Why does a closed-open pipe only produce odd harmonics?

The boundary condition requires a pressure node (zero pressure variation) at the open end and a pressure antinode (maximum pressure variation) at the closed end. This constraint means only a quarter-wavelength or odd multiples thereof can fit into the tube length. This results in the harmonic series f, 3f, 5f, etc., unlike the full integer series of an open-open pipe.

Is the speed of sound really constant in the simulator?

Yes, for simplicity, the model uses a fixed speed of sound (typically ~343 m/s at 20°C). In reality, the speed of sound in air depends on temperature. This simplification allows students to focus on the core relationship f ∝ 1/L without the added variable of temperature.

How does this relate to real musical instruments?

Wind instruments like flutes (approximately open-open) and clarinets (approximately closed-open) operate on these principles. The harmonic series determines the instrument's natural notes and timbre. However, real instruments have tone holes, bell flares, and player's embouchure that modify the ideal resonance, which this basic model does not include.

What exactly is being visualized—the displacement of air molecules or the pressure?

The graph shows the variation in acoustic pressure along the tube. At an open end, the pressure must equal atmospheric pressure, creating a pressure node. At a closed end, the pressure can oscillate maximally, creating a pressure antinode. This is the inverse of a displacement wave: where pressure is a node, displacement is an antinode, and vice versa.

Resonance Tube

Open vs closed pipe harmonics. Hear fₙ and see standing pressure wave.

Live graphs

About this model

Resonance in air columns is a fundamental phenomenon in acoustics, governed by the interference of sound waves. This simulator visualizes and sonifies the standing pressure waves that form within a cylindrical tube, depending on whether it is open at both ends or closed at one end. The core principle is that resonance occurs when the length of the tube equals an integer or half-integer multiple of the sound wavelength, creating constructive interference. For an open-open pipe, the resonant frequencies are given by f_n = n(v / 2L), where n = 1, 2, 3,... and v is the speed of sound. For a closed-open pipe, only odd harmonics are present: f_n = n(v / 4L), where n = 1, 3, 5,... The model plots the spatial variation of acoustic pressure (antinodes at open ends, nodes at closed ends) and allows you to hear the corresponding pure tone. Key simplifications include assuming a perfectly rigid tube with no thermal or viscous losses, a point source of sound at one end, and a constant speed of sound. By adjusting the tube length, selecting the boundary condition, and stepping through harmonics, students directly explore the relationship between boundary conditions, allowed wavelengths, harmonic series, and the resulting timbre.

Who it's for: High school and introductory undergraduate physics students studying wave mechanics, sound, and harmonics, as well as music students learning about the acoustics of wind instruments.

Key terms

Standing Wave
Harmonic Series
Resonant Frequency
Fundamental Frequency
Acoustic Pressure Node
Acoustic Pressure Antinode
Boundary Condition
Wavelength

How it works

Standing sound waves in a tube: allowed frequencies depend on boundary conditions. Open–open: fₙ = n·v/(2L). Open–closed (one closed end): odd quarter-wave modes, fₙ = (2n−1)·v/(4L). The animation shows a pressure-like pattern (schematic). Use Play to hear the selected harmonic.

Key equations

Open–open: fₙ = n v / (2L) · Open–closed: fₙ = (2n−1) v / (4L)

Frequently asked questions

Why does a closed-open pipe only produce odd harmonics?: The boundary condition requires a pressure node (zero pressure variation) at the open end and a pressure antinode (maximum pressure variation) at the closed end. This constraint means only a quarter-wavelength or odd multiples thereof can fit into the tube length. This results in the harmonic series f, 3f, 5f, etc., unlike the full integer series of an open-open pipe.
Is the speed of sound really constant in the simulator?: Yes, for simplicity, the model uses a fixed speed of sound (typically ~343 m/s at 20°C). In reality, the speed of sound in air depends on temperature. This simplification allows students to focus on the core relationship f ∝ 1/L without the added variable of temperature.
How does this relate to real musical instruments?: Wind instruments like flutes (approximately open-open) and clarinets (approximately closed-open) operate on these principles. The harmonic series determines the instrument's natural notes and timbre. However, real instruments have tone holes, bell flares, and player's embouchure that modify the ideal resonance, which this basic model does not include.
What exactly is being visualized—the displacement of air molecules or the pressure?: The graph shows the variation in acoustic pressure along the tube. At an open end, the pressure must equal atmospheric pressure, creating a pressure node. At a closed end, the pressure can oscillate maximally, creating a pressure antinode. This is the inverse of a displacement wave: where pressure is a node, displacement is an antinode, and vice versa.

Continue with similar topics in this category — or all 37 in Waves & Sound.

View category →

NewSchool