Standing waves (3D)
About this app
Two opposing waves become a standing wave: drag the frequency along the resonance curve and the string builds up only at matching frequencies. Air column open or closed at one end, microwaves in front of a metal plate, measure the node spacing and calculate c. Energy beads show that nothing is transported on average. With exam-style tasks.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
How a standing wave forms
When two opposing waves of equal frequency and amplitude overlap, the result is y = 2A cos(kx) cos(ωt). Position now appears only in the amplitude. At the nodes it is zero at all times, at the antinodes it is twice the amplitude of one component wave. Neighbouring nodes are λ/2 apart. On a medium of length L the ends decide which wavelengths fit: with equal ends f_n = n · c/(2L), with unequal ends f_n = (2n − 1) · c/(4L).
What the app shows
Forward and returning waves run on their own rails, and below them stands their sum with envelope, nodes and antinodes. On the string you drag the frequency along the resonance curve and see the amplitude grow once the reflections arrive in phase. In the air column, particles vibrate in a glass tube, open or closed at one end, with a choice of displacement or pressure. Microwaves at 10.5 GHz stand in front of a metal plate. Energy beads follow the calculated energy flow: they swing and never pass a node.
What you can try
Switch off the returning wave and see the nodes vanish and the energy set off at c. Drag the wall or piston and watch the natural frequencies shift. Compare a fixed and a free string end. Put the two markers on nodes and calculate c from λ and f. Move the receiver through the microwave field and measure across ten node spacings. Two predictions and three exam-style calculations give feedback on typical mistakes and show the worked solution.
Frequently asked questions
- How far apart are two nodes of a standing wave?
- Neighbouring nodes are λ/2 apart, and a node and the next antinode λ/4. If you measure the distance d across k node spacings, then λ = 2d/k. Together with the frequency this gives the wave speed c = λ · f.
- Does a standing wave transport energy?
- Not on average. The forward and returning waves carry the same energy in opposite directions. The energy swings within λ/4 between motion and tension. No energy flows through nodes and antinodes at any moment.
- What are the natural frequencies of a tube closed at one end?
- The closed end has a displacement node and the open end an antinode. Only odd multiples of λ/4 fit in, so f_n = (2n − 1) · c/(4L). The natural frequencies are f₁, 3 f₁, 5 f₁ and so on, and the even multiples are missing.
- How does the app help with exam preparation?
- You measure node spacings directly in the scene and work out c yourself. Two prediction questions and three calculations recognise typical mistakes such as a forgotten factor of 2 and show the worked solution. An explain mode guides you through all four states.
Subject: School physics
More from School physics
- The exposure triangle
- Density and buoyancy
- The Doppler effect
- Additive and subtractive colour mixing
- Franck–Hertz tube (3D)
- Grating interference
- Photoelectric effect (3D lab)
- Millikan experiment
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