequency, tension and the two linear densities, and choose a fixed, free or absorbing right end
3. Press Play and use slow motion and Step to catch the moment a pulse reaches the junction or the end; the grey dashed curve is the exact solution
4. Once the pulses separate, compare the measured r, t, R and T with theory; with a continuous wave, look for the nodes in the envelope

## Features

- Finite-difference solution of the wave equation on a string with varying density, CFL-stable, with optional damping
- Transparent source at the left so returning waves leave cleanly; fixed, free or first-order Mur absorbing right end
- Automatic measurement of amplitude reflection and transmission coefficients and energy shares, shown next to theory
- Exact solution overlaid as a dashed curve so you can see the simulation match theory
- Colliding pulses (same or opposite sign) for superposition, and a continuous wave on a fixed end that builds a standing wave with the predicted nodes marked
- Gaussian, square, single-sine and continuous waveforms, with slow motion and single-step

## FAQ

### Why does a pulse reflect upside down going from a light to a heavy string?

Displacement and slope must both be continuous at the junction. The heavy string has a larger impedance Z = √(Tμ) and is hard to move, so it holds the junction back like a partly fixed end, and the reflection is inverted: r = (Z₁−Z₂)/(Z₁+Z₂) < 0. In the limit Z₂ → ∞ you get a fixed end with r = −1, the “half-wave loss” of optics. Going from heavy to light, r > 0 and the reflection is upright, with a free end (r = +1) as the limit.

### The transmitted pulse can be taller than the incident one. Doesn’t that break energy conservation?

From a heavy to a light string t = 2Z₁/(Z₁+Z₂) exceeds 1, for example t = 4/3 when μ drops from 0.4 to 0.1. But the energy a wave carries is also proportional to the impedance: the transmitted share is (Z₂/Z₁)t², and the light string’s Z₂ is small, so no energy is created. The reflected share r² plus the transmitted share (Z₂/Z₁)t² always equals 1, and the tool lists both measured values next to theory.

### How can two pulses pass straight through each other?

The small-amplitude wave equation is linear, so the sum of two solutions is also a solution: the superposition principle. Where the pulses overlap the displacements add point by point. Same-sign pulses build a taller peak; opposite-sign pulses briefly cancel and the string looks flat, but it is still moving, so all the energy is kinetic at that instant. Afterwards each pulse carries on unchanged.

### How does a standing wave form, and where are the nodes?

A continuous sine wave reflects from the fixed end and travels back with the same frequency. The two counter-propagating waves add to 2A sin(kx) cos(ωt): some points never move (nodes) and some swing with amplitude 2A (antinodes). A fixed end is a node and neighbouring nodes are λ/2 = c/(2f) apart; a free end is an antinode. The tool marks the predicted nodes with purple triangles and shades each point’s amplitude envelope.

### How does the simulation stay stable and accurate?

It uses a centred finite-difference scheme with Δt = 0.98Δx/c_max, which satisfies the CFL condition cΔt ≤ Δx; otherwise errors would grow exponentially. The slower section runs at a lower Courant number and has slight numerical dispersion, so sharp square edges ring a little, which is why the square pulse edges are smoothed. The absorbing end uses a first-order Mur boundary that leaves around 10⁻⁶ of the energy behind, and for Gaussian pulses the measured coefficients typically match theory within 0.5%.

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