Standing Waves in an Air Column (2D)

Pipe Setup

Mode number n 1

Pipe length L 1.20 m

Speed of sound v 343 m/s

Readouts

Resonant frequency f

0 Hz

Wavelength λ

0 m

Displacement nodes

Displacement antinodes

Boundary condition

Open–Open

How it works (2D method)

This is the 2D-native counterpart to the 3D standing-wave simulator, but it swaps a vibrating string for a resonating column of air — a longitudinal wave, not a transverse one, with the opposite boundary condition at a closed end: a string end is always pinned (a displacement node), while a pipe end can be either open (air moves freely — a displacement antinode / pressure node) or closed (air can't move — a displacement node / pressure antinode).

Open–Open and Closed–Closed pipes support the same harmonic series as a string: f_n = n·v/(2L), all integers n = 1,2,3…
Closed–Open ("stopped") pipes only support odd harmonics: f_n = n·v/(4L), n = 1,3,5… — a quarter-wavelength must fit in the tube instead of a half-wavelength.
The top trace draws the longitudinal displacement envelope ±sin/cos(kx)·cos(ωt); the bottom trace (toggle) draws the 90°-shifted pressure envelope, since pressure is maximal exactly where displacement is zero and vice versa.

Open-Open / Closed-Closed:  f_n = n·v / (2L),  n = 1,2,3,...
Closed-Open (stopped):      f_n = n·v / (4L),  n = 1,3,5,...
Wavelength:                 λ = v / f_n
Displacement:  y(x,t) = A·shape(x)·cos(2π f_n t)

Same resonance-mode logic as the 3D harmonic-string version (mode number, medium speed, and boundary type set the standing pattern) — only the medium and the boundary physics are genuinely different, drawn as real 2D waveform envelopes rather than a rendered 3D object.

Displacement envelope Pressure envelope Node (fixed point) Antinode (max amplitude)