Standing waves in pipes

Air columns as resonators

Pipes and tubes can support standing sound waves. Musical instruments such as flutes, clarinets, organ pipes, and brass instruments rely on resonances of air columns. The allowed frequencies depend on the pipe length, sound speed, and whether the ends are open or closed.

Sound waves in air are longitudinal waves, but they can still form standing waves. The oscillating quantities include air displacement and pressure variation.

Open and closed ends

At an open end of a pipe, the air is free to move, so there is a displacement antinode. The pressure variation must stay close to atmospheric pressure, so there is a pressure node.

At a closed end, the air cannot move through the wall, so there is a displacement node. Pressure variation is largest there, so there is a pressure antinode.

This means displacement and pressure standing waves are shifted relative to each other.

Open-open pipe

For a pipe open at both ends, both ends are displacement antinodes. The allowed wavelengths satisfy

L=nrac{lambda_n}{2}

where

$n=1,2,3,ldots$

Thus

lambda_n=rac{2L}{n}

and

f_n=rac{nv}{2L}

All integer harmonics are allowed:

$f_n=nf_1$

where

f_1=rac{v}{2L}

This is mathematically similar to a string fixed at both ends, though nodes and antinodes refer to different physical quantities.

Open-closed pipe

For a pipe closed at one end and open at the other, one end is a displacement node and the other is a displacement antinode. The fundamental mode fits one quarter wavelength in the pipe:

L=rac{lambda_1}{4}

so

f_1=rac{v}{4L}

The allowed wavelengths satisfy

L=(2n-1)rac{lambda_n}{4}

where

$n=1,2,3,ldots$

The frequencies are

f_n=(2n-1)rac{v}{4L}

Only odd harmonics appear:

$f_1, 3f_1, 5f_1,ldots$

Pressure versus displacement

It is easy to confuse pressure nodes with displacement nodes. For sound in a pipe, an open end is a displacement antinode but a pressure node. A closed end is a displacement node but a pressure antinode.

This happens because pressure variation is related to compression of air, not simply to how far air elements move.

End correction

Real open pipes do not have antinodes exactly at the physical end. The air motion extends slightly beyond the pipe. This is called end correction. The effective length is a bit longer than the physical length.

For a simple open end, the correction is often roughly proportional to pipe radius. This is important in accurate instrument design, though the ideal formulas capture the main behavior.

Instruments

Flutes behave approximately like open-open pipes. Clarinets behave approximately like open-closed pipes for their lower resonances, which helps explain their strong odd-harmonic character. Brass instruments are more complicated because of mouthpieces and flared bells, but standing wave ideas still apply.

The big idea

Standing waves in pipes are determined by boundary conditions at open and closed ends. Open ends are displacement antinodes and pressure nodes; closed ends are displacement nodes and pressure antinodes. Open-open pipes support all harmonics, while ideal open-closed pipes support only odd harmonics.