Resonance in pipes occurs when sound waves reflect back and forth inside an air column and form a standing wave. Only certain wavelengths fit the pipe geometry, so only certain frequencies are reinforced strongly. These special frequencies are called resonance frequencies or harmonics.
The boundary conditions depend on whether the pipe end is open or closed. For air displacement:
an open end behaves like a displacement antinode, while a closed end behaves like a displacement node. This difference changes which wavelengths can fit inside the pipe.
For an open pipe (open at both ends), antinodes occur at both ends, so the allowed wavelengths satisfy
\[
L = \frac{n\lambda_n}{2}, \qquad n=1,2,3,\dots
\]
which gives
\[
\lambda_n = \frac{2L}{n}.
\]
Since frequency is related to wavelength by \(f=v/\lambda\), the resonance frequencies are
\[
f_n = \frac{n v}{2L}.
\]
Every positive integer harmonic is allowed for an ideal open pipe.
For a closed pipe (one end closed, one end open), there is a node at the closed end and an antinode at the open end. The allowed standing waves satisfy
\[
L = \frac{n\lambda_n}{4},
\]
but only for odd integers
\[
n=1,3,5,\dots
\]
Therefore,
\[
\lambda_n = \frac{4L}{n},
\qquad
f_n = \frac{n v}{4L},
\]
with odd \(n\) only. This is why a closed pipe does not contain the full harmonic sequence in the ideal model.
The sound speed in air depends on temperature. A common approximation for classroom problems is
\[
v = 331 + 0.6T,
\]
where \(T\) is in °C and \(v\) is in m/s. Warmer air gives a faster sound speed, which raises all resonance frequencies. For example, at \(20^\circ\text{C}\),
\[
v = 331 + 0.6(20) = 343\ \text{m/s}.
\]
If a closed pipe has length \(L=0.5\ \text{m}\) and we use the fundamental \(n=1\), then
\[
f_1 = \frac{v}{4L} = \frac{343}{4(0.5)} = 171.5\ \text{Hz}.
\]
Standing waves in pipes are important in musical acoustics, organ pipes, wind instruments, and resonance demonstrations in physics labs. The node–antinode structure also explains why the tone changes when the effective air-column length changes, such as when finger holes are opened or closed in a flute-like instrument.
This calculator uses the ideal pipe model. At university level, a more refined treatment includes end correction, viscosity, temperature gradients, non-uniform bores, and real acoustic impedance at the pipe ends. Still, the formulas used here capture the main resonance pattern very well and are the standard starting point for sound-wave and acoustics problems.
