Shattering a Wine Glass With Sound Alone: The Physics of Resonant Fracture

The glass is a lightly damped oscillator

To first approximation a wine glass rim behaves like a single driven, damped harmonic oscillator. Its equation of motion is

• m ẍ + b ẋ + k x = F₀ cos(ωt)

where x is the outward displacement of a point on the rim, m the effective modal mass, k the effective stiffness, b the damping coefficient, and F₀ cos(ωt) the acoustic pressure force from the loudspeaker. The undamped natural frequency is ω₀ = √(k/m), i.e. f₀ = ω₀/2π, and the sharpness of the resonance is set by the quality factor Q = ω₀m/b = √(mk)/b.

Solving for the steady-state amplitude when driven at angular frequency ω gives

• A(ω) = (F₀/m) / √[(ω₀² − ω²)² + (ω₀ω/Q)²]

Exactly on resonance (ω = ω₀) the bracketed term collapses to (ω₀²/Q)², so A(ω₀) = Q·(F₀/k). The static deflection F₀/k is what that same force would produce if you simply pushed on the glass. Resonance multiplies it by Q. For fine lead crystal Q is often 1000 or more (the tone rings for seconds), so a force that would deform the rim by a nanometre statically can drive it through a micron-to-millimetre swing at resonance. That factor of a thousand is the entire secret.

Why the rim ovals: the n = 2 shell mode

A wine glass is not a point mass but a thin-walled axisymmetric shell, so it has a whole spectrum of normal modes. The one that matters — the mode you excite when you run a wet finger around the rim, and the one a singer targets — is the fundamental flexural shell mode with n = 2 nodal diameters. In this mode the circular rim flexes into an ellipse and back: two opposite points bulge out while the two points 90° away pinch in, then the pattern swaps. There are four nodes around the rim, spaced 90° apart, where the glass barely moves.

• The tone lies typically between 400 Hz and 900 Hz for a stemmed wine glass, higher for smaller/thicker glasses.
• Adding liquid lowers the frequency (extra effective mass), which is why the pitch drops as you fill the glass — a live demonstration of ω₀ = √(k/m).
• Higher modes (n = 3 triangular, n = 4 square deformations) exist at higher frequencies but are harder to excite and less efficient to fracture.

Real glasses are never perfectly round, so the n = 2 mode usually splits into two nearly-degenerate modes a few hertz apart with fixed nodal orientations. Beating between them is audible as a slow throb, and it means the driving tone must land on one specific split frequency, not the average.

The numbers: how loud, how long, how far it bends

Let us put real magnitudes on it. Glass fractures when the local tensile strain reaches roughly ε ≈ 0.1%, corresponding to a stress σ = Eε ≈ (70 GPa)(0.001) ≈ 70 MPa — though surface flaws (Griffith cracks) drop the practical fracture stress of ordinary glass to 30–50 MPa. On a rim of radius R ≈ 3 cm bending into an ellipse, that strain is reached when the rim's peak radial displacement is on the order of tens to a few hundred micrometres to ~1 mm, depending on wall thickness.

To drive the rim that far you need the on-resonance amplitude A(ω₀) = Q·(F₀/k) to reach that displacement. Because Q ≈ 10³ does the heavy lifting, the required sound pressure is not extreme — an acoustic level of about 100–110 dB (sound intensity ≈ 0.01–0.1 W/m², pressure amplitude ≈ 3–9 Pa) at the glass is typically enough. That is loud, roughly a jackhammer at 1 m, but far below the ~194 dB ceiling of air. Crucially, the buildup is not instant: the amplitude approaches steady state on a timescale

• τ ≈ Q/(π f₀) ≈ 1000/(π·660) ≈ 0.5 s

so you must hold the exact tone for a fraction of a second while energy ratchets up cycle by cycle. Cut the tone early and the rim relaxes harmlessly.

Energy accumulation and the frequency you must hit

The reason a modest speaker can smash glass is coherent energy accumulation. Each cycle, the drive does a little positive work on the rim; a lightly damped oscillator only loses a fraction 2π/Q of its stored energy per cycle. So if you feed energy in phase for hundreds of cycles, stored vibrational energy climbs until dissipation (dominated by internal friction and acoustic radiation) balances input — but by then the amplitude is Q× larger than a single push would give.

Getting the phase right requires hitting the resonance sharply. The resonance bandwidth (full width at half power) is

• Δf = f₀/Q ≈ 660/1000 ≈ 0.66 Hz

Detune by even a couple of hertz and the response drops sharply and the phase slips away from the optimal 90° lag, so energy that went in on one cycle gets pulled back out on the next. This is why the classic feedback trick works: a microphone near the glass picks up its own ringing tone, an amplifier plays it back, and the loop automatically locks onto f₀ and its split partner — the system tunes itself to the mode instead of a human guessing to the nearest hertz.

From vibration to fracture: brittle cracks at the rim

Glass is a brittle amorphous solid with essentially no plastic yielding, so it cannot relieve stress by flowing — it stores elastic energy until a crack runs. Fracture initiates not in the pristine bulk but at a microscopic surface flaw on the outer rim, where the ovaling mode puts the maximum tensile strain. Griffith's criterion says a crack of length a becomes unstable when the stress reaches

• σ_c ≈ √(2Eγ / πa)

with E ≈ 70 GPa the Young's modulus and γ the surface energy (~few J/m²). A flaw only a few micrometres deep lowers σ_c from the theoretical ~10 GPa of pristine silica to the ~30–50 MPa actually seen. Once the resonant strain reaches σ_c at that flaw, the crack propagates at up to ~1500 m/s (roughly half the shear-wave speed in glass), branches, and the stored elastic energy from the whole vibrating shell releases at once — the glass appears to explode. This is why identical-looking glasses differ so much: a thin-walled, low-damping crystal glass with a well-defined mode and a convenient surface flaw is far easier to break than a thick, heavily-damped tumbler whose Q is only tens.

Where else resonant runaway shows up

The wine glass is a photogenic special case of a universal danger: driving any weakly-damped structure at a normal-mode frequency.

• Tacoma Narrows Bridge (1940) — wind-driven aeroelastic flutter fed energy into a torsional mode until the deck's amplitude ran away and it collapsed. The mechanism (energy input each cycle exceeding damping loss) is identical, even though the source was aerodynamic, not acoustic.
• Soldiers breaking step on bridges — marching cadence near a bridge's natural frequency is banned for exactly this reason (the 1831 Broughton Suspension Bridge failure).
• MRI and machinery — engineers hunt for and damp structural resonances so pumps, turbine blades, and gantries do not fatigue and fail.
• Ultrasonic and medical use — controlled resonant excitation is exploited constructively in ultrasonic cleaners, atomizers, and lithotripsy, where focused acoustic energy fractures kidney stones by driving their own resonances and cavitation.

In every case the lesson is the same as the glass: it is not the peak force that destroys, it is timing — delivering energy in step with the structure's own oscillation so that Q multiplies a small drive into a catastrophic amplitude.

Misconceptions, and why it's harder than TV suggests

Myth: any loud enough sound will do. No — a broadband bang or a mistuned tone deposits energy across many modes and detunes off the sharp resonance, so almost nothing accumulates. You need a pure, sustained tone within roughly Δf = f₀/Q ≈ 1 Hz of the mode, held for the ~0.5 s buildup time.

• Myth: a human voice alone routinely shatters glass. It can, but it is genuinely hard: a singer must match the split-mode frequency within about a hertz, sustain ~100+ dB at that pitch, and often relies on a glass that is already thin, low-damping, and slightly flawed. The MythBusters demonstration required a trained singer and a carefully selected glass; amplified feedback is far more reliable.
• Myth: it's about resonance being 'in tune' with the whole glass. It's specifically the rim's n = 2 flexural mode, not the air column or the stem, that does the work.
• Myth: filling the glass makes it easier. Liquid adds mass (lowering f₀) but also strongly damps the mode — cutting Q — so it usually makes fracture harder, not easier. Empty, thin crystal is the ideal target.

The deep point is that resonant shattering rewards precision over brute force. A speaker that can barely rattle a table can, if it plays exactly the right note, quietly pump a glass past its fracture strain in half a second.

Driving a wine glass on-resonance versus off-resonance — same loudspeaker, same power

Property	On-resonance (f = f₀)	Off-resonance (f ≠ f₀)
Steady rim amplitude	A ≈ Q·(F₀/k) ≈ 1000× static	A ≈ F₀/k (static push)
Phase of response	90° behind the drive	≈ 0° (in phase) or 180°
Energy per cycle	Absorbed ≈ dissipated at peak	Mostly reflected/reactive
Buildup time	τ ≈ Q/(πf₀) ≈ 0.5 s	Reaches tiny steady state fast
Outcome at 100 dB	Rim strain > 0.1% → fracture	Glass merely hums, survives