Singing Wine Glass

What is modelled

The lowest flexural mode of a wine glass: the rim breathes into an ellipse (azimuthal mode number n = 2, four nodes and four antinodes) at a pitch of about 500 Hz. A loudspeaker drives that one mode with a pure tone; its amplitude builds up, rings down, and when the surface bending strain at the rim exceeds what the glass surface can take, the glass fractures. Everything on screen except the drawing exaggeration of the motion comes from the numbers below.

Equations

• Natural frequency of the n-th flexural mode of a thin ring (Hoppe): f = (h / 2πR²) √(E′ / 12ρ) · n(n²−1)/√(n²+1), with E′ = E/(1−ν²), R = 42.5 mm, h = wall thickness, glass E = 63 GPa, ν = 0.22, ρ = 2500 kg/m³. For h = 1.5 mm this gives 527 Hz.
• Water in the bowl adds mass: for wall motion w cos nθ the potential-flow added mass is ρwR/n per wetted wall area. With the assumed mode shape s(z) = (z/H)² and water up to fraction φ of the height, the Rayleigh quotient gives f = f0 / √(1 + (ρwR/(nρh)) φ⁵).
• Modal oscillator for the rim displacement amplitude w: w″ + 2ζω0 w′ + ω0² w = c p0 cos(ωd t) / μ, with damping ζ = 1/(2Q), areal mass μ = ρh (times the water factor), peak sound pressure p0 = √2 · 20 µPa · 10SPL/20 and a coupling c = 0.08 (the part of the pressure that has the shape of the n = 2 mode).
• The page integrates the slowly varying complex amplitude z of w = Re(z eiωdt): z′ = [−ζω0 + i(ω0²−ωd²)/(2ωd)] z − iF/(2ωd), advanced exactly each step. Steady state on pitch: w = F/(2ζω0²); the resonance is only f0/Q (about 0.13 Hz) wide.
• Surface bending strain of the ring (inextensional): ε = w (n²−1) h / (2R²). The glass fractures when ε ≥ σ/E (default 50 MPa, a typical value for glass with surface flaws). The breaking sound level on pitch follows by solving the steady-state formula for the pressure: about 129 dB for the default glass.

Checks run in the page

• The envelope integrator against a brute-force RK4 integration of the full oscillator.
• The ring-down time against 2Q/ω₀ from the RK4 integration.
• The fracture level found by bisection on the time-stepped glass against the closed form.
• Power balance of the driven oscillator (power in = power dissipated).
• The numerical integral behind the water-loading factor, and the linear thickness scaling of the pitch.

Honest simplifications

• Only the n = 2 rim mode is modelled, as a thin inextensional ring; real glass has a bowl-shaped shell with a range of modes, and the real pitch can differ by tens of Hz. The mode shape in z (a quadratic that is largest at the rim) is an assumption, used for the drawing and for the water loading.
• The coupling of sound to the mode (c = 0.08), the quality factor (Q = 4000 by default) and the 50 MPa strength are representative, not measured values; real glasses differ and breaking needs very loud sound (the often quoted figure is about 100 to 130 dB). Fatigue and sub-critical crack growth are not modelled, only a strain limit.
• The water term is an estimate (potential flow, assumed mode shape); the real shift depends on fill height more strongly than a simple scaling suggests.
• The motion in the picture is exaggerated (default 25 times; the real rim moves well under one millimetre) and slowed to about 2 cycles per second so you can see the standing wave. The sound is a pure sine and a damped sine at the computed frequencies. The drive is a clean tone, whereas a singing voice also has overtones and wobble (set by the slider).
• Sound waves drawn from the speaker are decoration, not an acoustic simulation.

• Press the default preset and wait: on pitch at 126 dB the rim builds up slowly but stays just below the breaking strain; at 134 dB it cracks within seconds.
• Detune the tone by 0.4 Hz or add singer wobble: the resonance is only f0/Q = 0.13 Hz wide, so the amplitude collapses and the glass survives.
• Pour water to 85 %: the pitch drops (third graph) and you must retune. Tap the glass to hear and see it ring down freely.