Abstract
A cavity with an opening sounds at one pitch (Paper 888), but in a real body the wall plates move too and couple to the air in the neck through the air in the cavity. This paper takes the cavity of Paper 888 (volume $0.010$ m³, opening radius $0.05$ m), lets part of its wall be a plate of area $0.04$ m² and mass $0.2$ kg (set values) moving like a piston, varies the plate’s spring, and counts how close the two coupled resonances can come. No new theorem or law is claimed.
First, the two resonances are pushed outside the original two frequencies: holding the plate still returns Paper 888’s $163.086441$ Hz. With the plate moving, for all seven plates the lower resonance was below the smaller of the Helmholtz frequency and the frequency of the plate with the opening blocked, and the upper resonance above the larger (Section 2).
Second, the gap between the two frequencies is the hypotenuse of a right triangle: with $f_v$ the plate’s frequency on its own spring (no air), the gap is the hypotenuse whose sides are $f_v$ minus the Helmholtz frequency and the plate’s frequency on the air spring alone, $53.487201$ Hz, agreeing within $10^{-6}$ Hz for seven plates (Section 2).
Third, at their closest, the gap is the plate’s frequency on the air spring alone: searching the plate spring without the formula, the smallest gap was $\mathbf{53.487201}$ Hz, with $f_v$ at $163.086443$ Hz, level with the Helmholtz frequency. With plate masses of $0.1$ and $0.4$ kg the smallest gap was $75.642324$ and $37.821162$ Hz (Section 3).
Fourth, matching the blocked-opening plate frequency does not bring them closest: matching the plate frequency with the opening blocked, air spring included, to the Helmholtz frequency gave resonances of $133.694241$ and $187.936757$ Hz, a gap of $54.242515$ Hz, $\mathbf{1.014121}$ times the smallest. At the smallest gap the blocked-opening plate frequency was $171.633530$ Hz (Section 3).
Fifth, the separator is whether the plate’s frequency is counted with the cavity’s air spring included or left out: matching the frequency counted without it gives the smallest gap; matching the frequency counted with it gives more than the smallest. The difference between the two ways of counting is exactly the plate’s frequency on the air spring alone, and that became the smallest gap (Section 4).
Keywords: Helmholtz resonance, coupled oscillators, resonating body, air spring, natural frequency, instrument body.
Scope (what is not claimed)
No new theorem or law is claimed: representing the low range of an instrument body by a model of two coupled oscillators, the cavity air and a plate, that coupled resonances are pushed outside the original frequencies, and that the product and sum of squares of the two frequencies are fixed by the original values, are all known in vibration and musical acoustics. The plate moves in one shape: the plate moves uniformly like a piston; its bending shape, multiple vibration shapes, losses between plate and air, and sound radiation are not included. These are set values: the cavity is as in Paper 888 (speed of sound $343.0$ m/s, air density $1.2$ kg/m³, volume $0.010$ m³, opening radius $0.05$ m, plate thickness $0.003$ m, end correction $1.7$ times the radius), and the plate area $0.04$ m² and mass $0.2$ kg are set here. They correspond to no instrument body. Timbre is not treated: how strongly each resonance sounds, how it is heard, and values for real guitars or drums are not treated. Relation to earlier papers: Paper 888, in its scope note, did not include plate vibration, leaving out coupling of the body’s plates to the cavity and absorption at the walls; this paper sits in the seat of plates vibrating and coupling to the cavity (no earlier occupant). Paper 888 showed that the pitch of a cavity with an opening is set not by length, as in a pipe, but by volume and opening, whatever the shape. Here it is counted how, when the cavity’s wall moves, the pitch splits in two and how close the two can come. What is added: finding the two resonances with a coupled plate for seven plates, confirming that the gap between them is a hypotenuse, searching for the smallest gap without the formula, comparing with matching the blocked-opening plate frequency, and placing the separator on “air spring included, or left out”.
1. Notation and the Question
Setting: for the cavity of Paper 888 (speed of sound $c=343.0$ m/s, air density $\rho=1.2$ kg/m³, volume $V=0.010$ m³, opening radius $0.05$ m, plate thickness $0.003$ m, with $1.7$ times the radius added to the neck length), the neck air mass is $m_n=8.293805\times10^{-4}$ kg and the cavity spring coefficient is $\kappa=\rho c^2/V=1.411788\times10^{7}$ Pa/m³. With outward displacement $x$ of the neck air (opening area $S_n$) and outward displacement $y$ of the wall plate (area $S_p=0.04$ m², mass $m_p=0.2$ kg, plate spring $k_p$), the cavity pressure is $p=-\kappa(S_nx+S_py)$, and
$$m_n\ddot x=S_np,\qquad m_p\ddot y=-k_py+S_pp$$In angular frequency, with $\omega_H^2=S_n^2\kappa/m_n$ (Helmholtz), $\omega_v^2=k_p/m_p$ (plate spring alone, no air), $\omega_a^2=S_p^2\kappa/m_p$ (plate on the air spring alone) and $\omega_b^2=\omega_v^2+\omega_a^2$ (plate with the opening blocked), the two coupled angular frequencies satisfy
$$\omega_1^2+\omega_2^2=\omega_H^2+\omega_b^2,\qquad \omega_1\omega_2=\omega_H\omega_v,\qquad (\omega_2-\omega_1)^2=(\omega_H-\omega_v)^2+\omega_a^2$$In the check, eigenvalues were found numerically and compared with these.
Question: however the plate spring is chosen, how close can the two resonances come?
2. The Two Resonances and Their Gap
Computation 2.1: Helmholtz $f_H=163.086441$ Hz; plate on the air spring alone $f_a=53.487201$ Hz. The two resonances for seven choices of the plate’s own spring frequency $f_v$:
| $f_v$ (Hz) | $f_b$ (Hz) | lower | upper | gap | hypotenuse |
|---|---|---|---|---|---|
| $80.000000$ | $96.233469$ | $75.043614$ | $173.857769$ | $98.814156$ | $98.814156$ |
| $120.000000$ | $131.380671$ | $109.706192$ | $178.388956$ | $68.682764$ | $68.682764$ |
| $140.000000$ | $149.869545$ | $124.756476$ | $183.013359$ | $58.256883$ | $58.256883$ |
| $163.086441$ | $171.633528$ | $138.521059$ | $192.008259$ | $\mathbf{53.487201}$ | $53.487201$ |
| $180.000000$ | $187.778808$ | $145.566533$ | $201.664207$ | $56.097675$ | $56.097675$ |
| $220.000000$ | $226.408658$ | $154.349847$ | $232.452560$ | $78.102713$ | $78.102713$ |
| $300.000000$ | $304.730833$ | $159.587324$ | $306.577811$ | $146.990487$ | $146.990487$ |
The hypotenuse is $\sqrt{(f_H-f_v)^2+f_a^2}$. For seven plates the gap agreed with it within $10^{-6}$ Hz, the product of the two resonances agreed with $f_Hf_v$ within a relative $10^{-9}$ ($13046.915309$ and $13046.915309$ at $f_v=80$ Hz), and the lower resonance was below the smaller of $f_H$ and $f_b$ and the upper above the larger. With a plate spring of $10^{12}$ N/m the lower resonance was $163.086439$ Hz.
Observation 2.2 the two resonances are pushed apart, and their gap is a hypotenuse. The gap was smallest in the row with $f_v$ level with Helmholtz, and larger in rows further away: the gap was $146.990487$ Hz at $f_v=300$ Hz and $98.814156$ Hz at $f_v=80$ Hz. In the row with $f_v$ level with Helmholtz, the gap was the same $53.487201$ Hz as $f_a$.
3. The Plate That Comes Closest
Computation 3.1: without the formula, $f_v$ was searched between $50$ and $400$ Hz by golden-section search ($200$ steps) to minimize the gap between numerical eigenvalues.
| matched frequency | two resonances (Hz) | gap (Hz) | ratio to smallest |
|---|---|---|---|
| smallest gap ($f_v=163.086443$ Hz, $f_b=171.633530$ Hz) | $138.521060$, $192.008260$ | $\mathbf{53.487201}$ | $1$ |
| blocked-opening plate $f_b=f_H$ | $133.694241$, $187.936757$ | $54.242515$ | $\mathbf{1.014121}$ |
The smallest gap found agreed with $f_a$ within $10^{-6}$ Hz, and $f_v$ there with $f_H$ within $10^{-3}$ Hz. With plate masses of $0.1$ and $0.4$ kg and $f_v=f_H$, the gap was $75.642324$ and $37.821162$ Hz, within $10^{-6}$ Hz of each $f_a$ ($75.642324$ and $37.821162$ Hz).
Observation 3.2 the two resonances come no closer than the plate’s frequency on the air spring alone. They come closest when the plate’s frequency without air, $f_v$, is level with Helmholtz: matching the blocked-opening plate frequency $f_b$ instead gave a gap of $54.242515$ Hz, $1.014121$ times the smallest. A lighter plate raises $f_a$, and the smallest gap widened to $75.642324$ Hz at $0.1$ kg.
4. The Separator
Observation 4.1 the separator is whether the plate’s frequency is counted with the cavity’s air spring included or left out. Matching $f_v$, counted without it, to Helmholtz gave the smallest gap of $53.487201$ Hz: matching $f_b$, counted with it, gave $54.242515$ Hz. The difference of the squares of the two ways of counting, $\omega_b^2-\omega_v^2$, is $\omega_a^2$ of the plate on the air spring alone, and its $f_a$ became the smallest gap.
Observation 4.2 what is not said. Plate bending shapes, multiple vibration shapes, losses and radiation, how strongly each resonance sounds, and values for real instruments are not treated. What is shown is that in a model coupling a piston-like plate to the cavity of Paper 888, the gap between the two resonances is a hypotenuse and comes no closer than the plate’s frequency on the air spring alone.
5. Roll Call and Closing
- sum, product and gap of the two angular frequencies (Section 1): the gap is a hypotenuse.
- gap and hypotenuse within $10^{-6}$ Hz for seven plates (Computation 2.1): pushed apart.
- smallest gap $\mathbf{53.487201}$ Hz (Computation 3.1): the plate on the air spring alone.
- matching the blocked-opening plate gives $\mathbf{1.014121}$ times (Computation 3.1): not the smallest.
- $75.642324$ and $37.821162$ Hz at $0.1$ and $0.4$ kg (Computation 3.1): the smallest gap is still $f_a$.
- air spring included, or left out (Observation 4.1): the separator; the scope note of Paper 888.
Closing: The air and plate resonances of a body never come closer than the plate’s frequency on the air spring alone. Coupling a plate of area $0.04$ m² and mass $0.2$ kg to the $163.086441$ Hz cavity of Paper 888, the gap between the two resonances became the hypotenuse whose sides are the plate’s own spring frequency minus Helmholtz and the plate’s frequency on the air spring alone, $53.487201$ Hz: the smallest gap was $53.487201$ Hz, reached when the plate’s frequency without air was level with Helmholtz. Matching the blocked-opening plate frequency gave a gap of $54.242515$ Hz. The separator is whether the plate’s frequency is counted with the cavity’s air spring included or left out. Placed among the earlier papers: to the pitch of the cavity that Paper 888 counted with the plates still, the plate vibration that Paper 888 did not include was coupled. To be honest: real body plates bend and vibrate in many shapes. With one piston-like plate, how close the two resonances can come, and no further.
References
- Fletcher, N. H., & Rossing, T. D. (1998). The Physics of Musical Instruments (2nd ed.). Springer.
- Kinsler, L. E., Frey, A. R., Coppens, A. B., & Sanders, J. V. (2000). Fundamentals of Acoustics (4th ed.). Wiley.
- Yamagishi, Y. (2026). A Resonating Body’s Pitch Is Set Not by Length, as in a Pipe, but by Volume and Opening, Whatever the Shape [Paper 888]. YOI: https://www.yamagishiyuuki.com/yoi/888
Reference
Yamagishi, Y. (2026). The Air and Plate Resonances of a Body Never Come Closer Than the Plate’s Frequency on the Air Spring Alone [Paper 962]. YOI: https://www.yamagishiyuuki.com/yoi/962
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 962. This paper is a preprint and has not been peer reviewed.