Causal limits and optimal allocation for passive frequency-selective hearing protection

Geonhwi Hwang; Jaewoo Lee; Hyunjun Kim; Hyeonwoo Na

cs.SDSep 18, 2026

Causal limits and optimal allocation for passive frequency-selective hearing protection

Authors: Geonhwi Hwang, Jaewoo Lee, Hyunjun Kim, Hyeonwoo Na

Organizations: Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan · CHA University School of Medicine, Gyeonggi-do 13488, Republic of Korea · School of Computing, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea · Division of Mechanical and Space Engineering, Hokkaido University, Sapporo 060-8628, Japan

Abstract

Passive Helmholtz-resonator earplugs aim to attenuate hazardous noise while keeping speech audible; the air volume of an ear-sized shell is finite, and we treat it as the resource a design spends. For a vented bore transparent at zero frequency and loaded by reactive side branches, causality fixes a conserved integral of transmission loss. We derive it in device variables, add viscous and flanking constraints, and allocate the budget by hazard-weighted water-filling with a continuous speech-cost price. The audio-band budget B_ad = (20/ln10) pi^2 V / 2cS is set by the ratio of total cavity volume V to bore area S; occluding plugs are exempt. The f^2 price signal favours frequencies well above the source peak. Replacing strict transparency by a speech-cost price gives an exchange rate of about 1.3 dB of protection per decibel of speech cost under the present model, with the budget setting where it saturates. Across three form factors, low-frequency-selective designs stay near a decibel while hazard-matched mid/high-frequency designs reach twelve. The binding variables are bore area and speech allowance, not resonator count.

Figures & tables

	class	$V$ ( ${\mathrm{mm}}^{3}$ )	$N$	bore $\varnothing$ ( $\mathrm{mm}$ )	$S$ ( ${\mathrm{mm}}^{2}$ )	$\mathcal{B}_{\mathrm{ad}}$ ( $\mathrm{dB}$ )
F1	canal plug	150	3	2.0	3.14	0 5.97
F2	deep canal plug	400	4	2.0	3.14	15.91
F3	concha+canal mould	1200	6	2.0	3.14	47.73

Table 1: Representative device classes. No shell geometry is assumed; a class is fixed by its total cavity volume

V

, the number of cavities

N

it is divided into, and the open bore area

S

. The tabulated budget is the audio-band value

\mathcal{B}_{\mathrm{ad}}

of Eq. ( 5 ).

Figure 1: (a) The audio-band budget

\mathcal{B}_{\mathrm{ad}}

of Eq. ( 5 ) over the design plane, with the three representative classes of Table 1 marked; contours are labelled in

\mathrm{dB}

. Because

\mathcal{B}_{\mathrm{ad}}\propto S^{-1}\propto d^{-2}

, reducing the bore diameter raises the budget as effectively as enlarging the volume, at no cost in packaging space. (b) Transmission loss of four dissimilar allocations of the same

400\text{\,}{\mathrm{mm}}^{3}

. (c) Their running integrals

\int_{0}^{f}\mathrm{TL}f^{\prime-2}\mathrm{d}f^{\prime}

, all converging to

\mathcal{B}_{\mathrm{ad}}

(dotted). The

220\text{\,}\mathrm{Hz}380\text{\,}\mathrm{Hz}

array spends its budget below

100\text{\,}\mathrm{Hz}

, in an overdamped and inaudible tail.

band	F1	F2	F3
$150\text{\,}\mathrm{Hz}400\text{\,}\mathrm{Hz}$	0 1.4	0 3.8	0 11.5
$300\text{\,}\mathrm{Hz}1000\text{\,}\mathrm{Hz}$	0 2.6	0 6.8	0 20.5
$1\text{\,}\mathrm{kHz}8\text{\,}\mathrm{kHz}$	0 6.8	18.2	54.6
$2\text{\,}\mathrm{kHz}6\text{\,}\mathrm{kHz}$	17.9	47.7	143.2
$2\text{\,}\mathrm{kHz}4\text{\,}\mathrm{kHz}$	23.9	63.6	190.9
$3\text{\,}\mathrm{kHz}6\text{\,}\mathrm{kHz}$	35.8	95.5	286.4

Table 2: Flat transmission loss sustainable on the audio-band budget

\mathcal{B}_{\mathrm{ad}}

, Eq. ( 6 ). Entries in italics exceed the flanking floor of Sec. 3.3 and are not physically realisable; they indicate only that the causal budget has ceased to be the binding constraint.

$f_{0}$ (Hz)	$a_{\mathrm{req}}$ ( $\mathrm{\SIUnitSymbolMicro m}$ )	$\delta_{v}$ ( $\mathrm{\SIUnitSymbolMicro m}$ )	$a_{\mathrm{req}}/\delta_{v}$	F1 (dB)	F2 (dB)	F3 (dB)
150	22	179	0.12	0.00	0.00	0 0.01
300	45	127	0.35	0.01	0.02	0 0.10
500	75	98	0.77	0.05	0.20	0 0.79
800	123	78	1.58	0.32	0 1.28	0 4.38
1200	189	63	2.98	0 1.57	0 4.96	11.52
2000	331	49	6.74	0 6.84	14.18	23.67

Table 3: Viscous accessibility.

a_{\mathrm{req}}

is the neck radius that places the resonance of one F2 cavity (

V/N=

100\text{,}{\mathrm{mm}}^{3}

) at $f_{0}$ , computed with the flanged effective length $L_{\mathrm{eff}}=L+1.7a_{\mathrm{req}}$ for a geometric neck length $L=$2\text{\,}\mathrm{mm}

(solved self-consistently);

\delta_{v}

is the viscous boundary-layer thickness there. The last three columns give the peak transmission loss attainable from a single cavity of

V/N

in each class.

Figure 2: (a) Constraints 1–3. Solid: the causal ceiling of Eq. ( 6 ) for a one-octave band, for the three classes of Table 1 . Dashed: the viscously attainable peak transmission loss of a single F2 cavity. Shaded: the flanking floor of Sec. 3.3 . The

150\text{\,}\mathrm{Hz}400\text{\,}\mathrm{Hz}

band targeted by low-frequency-selective designs lies beneath both lower constraints. (b) Recovering the thermally stranded budget: transmission loss of the same four-resonator array with adiabatic cavities and with fibre-filled cavities of effective thermal radius

40\text{\,}\mathrm{\SIUnitSymbolMicro m}

. The strict dc-anchored budget is

\mathcal{B}_{\mathrm{dc}}=\gamma\mathcal{B}_{\mathrm{ad}}

; filling converts part of that excess from inaudible frequencies into the audio band.

	unweighted	A-weighting	damage-risk
value-density peak ( $\mathrm{kHz}$ )	2.03	2.21	2.68
$R_{\infty}$ , $[300,3400]$ $\mathrm{Hz}$ (dB)	0.71	0.51	0.33

Table 4: Robustness of the headline results to the hazard weighting

W(f)

, for the

0.3\text{\,}\mathrm{ms}

blast. The damage-risk surrogate is an outer/middle-ear pressure transfer peaking near

3.5\text{\,}\mathrm{kHz}

times the A-weighting skirt; its exact form is in the released code, so a reviewer may substitute a validated model. In every row the weighting moves the result in the same direction, so the conclusions do not depend on the choice.

Figure 3: (a) Source spectra (dotted) and hazard-weighted value densities

f^{2}P(f)W_{A}(f)

(solid), each normalised to its own maximum; triangles mark the value-density peaks. (b) Budget-optimal allocations from Eq. ( 11 ) for class F2 and the

0.3\text{\,}\mathrm{ms}

blast, at four speech allowances; the legend gives the realised A-weighted protection. The narrow spikes at the edges of the speech-importance support are the residue of the strict constraint: budget accumulates where speech importance vanishes, and buys almost nothing there.

Figure 4: Protection–speech-cost Pareto fronts under Eq. ( 11 ) with the flanking floor of Eq. ( 7 ), for (a) the

0.3\text{\,}\mathrm{ms}

blast and (b) continuous broadband noise. Stars mark the strict-transparency ceiling

R_{\infty}

of Eq. ( 9 ). Dotted lines mark the saturation set by each class’s budget. The near-common initial slope is specific to the present source and speech-cost model (see text).

class	archetype	peak $\mathrm{TL}$ (dB)	$\Delta L_{A}$ blast	$\Delta L_{A}$ cont.	$\mathcal{C}$ (dB)
F1	A low-frequency selective	0 0.1	0.11	0.09	0 0.10
F1	B speech band-pass	0 2.7	1.30	0.88	0 1.02
F1	C protection notch	14.9	2.66	3.44	0 4.78
F1	D mid+high protection	0 8.9	3.16	3.17	0 3.82
F1	E hazard-optimal	0 7.7	3.10	2.89	0 3.39
F2	A low-frequency selective	0 1.0	0.46	0.34	0 0.45

Table 5: Design atlas.

\Delta L_{A}

is the A-weighted energy reduction against the

0.3\text{\,}\mathrm{ms}

blast and against continuous noise;

\mathcal{C}

is the articulation-weighted speech cost of Eq. ( 10 ). All entries are realised finite-

Q

designs including the flanking path.

Figure 5: (a) Realised transmission loss of the five archetypes in class F2 (four resonators,

400\text{\,}{\mathrm{mm}}^{3}

), including the flanking path; the dotted line is the flanking floor. (b) Realised A-weighted protection against the

0.3\text{\,}\mathrm{ms}

blast for each archetype and class; the small figures above each bar give the articulation-weighted speech cost incurred.

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Open paper AskPDF arXiv

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Abstract

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