What it does
A wave is a pattern that moves while the stuff it moves through stays where it is. This page draws a wave travelling along a string, y = A sin(kx − ωt), and marks one particle of the string in orange: it only bobs up and down as the wave passes. Change the amplitude, wavelength, period and direction and the equation is rewritten with your numbers. Two more views show what happens when waves meet: Two waves adds a second wave (superposition), which makes a standing wave when equal waves go opposite ways, and Beats shows and plays two close notes that swell and fade.
It follows the school topic of waves: Class 11 Physics, Waves (transverse and longitudinal waves, displacement relation y = A sin(kx − ωt), amplitude and phase, wavelength and wave number, period and angular frequency, the speed of a travelling wave, superposition, reflection and standing waves, beats). Before this, see SHM circle for the up-and-down motion of each particle; afterwards, standing waves calculator works out the notes of strings and pipes and doppler and beats covers the Doppler effect.
How to use it
- One wave: move the sliders. A is the height of a crest, λ the distance from crest to crest, and T the time for one full up-and-down of a particle. Choose the direction: kx − ωt moves right, kx + ωt moves left.
- Move the tracked particle slider to follow a different point of the string. Its height is shown under the picture.
- Two waves: set the second wave’s amplitude, wavelength, direction and phase. Thin lines are the two waves; the thick line is their sum. With equal waves going opposite ways, the dots mark the nodes that never move.
- Beats: set two tones a few hertz apart. The picture shows one second of sound at one place, with the loud–soft envelope. Press Play the beats to hear them (nothing plays until you press).
- Pause stops the picture so you can study one moment.
Worked example
With the page as it opens: A = 5 cm, λ = 2 m, T = 1 s, moving right. The wave number is k = 2π / 2 = 3.142 rad/m and the angular frequency is ω = 2π / 1 = 6.283 rad/s, so y(x, t) = 5 sin(3.142 x − 6.283 t) cm. The wave speed is v = ω / k = 2 m/s, but the orange particle never moves sideways: its top speed up and down is Aω = 31.42 cm/s.
In Two waves, with a second wave of 5 cm and 2 m going left, the sum is a standing wave: nodes every λ/2 = 1 m and antinodes swinging to 10 cm. In Beats, tones of 440 Hz and 444 Hz give |440 − 444| = 4 beats a second, a loud moment every 0.25 s, at a pitch of about 442 Hz.
Check yourself
Five questions with fresh numbers: the speed and wavelength from an equation y = A sin(kx − ωt), the frequency from the speed and wavelength, the number of beats between two tuning forks, and which way a wave travels from the sign in its equation. Nothing is saved.
Tips
- Wave speed (how fast the pattern moves) and particle speed (how fast a bit of string moves up and down) are different things.
- On one string every wave travels at the same speed, so when you change the second wave’s wavelength, its period changes with it.
- Musicians tune by listening for beats: as two notes get closer, the beats slow down and vanish when they match.
FAQ
Why does the standing wave not move along? The two waves carry energy in opposite directions, so the pattern stays put: y = 2A sin kx cos ωt. Each point swings with its own amplitude, zero at the nodes.
What does the phase slider do? It shifts the second wave along by part of a wavelength. Going the same way, a phase of 180° makes equal waves cancel; going opposite ways it slides the nodes along.
Is this from NCERT? No. It is an independent study aid that follows the same school topics, with its own pictures and numbers.
Where the numbers come from
- Travelling waves y = A sin(kx ∓ ωt), k = 2π/λ, ω = 2π/T, v = ω/k = λ/T, the principle of superposition, standing waves 2A sin kx cos ωt with nodes λ/2 apart, interference of two waves going the same way, and beats |f₁ − f₂|: standard physics.
- Checked against an independent Python calculation and the textbook identities for standing waves and beats, written for the tests.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.