BOTTLE ORGAN — credits, provenance and what is not mine ======================================================= THE INSTRUMENT -------------- A row of tuned bottles is folk apparatus. It has no author, no first edition and no publisher, and it would be dishonest to invent one in order to have somebody to credit. It has been a school science-room and music-room fixture for well over a century, and it is the subject rather than the source of this app. This bundle is an INDEPENDENT IMPLEMENTATION written from the acoustics. No code, art, sound, layout or text was taken from any product, site or teaching kit. Specifically: - every frequency is COMPUTED from the bottle's geometry, never sampled or tabulated; - there is no audio file anywhere in this bundle — every note is built at play time from an oscillator or from a noise buffer the page fills itself; - the renderer, the synthesiser, the two numerical oracles and the page are original; - there is no network call, no account, no analytics and no third-party script. THE PHYSICS, WHICH IS NOT MINE ------------------------------ The closed forms this app computes with are classical results. They are credited here and tagged in facts.md, and every one of them is independently RE-DERIVED numerically by tools-harness.js from a simulation that is never shown the formula. Helmholtz resonance f = (c / 2pi) sqrt( A / (V L_eff) ) Hermann von Helmholtz, "Die Lehre von den Tonempfindungen" (1863). The modern textbook treatments used here are Lord Rayleigh, "The Theory of Sound" (2nd ed., 1894-96) and Kinsler, Frey, Coppens & Sanders, "Fundamentals of Acoustics". End correction, FLANGED end (a pipe flush in a large rigid plane) alpha = 0.82159 A. N. Norris and I. C. Sheng, "Acoustic radiation from a circular pipe with an infinite flange", Journal of Sound and Vibration 135(1), 85-93 (1989). THIS IS NOT 8/(3 pi) = 0.8488, AND THIS BUNDLE SHIPPED THAT WRONG VALUE FIRST. 8/(3 pi) is Rayleigh's UPPER BOUND, obtained by assuming a uniform axial velocity across the mouth, and Rayleigh says in the same section (Theory of Sound II, sec. 307) that it is never attained. He proved pi/4 = 0.7854 as a lower bound, tightened the upper bound to 0.82422 in Appendix A, and conjectured "the real value of alpha is probably not far from .82 R" — which Norris and Sheng confirmed 93 years later. Several textbooks print 0.85 as though it were the flanged value. This one does not, and keeps the bound as a separate named constant so the two cannot be confused again. Note also INFINITE: this app measures 0.7872 for a FINITE baffle, and shows the measurement climbing from 0.7465 toward 0.8216 as the baffle is widened. The FLANGE'S CONTRIBUTION, measured before either exact value existed The modern exact difference is 0.82159 - 0.6133 = 0.2083 radii. Rayleigh measured it experimentally with two organ pipes and beat counting and got about 0.2R; Bosanquet repeated the experiment and got 0.25R (Theory of Sound II, sec. 314, 1877). WHAT TO ADD TO A BOTTLE NECK — an open question, not a constant Seven prescriptions are in print and they span 1.4a to 1.7a: 1.4 (Jyvaskyla / DigiPhysLab), 1.4349 (the exact flanged + exact unflanged pair, this app's default), 1.45 (the rounded constants summed), 1.5 (Adhikari 2021), 1.538 (UNSW Music Acoustics' own worked example, "one baffled and one unbaffled end effect"), 1.6 (UNSW prose on the same page, contradicting its own example), 1.6977 (Fletcher & Rossing, two flanged ends) and 1.7 (UIUC Physics 406 handout). That is a 21% spread, worth about 33 cents of pitch on a wine bottle. Both 1.7a sources build it as TWICE 8/(3 pi) — twice a bound. The app offers all of them in a selector rather than picking one silently. The end correction also drifts with ka, DECREASING as frequency rises F. Silva, Ph. Guillemain, J. Kergomard, B. Mallaroni and A. N. Norris, "Approximation formulae for the acoustic radiation impedance of a cylindrical pipe", Journal of Sound and Vibration 322(1-2), 255-263 (2009); arXiv:0811.3625, Table 1. At a bottle's mouth ka is a few hundredths and the drift is under 0.1%, which is why this app uses a constant — and prints the drift so that choice is visible rather than assumed. End correction, UNFLANGED end (a thin-walled pipe radiating into free air) alpha = 0.6133 in the low-ka limit H. Levine and J. Schwinger, "On the Radiation of Sound from an Unflanged Circular Pipe", Physical Review 73, 383 (1948). This app's own 2-D solver, which is never given the number, measures 0.6167. Inextensional thin-ring in-plane bending (the struck bottle) f_n = (1/2pi) n(n^2-1)/sqrt(n^2+1) sqrt( E I / (rho A R^4) ) Lord Rayleigh, "The Theory of Sound". The same mode family that makes a wine glass sing; see also A. P. French, "In vino veritas: A study of wineglass acoustics", American Journal of Physics 51, 688 (1983), and T. D. Rossing's writing on wine glasses and bell modes. Added mass of a liquid inside a cylinder vibrating in circumferential mode n mu = rho_w R / n per unit wall area A classical potential-flow result. It is re-derived here twice: analytically in the comments of js/ring.js, and numerically by a one-dimensional boundary-value solve in addedMassCoefficient(), which recovers 1/n to about 1e-4 relative. The DEPTH LAW for a partly filled vessel omega^2 = omega_0^2 / (1 + alpha h^n), with the liquid's kinetic energy proportional to the integral of S(z) z^(2 beta) dz and the wall's modal amplitude growing as z^beta. G. Jundt, A. Radu, E. Fort, J. Duda, H. Vach and N. H. Fletcher, "Vibrational modes of partly filled wine glasses", Journal of the Acoustical Society of America 119(6), 3793-3798 (2006), Eqs. (6)-(8). Their measured exponent for the (2,0) mode of a wine glass is n = 5.5. QUALIFIED for this app: the measurement is on a thin-walled open wine glass, not a thick-walled closed bottle. The MECHANISM carries over; the exponent very probably does not, so the app offers three mode shapes and reports their spread. That the struck bottle is the GLASS and not the water or an air column Stated correctly, in quotable words, by the Department of Physics, University of Guelph ("the tone is produced by the glass vibrating, not the air inside it") and by the University of Illinois Physics Van ("the water adds mass without changing the stiffness"). Both are outreach pages rather than refereed sources, and are credited as such. The air-column hypothesis is ruled out by SIGN, not by authority: an air column above a rising water level gets shorter, so it would rise. Speed of sound in air c(T) = 331.3 sqrt(1 + T/273.15) m/s; the linear form 331.3 + 0.606 T is its tangent at 0 C and the two agree to better than 0.4% over -20 to +50 C, which this app measures rather than asserts. Equal temperament and concert pitch A4 = 440 Hz (ISO 16); twelve-tone equal temperament; cents = 1200 log2(f2/f1). WHAT DIFFERS FROM THE ORDINARY CLASSROOM DEMONSTRATION ------------------------------------------------------ 1. It gives you BOTH oscillators on the same object and the same axes. The usual demonstration does one or the other; the point here is that they have opposite signs. 2. It refuses to answer where it is not entitled to. When kL passes 0.5 the neck has stopped being a plug of air, and the app says so and draws the curve dashed instead of quietly extrapolating a formula past its own assumption. 3. It reports that the textbook lumped formula is 28% high on its wine profile and 14% high on an abrupt-shouldered control, and shows a full transfer-matrix solve of the same profile closing most of the gap. The difference between those two figures is the shoulder, whose air the lumped model files as spring when most of it acts as mass. This app's four-segment profile has no punt, so its shoulder is longer than a real Bordeaux's and the larger figure is an upper bound, not a measurement of a real bottle. 4. It shows the WRONG explanation next to the right one. The air column above the water is what people usually reach for to explain the struck bottle; it predicts the opposite sign, and you can switch it on and watch it be wrong. 5. It treats the bottle's shoulder as the truncated cone it is, integrated exactly, rather than as a cylinder of the mean radius. 6. It is silent about absolute struck pitch beyond what a thin-ring model deserves. A bottle is not a free ring: it has a base, a shoulder and a finite height. TRADEMARK — SEARCHED AND REPORTED, NOT CLEARED ----------------------------------------------- Searched on TMview (https://www.tmdn.org/tmview/) on 2026-09-21, across the offices whose indexes that service covers. Reproduced as it came back: - EXACT-match search, "bottle organ": 0 results. - EXACT-match search, "BOTTLE ORGAN": 0 results. - Contains-search, "bottle organ", all classes: 53 results, and NOT ONE of them is the phrase "bottle organ". Every hit is a fuzzy match on "bottle" together with "organic", "ORGANIKA" or "organizer". Representative entries, verbatim: "ORGANIC BEST ALE BOTTLED BY SAM.SMITH OLD BREWERY (TADCASTER)" EM 000608620, Registered, class 32, SAMUEL SMITH OLD BREWERY (TADCASTER) "WATER BOTTLE ORGANIZER" US 98159535, Ended, class 20, Hong, Qiuna "ORGANIKA TRUTH IN A BOTTLE HEALTH PRODUCTS INC" EM 005712567, Registered, classes 5 and 3, Organika Health Products Inc. "Bottle Tree Hill Organics" AU 2518237, Registered, class 31, William Stafford Seiler "Danish Organic Bottles" EM 018799407, Ended, class 40, Laursen, Poul - Contains-search, "bottle organ", restricted to Nice class 9: 0 results. - Contains-search, "bottle organ", restricted to Nice class 15: 0 results. - Contains-search, "bottle organ", restricted to Nice class 28: 0 results. - Contains-search, "bottle organ", restricted to Nice class 41: 0 results. - Contains-search, "bottleorgan": 0 results. - Contains-search, "tuned bottles": 0 results. - Contains-search, "bottle phone": 0 results. - EXACT-match search, "bottle band": 4 results, all of them the single word BOTTLEBAND in class 21 (glassware), which is not this: BOTTLEBAND US 97090490 Registered class 21 Zorba LLC BOTTLEBAND US 98327116 Registered class 21 Zorba LLC BOTTLEBAND US 86085955 Ended class 21 Hancey, Nicholas S. BOTTLEBAND US 76529109 Ended class 21 Lorimor, Jennifer - Contains-search, "glass harp": 4 results, none of which contains the phrase "glass harp" in its mark text. THIS IS NOT A CLEARANCE OPINION. It is one database, on one day, covering only the offices TMview indexes, matching only registered and applied-for marks, and saying nothing about unregistered or common-law rights. No clearance is claimed or implied. LICENCE ------- See LICENSE.txt. The classical results above are not licensed by anyone; the code is.