Minnaert's Bubble

Plate 90 speaks for itself: “Every bubble sings at a pitch set by its size.”

The Physics

A gas bubble in water rings like a spring: the gas is the spring and the surrounding water is the mass. The pitch is set by its radius, which is why streams babble.

f = (1 / 2πa) √(3γp / ρ)

• f: resonant frequency, Hz
• a: bubble radius, m
• γ: heat capacity ratio (air 1.4), dimensionless
• p: ambient pressure, Pa
• ρ: water density, kg m⁻³

Air in water at 1 atm: f·a ≈ 3.3 m/s. A 1 mm radius bubble sings near 3.3 kHz.

Honest caveat: Small bubbles are closer to isothermal and surface tension matters, so the formula is best for millimetre-size bubbles.

Sources

• M. Minnaert, Phil. Mag. 16, 235 (1933).

On the poster

Plate 090, “Minnaert's Bubble”: the pixel-art plate with its slogan “Every bubble sings at a pitch set by its size.”, the key equation and the facts panel shown above. Pixelated Physics. Physics. Personified.