Helmholtz Resonator Bottle (2D)

What you are looking at

Blowing across a bottle makes it hum. The air in the neck behaves like a small piston (a mass), and the air trapped in the body behaves like a spring that is squeezed when the piston moves in. A mass on a spring has one natural frequency, so the bottle has one pitch: the Helmholtz resonance.

Equations used

• Plug mass m = rho A Leff, with A = pi r^2 and Leff = L + 1.7 r (the moving air extends a little beyond both ends of the neck; the end correction is a slider).
• Cavity stiffness for adiabatic compression k = gamma P0 A^2 / V = rho c^2 A^2 / V, with c = sqrt(gamma R T / M).
• Resonance f0 = (1/2pi) sqrt(k/m) = (c/2pi) sqrt(A / (V Leff)).
• Equation of motion m x'' + b x' + k x = A p_drive(t), integrated with RK4 at 40 steps per period. The cavity pressure change is p' = rho c^2 A x / V.
• Damping b = b_rad + b_visc: radiation into the room b_rad = rho w^2 A^2 / (2 pi c) (flanged piston) and the laminar boundary-layer shear in the neck b_visc = 2 pi r L sqrt(rho w eta / 2), multiplied by an "extra loss" factor for turbulence at the lip.
• Quality factor Q = sqrt(k m) / b, ring-down time tau = 2m/b, driven gain G(w) = w0^2 / sqrt((w0^2 - w^2)^2 + (b w / m)^2) (equals Q at resonance).

What every control does

• Bottle volume, water poured in: the air volume is the bottle minus the water. Less air means a stiffer spring, so the pitch rises as you fill the bottle (f0 ~ 1/sqrt(V)).
• Neck length, neck radius, end correction: a longer neck is a heavier plug (lower pitch); a wider neck has more area, so a stronger spring and heavier plug at once (higher pitch, f0 ~ r / sqrt(L + 1.7 r)).
• Gas and temperature: they change the sound speed. Helium raises c by about 2.7 times, so the pitch rises by the same factor.
• Extra loss: real bottles are damped far more than laminar theory says because the air jet separates at the lip; this slider scales the viscous term.
• Excitation: a pure tone from a loudspeaker, a frequency sweep that plots the measured response, breath noise (the bottle picks its own frequency out of the noise) or a tap and free ring-down.
• Slow motion and exaggeration: the real motion is 100 times a second; the animation slows time and enlarges the plug swing (a few millimetres). The graphs and readouts are in real units. Sound output is at the true frequency.

Honest simplifications

The model is linear and uses a lumped plug, so it ignores the shape of the shoulder, wall flexing, the nonlinear jet that really drives a blown bottle, and the fact that the wavelength must be much longer than the bottle (valid for f0 up to roughly c / (2 pi) / bottle size, which holds for the presets). The 1.7 r end correction is the flanged-opening value; an unflanged opening is closer to 1.45 r. The damping is a model, not a measurement of a particular bottle.

Checks run in the page

(1) An RK4 ring-down measures the frequency by zero crossings and the decay rate from successive peaks and compares them with the analytic damped values. (2) Nine separate driven runs find the steady-state displacement peak, compared with w_pk = sqrt(w0^2 - b^2/2m^2). (3) Ring-downs at three water levels confirm that w0^2 V_air stays constant.