# Wave and Vibration Graphs - direction rules, multiple solutions

> Synchronised wave (y–x) and vibration (y–t) graphs with direction rules, multiple-solution problems, two-source interference, standing waves and quizzes.

- URL: https://qilingtools.com/en/physics/wave-graphs
- 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

On the left is the waveform y–x showing every particle at one instant; on the right is the vibration graph y–t of a chosen particle P. Both update together as time runs, so you can see exactly how one graph follows from the other. Turn on the same-side, follow-the-leader or small-shift rule to draw helper arrows at P showing how its motion relates to the travel direction. Extra tabs cover multiple-solution problems, two-source interference, standing waves and a randomised quiz.

## How to use

1. Set amplitude, wavelength, period, direction and initial phase, then move particle P
2. Pick a direction rule to see the helper arrows at P, then press Play
3. In Multiple solutions enter the shift between two waveforms and the time gap to list periods and speeds for each n and both directions
4. Tap any point in Interference to see its path difference, view nodes in Standing waves, then try the quiz

## Features

- Synchronised waveform and vibration graph; read λ, A, T and get v = λ/T
- Three direction rules demonstrated on the graph
- Multiple solutions: Δt = (n + k)T for both directions, in a table
- Interference intensity map with path difference at any tapped point
- Standing waves with nodes, antinodes and the two component waves
- Random quiz on travel direction, particle motion, displacement after Δt and wave speed

## FAQ

### What is the difference between a wave graph and a vibration graph?

A wave graph has position x on the horizontal axis and shows every particle at one instant, giving the wavelength and amplitude. A vibration graph has time t and shows one particle over time, giving the period and amplitude. Together they give the wave speed v = λ/T.

### How does the same-side rule work?

At a particle on the waveform draw a horizontal arrow in the travel direction and a vertical arrow in the particle’s direction of motion. Both arrows always lie on the same side of the curve, so knowing one tells you the other.

### Where do multiple solutions in wave problems come from?

First, periodicity: the waveform after Δt looks the same as after Δt + nT, so Δt = (n + Δx/λ)T. Second, direction: if the travel direction is not given, both must be considered. A condition such as T > Δt is needed to fix n.

### Does a reinforced point in interference always have maximum displacement?

No. It has maximum amplitude, 2A, but it still oscillates, so its displacement swings between −2A and 2A and is sometimes zero. With in-phase sources, points whose path difference is a whole number of wavelengths are reinforced and odd half-wavelengths cancel.

### How far apart are the nodes of a standing wave?

Adjacent nodes (or antinodes) are half a wavelength apart. On a string fixed at both ends with n loops, L = nλ/2. Particles between two adjacent nodes move up and down together, while particles on opposite sides of a node move in opposite directions.

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