Whistle Acoustics Explorer

Purpose of This Tool

This tool models the interaction between a resonator's natural frequencies and the edge-tones produced by a jet of air striking a sharp edge (the labium). A stable musical note occurs only when the edge-tone frequency matches one of the resonator's natural modes, creating "acoustic locking."

Two Resonator Models

1. Pipe (Open-Closed Tube):

An open-closed tube has one end open to the atmosphere and one end closed (or effectively closed). Examples include flutes (embouchure end is effectively closed by the air column), tin whistles, clarinets, and some organ pipes.

Key characteristic: Open-closed tubes produce only odd harmonics (1st, 3rd, 5th, 7th...). This is why the frequency formula contains (2n-1), which generates odd numbers. The fundamental is at frequency f₁ = c/(4L), and subsequent modes are at 3f₁, 5f₁, 7f₁, etc. This odd-harmonic series gives instruments like the clarinet their distinctive sound, different from open-open tubes like pan pipes.

2. Helmholtz Resonator:

A Helmholtz resonator is a cavity with a narrow neck opening—like blowing across a bottle. The air in the cavity acts as a spring, and the air in the neck acts as a mass, creating a single dominant resonance frequency. Unlike pipes, Helmholtz resonators typically have just one strong mode.

End Correction (Pipes)

Why is end correction needed? The effective acoustic length of a pipe is slightly longer than its physical length. At the open end, sound waves don't stop abruptly—they extend slightly beyond the pipe opening due to acoustic radiation and diffraction.

Without end correction, calculated resonant frequencies would be too high (the pipe would seem shorter than it acoustically behaves). The correction accounts for the pressure antinode occurring outside the physical boundary.

Typical values:

• Unflanged pipe (most instruments): δ ≈ 0.6 × radius ≈ 0.3–0.4 × diameter
• Flanged pipe: δ ≈ 0.8 × radius
• For a 15mm bore instrument: δ ≈ 4–6 mm (use 0.005 m)
• For a 10mm bore: δ ≈ 3–4 mm (use 0.003–0.004 m)

The effective length becomes L_eff = L_physical + δ, lowering all resonant frequencies proportionally.

Edge-Tone Physics

When a thin jet of air strikes a sharp edge, it creates oscillating vortices at specific frequencies called edge-tones. The frequency is given by: f = St × U / d, where U is jet velocity, d is jet thickness, and St is the Strouhal number.

Strouhal Numbers (Edge-Tone Stages): The Strouhal number characterizes different oscillation modes of the jet. Edge-tones don't occur at all frequencies—only at discrete "stages" with specific Strouhal numbers:

• Stage 1 (St ≈ 0.2): Fundamental mode, lowest frequency, most stable
• Stage 2 (St ≈ 0.4): First overtone
• Stage 3 (St ≈ 0.7): Second overtone
• Stage 4 (St ≈ 1.0): Third overtone
• Stage 5 (St ≈ 1.3): Fourth overtone

These empirically-determined values (from Powell, 1961) represent the natural vortex shedding patterns. Higher stages require higher jet velocities but are less stable in practice.

Acoustic Locking

Locking occurs when the edge-tone frequency exactly matches a resonator mode frequency. The tool calculates the required jet velocities for each combination of resonator mode and edge-tone stage, showing you which velocities will produce stable tones.