For a tube closed at one end, the resonant standing waves have a node for displacement at the closed end (no motion there) and an anti-node at the open end. Once again, the pressure change is 90° out of phase with the displacement, so there's no change from atmospheric pressure at the open end and the pressure fluctuations have maximum amplitude at the closed end.

To satisfy these conditions the relationship between the wavelengths of the resonant standing waves and the length of the tube is:

n l 4

= L,     where n = 1, 3, 5, ...

Note that only the odd harmonics satisfy the conditions for resonance.

The resonant frequencies for the tube closed at one end are then given by:

fn

=

n v 4L

, where n = 1, 3, 5, ...

Many musical instruments are essentially tubes with one end open and one end closed.

Here's what it looks like as a longitudinal wave.