Standing waves in pipes

The boundary conditions for a pipe

Standing waves in a pipe are sound waves reflecting from the ends. What each end forces depends on whether it is open or closed:

• An open end is an antinode — the air there is free to move, so it oscillates with maximum amplitude.

• A closed end is a node — the air cannot move against the solid barrier, so its displacement is always zero.

• These two rules generate everything on this page. As with strings, sketch the pattern first and read the wavelength off the geometry.

Open pipes (open at both ends)

• Antinode at each end.
• The simplest pattern that fits has an antinode–node–antinode arrangement: half a wavelength.

$$\lambda_n = \frac{2L}{n} \qquad f_n = \frac{nv}{2L} = nf_1 \qquad (n = 1, 2, 3, \dots)$$

• All harmonics are present — the same series as a string, for the same reason: both ends impose the same kind of boundary, so a whole number of half wavelengths always fits.
• $v$ is the speed of sound in air, usually taken as $340$ m s⁻¹ unless a question states otherwise.

Closed pipes (closed at one end)

• Node at the closed end, antinode at the open end.
• The shortest pattern joining a node to an antinode is a quarter wavelength.

$$\lambda_n = \frac{4L}{n} \qquad f_n = \frac{nv}{4L} = nf_1 \qquad (n = 1, 3, 5, \dots \text{ only})$$

• Only odd harmonics are present. An even harmonic would require the same type of boundary at both ends, which a closed pipe does not have.
• The fundamental is $f_1 = \dfrac{v}{4L}$ — half that of an open pipe of the same length, so a closed pipe sounds an octave lower.

The three systems compared

	String (both ends fixed)	Open pipe	Closed pipe
Ends are	node, node	antinode, antinode	node, antinode
Fundamental wavelength	$2L$	$2L$	$4L$
Fundamental frequency	$\dfrac{v}{2L}$	$\dfrac{v}{2L}$	$\dfrac{v}{4L}$
Harmonics present	all	all	odd only
$n$th wavelength	$\dfrac{2L}{n}$	$\dfrac{2L}{n}$	$\dfrac{4L}{n}$ ($n$ odd)

• The pattern to remember: matching ends give all harmonics and $\lambda_1 = 2L$; mismatched ends give odd harmonics only and $\lambda_1 = 4L$.

Counting harmonics in a closed pipe

• The harmonics are $f_1$, $3f_1$, $5f_1$, $7f_1$ — so the third harmonic is the second one that exists.
• Questions exploit this. "The next harmonic above the fundamental" in a closed pipe is the third harmonic at $3f_1$, not $2f_1$.
• If a calculation for a closed pipe produces an even multiple of $f_1$, something has gone wrong — go back and check which end you made the node.