Guitar String
z = 3.5 · cos(π · x/8) · cos(2.5(y − t))
Open in the app The dials and keys named below are the app's.
What it draws
This sheet is not a landscape: x is position along a string, and y is time. A single row across the sheet, at one value of y, is the string as it stands at that instant. The direction of y is that instant repeated a little later and a little later again.
cos(πx/8) fixes the shape of the string. It is zero at x = ±4, so the string is clamped at both ends, 8 long, and this is its fundamental, one arch with no node in between. The leading 3.5 is the height of that arch.
cos(2.5(y − t)) supplies the timing. It repeats every 2π/2.5 ≈ 2.51 along y, so the string swings up and down with a period of 2.51 seconds at Speed 1. Because t enters only as y − t, the whole pattern slides along y at one unit per second, which is one unit of the time axis per second of clock.
Why it stands still
The two factors never mix: one holds x, the other holds y and t. That is what makes the picture a standing wave rather than a travelling one. The zeros at x = ±4 are nodes and stay put for all time. The crest at x = 0 is the antinode and swings the full ±3.5. Twice a period the whole string is momentarily straight, which shows on the sheet as a line at zero height running the full width. Splitting the product gives ½cos(πx/8 + 2.5(y − t)) + ½cos(πx/8 − 2.5(y − t)), the two waves running in opposite directions along the string whose sum the standing shape is.
History
Brook Taylor found the shape of the fundamental in 1713. Jean le Rond d'Alembert wrote down the wave equation for the string and its general solution in 1747. Daniel Bernoulli argued in 1753 that every motion of a string is a sum of modes like this one, a claim Euler and Lagrange refused and Fourier's work of 1822 vindicated.
Try
- Front in the deck looks along the time axis, so what you see is the string's own profile, the single arch.
- Ask for the third harmonic, 3.5·cos(3π·x/8)·cos(7.5(y − t)): two more nodes appear, at x = ±4/3, and the string swings three times as fast.
- Turn the history round, 3.5·cos(π·x/8)·cos(2.5(y + t)): the sheet slides the other way.
- Raise Span of x, y above 4 and the ends stop being ends: the formula keeps going past the clamps, into the next arch.
- Pull Mesh to its top stop: on a lit solid the still lines across the sheet are easiest to pick out.