# Bottle Organ

> A playable row of tuned bottles that settles a folk claim by making both halves of it
> testable on one object. Blow across a bottle and pouring water in RAISES the note; strike
> the same bottle and pouring water in LOWERS it. Two oscillators, one piece of glass,
> opposite signs.

URL: https://bottle-organ.skillsafe.ai/

## What it does

Models a row of 3-12 bottles of a chosen commercial format, each with an adjustable water
level, and sounds them two ways. Everything runs in the browser: no account, no network
call, no server, no audio files. Every note is synthesised at play time from the frequency
the physics gives.

## The two mechanisms

- **Blown (Helmholtz).** The air in the neck is the mass; the compressibility of the air in
  the body is the spring. f = (c/2pi) sqrt(A / (V L_eff)). Adding water shrinks V, which
  stiffens the spring without changing the mass, so the pitch rises.
- **Struck (shell mode).** The glass rings in its n = 2 ovalling mode, as a wine glass does.
  Water adds mass and changes no stiffness, so the pitch falls:
  f = f_dry / sqrt(1 + phi beta), beta = rho_water R / (n rho_glass t). The wetted modal
  fraction phi follows the published law of Jundt et al., JASA 119, 3793 (2006): the liquid's
  kinetic energy goes as the integral of S(z) z^(2 beta) dz, so phi is a steep POWER of depth
  rather than a ratio of depths. Their measured exponent on a wine glass is 5.5; a bottle is
  not a wine glass, so that number is quoted, not used.

The explanation people usually give for the struck bottle — an air column above the water —
predicts the OPPOSITE sign, and the app will draw it on the same axes so you can see it fail.

## What has been measured rather than asserted

Two simulations run behind the app, neither given the formula it checks.

- A 2-D axisymmetric FDTD of the wave equation, over the bottle's air and a region of free
  air outside its mouth. It contains no lumped element and no end correction. Validated
  first on a closed rigid cylinder, whose modes are exact: 0.002% on the axial mode and
  0.06% on the radial one, the radial one being the test of the 1/r curvature term.
- A ring eigensolver (finite-difference membrane and bending strains, Jacobi rotations) plus
  a one-dimensional potential-flow solve for the water's added mass.

Results:

- **Unflanged end correction, measured 0.6167 bore radii** against Levine & Schwinger's
  published 0.6133 — agreement to 0.56%.
- **Flanged 0.7872** against the exact 0.82159 of Norris & Sheng (1989). Note this is NOT
  8/(3pi) = 0.8488: that is Rayleigh's rigid-piston UPPER BOUND, which he proved is never
  attained, and which several textbooks print as though it were the value. The remaining
  shortfall here is a finite baffle, and widening the baffle moves the measurement from
  0.7465 toward 0.8216.
- **Flanged and unflanged genuinely differ**, by 0.1705 radii measured against an exact
  0.2083, with flanged the larger at every grid spacing. Rayleigh MEASURED the flange's
  contribution with organ pipes in 1877 and got 0.2R; Bosanquet got 0.25R.
- **What to add to a BOTTLE neck is unsettled.** Seven published prescriptions span 1.4a to
  1.7a — a 21% spread worth 33 cents of pitch — and one source contradicts itself on the
  same page. The app defaults to the exact pair (1.4349a) and offers all of them.
- **The lumped Helmholtz formula runs 28% high on this app's wine profile and 14% high on
  an abrupt-shouldered control.** The difference is the conical shoulder, whose air the lumped
  model files as spring when most of it acts as mass. A university lab handout warns that real
  bottles disagree with the closed form by 20-30%, and both figures sit inside that.
- The blown and struck curves for a wine bottle DO cross, but only well outside the lumped
  model's validity. That crossing is where the engine stops being entitled to an answer, not
  a prediction.

## Provenance

A row of tuned bottles is folk apparatus with no author, year or publisher; this is an
independent implementation written from the acoustics. The classical results it computes
with (Helmholtz; Rayleigh for the inextensional ring and for the bounds on the flanged end
correction; Norris and Sheng for its exact value; Levine and Schwinger for the unflanged pipe;
Jundt et al. for the liquid added-mass scaling law; ISO 16 for A4 = 440 Hz) are credited in CREDITS.txt,
which also reproduces a TMview trademark search verbatim and states plainly that no
clearance is claimed.

## Pages

- / — the app, the plot, the physics panel, the findings panel and the help/credits panel
- /CREDITS.txt — provenance, sources, the trademark search, and what differs from the
  ordinary classroom demonstration
- /LICENSE.txt — MIT