Standing Wave
y = sin(2x) · cos(2t) · 1.5
Open in the app The dials and keys named below are the app's.
What it draws
The formula is a product of one factor in x and one in t, and the two never mix. sin(2x) fixes the shape. The 2 is the wavenumber, so crests are 2π/2 = π ≈ 3.14 apart and the curve crosses zero every π/2 ≈ 1.57. Across ±6 that puts seven crossings on the plot, at 0, ±1.57, ±3.14, ±4.71, with eight peaks and troughs alternating between them.
cos(2t) fixes the timing. Every point returns to where it started after 2π/2 = π ≈ 3.14 seconds at Speed 1, and the trailing 1.5 is the height of the tallest crest.
Why the pattern
Because x and t sit in separate factors, nothing travels. The curve swells and collapses in place, and the seven crossings stay pinned at zero for all time. Those are the nodes, and the eight points of greatest swing are the antinodes.
The identity sin(2x)·cos(2t) = ½sin(2x − 2t) + ½sin(2x + 2t) says where the stillness comes from. This one curve is two identical waves passing through each other in opposite directions, each moving at 2/2 = 1 unit per second. A node is where the two are always out of step, an antinode where they always agree. Twice per period, at t = π/4, 3π/4, …, cos(2t) is zero and the whole line is momentarily straight, with all of its energy in the motion rather than the shape. A plucked string, an organ pipe and a particle in a box stand still for this reason.
History
Michael Faraday described standing waves in 1831, on the surface of a liquid in a shaking vessel. Franz Melde produced them on a string driven by a tuning fork in 1859 and gave them the name stehende Welle.
Try
- Front in the deck: straight on, the textbook figure, with the nodes as the points that never leave the line.
- Write the two travelling waves out in full, sin(2x − 2t)·0.75 + sin(2x + 2t)·0.75: the picture does not change.
- Then drop one of them. sin(2x − 2t)·0.75 alone slides steadily to the right and the nodes are gone.
- Change sin(2x) to sin(3x) for eleven nodes, π/3 ≈ 1.05 apart, at the same rate of flapping.
- Raise Ribbon depth until the ribbon's recent past covers one whole cycle, and its far edge repeats its near one.