What you are looking at
A string is clamped at both ends (a guitar or violin string, a jump-rope, a washing line). Because the ends
cannot move, only certain wave shapes fit — the ones with a whole number of half-wavelengths between the
clamps. These are the
standing waves, or
harmonics. Points that never move
are
nodes (always the two ends, plus more inside); points that swing the most are
antinodes.
Which shapes fit
To vanish at both ends, the string must hold a whole number n of half-wavelengths:
L = n · λ/2 → λₙ = 2L / n (n = 1, 2, 3, …)
n = 1 is the
fundamental (one big belly, the lowest note); n = 2, 3, … are the
overtones, each with one more loop. A wave travels along the string at a speed set by how
tight and how heavy it is:
v = √(T / μ) → fₙ = v / λₙ = n · v / (2L)
So the harmonics are evenly spaced in frequency — f₂ = 2f₁, f₃ = 3f₁ — which is exactly what makes a
plucked string sound musical.
Tighten the string (more T) and every frequency rises (the
wave travels faster); use a
heavier string (more μ) and they all fall.
A standing wave is two travelling waves
Switch on
travelling waves and you'll see the secret: a standing wave is the sum of two
identical waves running in opposite directions. Where they always cancel, you get a node; where they always
reinforce, an antinode:
A sin(kx)cos(ωt) = ½A sin(kx−ωt) + ½A sin(kx+ωt)
On a real string, the backward wave is the forward wave
reflected off the fixed end — which is why
only the resonant frequencies build up into a big, steady pattern instead of a messy jumble.
Things to try
Step through the harmonics and count the loops — n loops, n antinodes, n+1 nodes. Watch the frequency jump
to 2×, 3× the fundamental. Then adjust tension and density and see the wave speed and every frequency shift
together while the shape stays the same. Turn on the two travelling waves to see the standing pattern
assemble itself from them.