Why does a stopped pipe only produce odd harmonics?

The closed end must be a pressure antinode (point of maximum pressure variation), and the open end must be a pressure node. The simplest standing wave that fits is a quarter-wavelength. To fit more waves, you must add half a wavelength at a time, which always adds an even number of quarter-wavelengths to the initial quarter. This results in lengths of 1/4, 3/4, 5/4... of a wavelength, corresponding to the odd harmonics.

Is the 'stopped' pipe model realistic for real instruments?

Yes, it's a good first-order model for instruments like the clarinet or a soda bottle when you blow across the top. Real instruments have complications like tone holes, end corrections (because the pressure node isn't exactly at the physical open end), and material damping, but the core principle of missing even harmonics defines their characteristic timbre.

What does the speed of sound (v) depend on, and why does it matter?

The speed of sound in air depends primarily on temperature (v ≈ 331 + 0.6T m/s, with T in °C). Since the resonant frequency formulas are directly proportional to v, a warmer pipe will produce a higher pitch for the same length. This is why wind instruments go sharp when warm and flat when cold.

Why do we see a pressure wave shape instead of a displacement shape?

In sound waves, pressure and particle displacement are 90 degrees out of phase. Where the pressure variation is maximum (antinode), the particle displacement is zero (node), and vice versa. For organ pipes, it's often more intuitive to discuss pressure because the closed end is a pressure antinode. The simulator shows pressure to clearly align with these boundary conditions.

Organ Pipe (harmonic series)

Organ pipes are classic examples of standing sound waves in air columns, which this simulator explores by comparing two fundamental types: open-open pipes and stopped (closed-open) pipes. The core physics principle is that a standing wave forms when reflections at the boundaries create constructive interference. For an open-open pipe, both ends are pressure nodes (and displacement antinodes), requiring a half-wavelength to fit between the ends. This leads to the harmonic series f_n = n(v / 2L), where n = 1, 2, 3..., v is the speed of sound, and L is the pipe length. All harmonics are present. A stopped pipe has one closed end (a pressure antinode) and one open end (a pressure node), requiring an odd number of quarter-wavelengths. Its resonant frequencies are given by f_n = n(v / 4L), where n = 1, 3, 5..., producing only the odd harmonics. The simulator visually represents the pressure amplitude variation along the pipe's length for different modes, often showing the fundamental and several overtones. It also typically includes a table comparing the harmonic numbers and frequencies for both pipe types. Key simplifications include assuming perfectly rigid, smooth walls with no energy loss, an ideal speed of sound in air (e.g., 343 m/s at 20°C), and that the pipe's diameter is negligible compared to its length (so end corrections are ignored). By interacting, students learn to predict and compare harmonic series, connect mathematical formulas to visual wave shapes, and understand how boundary conditions dictate the allowed frequencies of musical instruments.

Who it's for: High school or introductory college physics students studying waves, sound, and harmonics, as well as music students learning the acoustical principles of wind instruments.

Key terms

Standing Wave
Harmonic Series
Fundamental Frequency
Overtone
Node
Antinode
Resonance
Boundary Condition

Harmonic series (same L, v)

n	Label	f (Hz)
1	Fundamental (all harmonics)	65.96
2	2th harmonic	131.92
3	3th harmonic	197.88
4	4th harmonic	263.85
5	5th harmonic	329.81
6	6th harmonic	395.77
7	7th harmonic	461.73
8	8th harmonic	527.69

How it works

A flue organ pipe is often modeled as a tube with ideal open ends (pressure nodes) or a stopped pipe with one closed end (pressure antinode there). The allowed standing waves set the harmonic series; the simulator plays a sine at the chosen mode frequency and draws the pressure shape along the pipe.

Key equations

Open–open: f_n = n v/(2L), n = 1,2,3,…

Stopped (open–closed): f_n = (2n−1) v/(4L) — only odd harmonics of the open pipe’s fundamental.

Frequently asked questions

Why does a stopped pipe only produce odd harmonics?: The closed end must be a pressure antinode (point of maximum pressure variation), and the open end must be a pressure node. The simplest standing wave that fits is a quarter-wavelength. To fit more waves, you must add half a wavelength at a time, which always adds an even number of quarter-wavelengths to the initial quarter. This results in lengths of 1/4, 3/4, 5/4... of a wavelength, corresponding to the odd harmonics.
Is the 'stopped' pipe model realistic for real instruments?: Yes, it's a good first-order model for instruments like the clarinet or a soda bottle when you blow across the top. Real instruments have complications like tone holes, end corrections (because the pressure node isn't exactly at the physical open end), and material damping, but the core principle of missing even harmonics defines their characteristic timbre.
What does the speed of sound (v) depend on, and why does it matter?: The speed of sound in air depends primarily on temperature (v ≈ 331 + 0.6T m/s, with T in °C). Since the resonant frequency formulas are directly proportional to v, a warmer pipe will produce a higher pitch for the same length. This is why wind instruments go sharp when warm and flat when cold.
Why do we see a pressure wave shape instead of a displacement shape?: In sound waves, pressure and particle displacement are 90 degrees out of phase. Where the pressure variation is maximum (antinode), the particle displacement is zero (node), and vice versa. For organ pipes, it's often more intuitive to discuss pressure because the closed end is a pressure antinode. The simulator shows pressure to clearly align with these boundary conditions.

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