Standing Waves in Air Columns

When sound travels through a pipe or tube, it reflects off the open and closed ends, creating two waves travelling in opposite directions. These waves superpose (add together) to form a standing wave — a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement) that appears stationary. Unlike a travelling wave, a standing wave stores energy in a fixed spatial pattern rather than transporting it. This phenomenon underpins how musical wind instruments work, from flutes and clarinets to organ pipes. The key to understanding standing waves in air columns lies in the boundary conditions — the rules that the wave must obey at each end of the pipe. At a closed end, air molecules cannot move freely, so a displacement node must form there. At an open end, air molecules can move with maximum freedom, so a displacement antinode forms. These constraints severely restrict which frequencies can resonate inside the pipe, producing a discrete set of allowed frequencies called harmonics or overtones. The collection of allowed resonant frequencies is called the harmonic series. Different pipe configurations — open at both ends, or closed at one end — produce different harmonic series. Understanding which harmonics are present determines the timbre (tonal quality) of a musical instrument, making this topic central to both physics and music theory.

Key takeaways
  • Boundary Conditions Explained
  • Open-Open Pipes: Both Ends Open
  • Open-Closed Pipes: One End Closed
  • The Speed of Sound and Temperature
  • Resonance and Amplitude Amplification

Boundary Conditions Explained

At a closed end, the rigid wall prevents air molecules from displacing longitudinally — the wall blocks their motion. This means a displacement NODE (zero amplitude) must exist at every closed end. At an open end, the pressure equalises with atmospheric pressure outside, so a displacement ANTINODE (maximum amplitude) forms. A pressure node coincides with a displacement antinode, and a pressure antinode coincides with a displacement node — they are always 90° out of phase spatially.

Open-Open Pipes: Both Ends Open

An open pipe (open at both ends) has antinodes at both ends. The simplest standing wave — the fundamental (1st harmonic) — fits exactly half a wavelength in the pipe: L = λ/2, so λ₁ = 2L. The nth harmonic has wavelength λₙ = 2L/n, giving frequency fₙ = nv/(2L) where v is the speed of sound and n = 1, 2, 3, … All harmonics (n = 1, 2, 3, 4, …) are present. This is why open-ended instruments like the flute have a richer, brighter tone — they produce a full harmonic series.

Open-Closed Pipes: One End Closed

A stopped pipe (one end closed) has a node at the closed end and an antinode at the open end. The fundamental fits a quarter wavelength: L = λ/4, so λ₁ = 4L. Only ODD harmonics are allowed: λₙ = 4L/n and fₙ = nv/(4L) where n = 1, 3, 5, 7, … The missing even harmonics give stopped pipes (like the clarinet) a distinctly hollow, woody timbre. At the same physical length, a closed pipe produces a fundamental one octave lower than an open pipe.

The Speed of Sound and Temperature

The speed of sound in air at room temperature is approximately 343 m s⁻¹ at 20 °C. More precisely, v ≈ 331 + 0.6T m s⁻¹, where T is in degrees Celsius. Because resonant frequencies depend on v, instruments go sharp (higher pitch) as temperature increases — a practical concern for orchestral tuning. End corrections also apply in practice: an open end radiates sound into free space, so the effective acoustic length is slightly greater than the physical pipe length (typically by about 0.6 × the pipe radius).

Resonance and Amplitude Amplification

When a sound source drives air in a pipe at one of the pipe's natural frequencies, resonance occurs. The energy input from the source adds constructively each cycle, building up very large amplitude oscillations. This is why blowing across the top of a bottle at just the right speed produces a loud note — you are exciting the fundamental mode of a closed-end air column. The same principle explains why the resonant frequencies of a concert hall must be considered in architectural acoustics.

The Key Equation

fn=nv2L(n=1,2,3,…)f_n = \frac{nv}{2L} \quad (n = 1, 2, 3, \ldots)
  • f_n — Harmonic frequency (Hz): The frequency of the nth standing wave resonance in the pipe.
  • n — Harmonic number (dimensionless): A positive integer (1, 2, 3, …) identifying which harmonic mode. For a closed-open pipe, only odd integers are used.
  • v — Speed of sound (m s⁻¹): The speed at which sound propagates through the medium (≈ 343 m s⁻¹ in air at 20 °C).
  • L — Pipe length (m): The physical length of the air column. For a closed-open pipe, the formula becomes fₙ = nv/(4L) with n odd.

Worked Example

Problem. A clarinet can be modelled as a cylindrical pipe closed at one end (the reed) and open at the bell. The pipe length is 0.60 m. Calculate (a) the fundamental frequency and (b) the frequency of the next allowed harmonic. Take the speed of sound as 340 m s⁻¹.

f1≈142 Hz,f3≈425 Hzf_1 \approx 142\,\text{Hz},\quad f_3 \approx 425\,\text{Hz}

Answer. Fundamental frequency ≈ 142 Hz; next allowed harmonic ≈ 425 Hz (third harmonic).

Figure
Diagram
Standing Waves in Air ColumnsOpen-Open Pipe (n=1)openopenλ = 2L | antinode–antinodeAAAf₁ = v / 2LOpen-Open Pipe (n=2)ANAλ = L | node in middlef₂ = v / L = 2f₁Closed-Open Pipe (n=1)closedopenNAλ = 4L | node–antinodef₁ = v / 4LClosed-Open Pipe (n=3)NNNA3rd harmonic (n=3, 1st overtone)f₃ = 3v / 4L = 3f₁Key: Node (N) vs Antinode (A)Node — zero displacement, max pressure variationAntinode — maximum displacement, min pressure variationClosed-open pipes: only odd harmonics (n = 1, 3, 5, …) are allowed
Origins · 1 minDaniel Bernoulli and how this was discovered
DB
Daniel Bernoulli
1700–1782 · Swiss

Daniel Bernoulli made foundational contributions to the theory of vibrating strings and air columns in the 18th century. He proposed the principle of superposition of modes — the idea that a vibrating system can simultaneously sustain multiple harmonic modes — which is essential to understanding why real pipes and strings produce complex tones rather than pure sinusoids. His work laid the mathematical groundwork for the harmonic series observed in resonating air columns. Bernoulli came from an extraordinary mathematical family: his father Johann and uncle Jakob were both celebrated mathematicians, and the family reportedly had bitter priority disputes over their discoveries — Daniel's father even plagiarised one of Daniel's books by backdating his own work.

  • Superposition of Waves — Standing waves arise directly from the superposition of two identical waves travelling in opposite directions — understanding superposition is a prerequisite for standing wave formation.
  • Standing Waves on Strings — Strings and air columns both form standing waves via reflection at boundaries; comparing the two reinforces the concept of nodes, antinodes, and harmonic series.
  • Wave Reflection and Transmission — The standing wave pattern in a pipe depends on how sound reflects at open and closed ends — reflection with or without phase inversion determines the boundary conditions.
  • Resonance — Air columns resonate when driven at their natural harmonic frequencies; resonance explains the large amplitude build-up exploited in all wind instruments.