Sum Rules for Passive Systems

• Sum rules for passive systems are integral constraints that connect frequency-dependent absorption to inherent physical parameters such as effective mass and static stiffness.
• They quantify trade-offs in passive acoustic designs, showing that enhancements in absorption or bandwidth necessitate compensatory reductions elsewhere through the waterbed effect.
• These principles underpin design guidelines for resonator arrays, membranes, and subwavelength absorbers by establishing rigorous performance limits.

Sum rules for passive systems express integral constraints on frequency-dependent response functions, fundamentally linking the macroscopic behavior of systems to their underlying passive, linear, and time-invariant (LTI) structure. In acoustics, such sum rules precisely quantify the trade-offs between absorption efficiency, bandwidth, and physical realizability for unidimensional waveguides with passive Lorentz-type loading, including resonator arrays, membranes, and other practical subwavelength absorbers. These relations provide fundamental limits that cannot be surpassed by any passive design, underpinning key concepts such as the waterbed effect.

1. Theoretical Framework: Herglotz Functions and Integral Identities

Consider a passive, LTI acoustic system in a 1D waveguide characterized by its complex pressure-reflection coefficient $R(\omega)$, with the normal-incidence absorption coefficient $a(\omega) = 1 - |R(\omega)|^2$ and $\theta(\omega) = \mathrm{Re}Z(\omega)$ as the real part of the normalized input impedance. The analysis constructs a Herglotz function,

$H_1(\omega) = i[1 - R(\omega)],$

analytic in $\Im \omega > 0$ and satisfying $H_1(-\omega^*) = -H_1(\omega)^*$, mapping the upper half-plane into itself.

The boundary value

$$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$

connects absorption to the system's impedance. The general Herglotz representation theorem [Bernland et al. '11] yields, for appropriate low- and high-frequency asymptotics,

$$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$

where $a_j$ and $b_j$ are coefficients of the low- and high-frequency expansions.

For $a(\omega) = 1 - |R(\omega)|^2$0, the relevant coefficients are

• $a(\omega) = 1 - |R(\omega)|^2$1, $a(\omega) = 1 - |R(\omega)|^2$2
• $a(\omega) = 1 - |R(\omega)|^2$3, $a(\omega) = 1 - |R(\omega)|^2$4

where $a(\omega) = 1 - |R(\omega)|^2$5 is the effective dynamic mass (large-$a(\omega) = 1 - |R(\omega)|^2$6 limit) and $a(\omega) = 1 - |R(\omega)|^2$7 the static stiffness (zero-frequency limit). Thus, the alternative linear sum rules for absorption are:

$a(\omega) = 1 - |R(\omega)|^2$8

$a(\omega) = 1 - |R(\omega)|^2$9

where $\theta(\omega) = \mathrm{Re}Z(\omega)$0. When $\theta(\omega) = \mathrm{Re}Z(\omega)$1 is bounded, two-sided bounds refine these:

$\theta(\omega) = \mathrm{Re}Z(\omega)$2

$\theta(\omega) = \mathrm{Re}Z(\omega)$3

These relations rigorously constrain broadband absorption, offering direct integral bounds without logarithmic or nonlinear terms, and are exact for any passive LTI loading with Lorentz-type (causal, no poles in $\theta(\omega) = \mathrm{Re}Z(\omega)$4, rational) impedance (Mo et al., 2024).

2. Application to Lorentz-Resonator and Composite Systems

For a fundamental Lorentz resonator with impedance

$\theta(\omega) = \mathrm{Re}Z(\omega)$5

characterized by mass $\theta(\omega) = \mathrm{Re}Z(\omega)$6, damping $\theta(\omega) = \mathrm{Re}Z(\omega)$7, stiffness $\theta(\omega) = \mathrm{Re}Z(\omega)$8, and cavity depth $\theta(\omega) = \mathrm{Re}Z(\omega)$9, the absorption bounds adapt to practical configurations.

Arrays of Parallel Resonators

For $H_1(\omega) = i[1 - R(\omega)],$0 parallel resonators each with area fraction $H_1(\omega) = i[1 - R(\omega)],$1 ($H_1(\omega) = i[1 - R(\omega)],$2) and depth $H_1(\omega) = i[1 - R(\omega)],$3:

• $H_1(\omega) = i[1 - R(\omega)],$4
• $H_1(\omega) = i[1 - R(\omega)],$5

Substituting these into the sum rules yields

$H_1(\omega) = i[1 - R(\omega)],$6

Cascaded Multi-Layer Systems

For $H_1(\omega) = i[1 - R(\omega)],$7 cascaded layers,

• $H_1(\omega) = i[1 - R(\omega)],$8
• $H_1(\omega) = i[1 - R(\omega)],$9

A plausible implication is that increasing the stiffness (via reduced total cavity depth) or layering more partitions tightens the sum-rule constraint, potentially degrading broadband absorption performance.

Other Architectures

• Membranes: For tension $\Im \omega > 0$0, radius $\Im \omega > 0$1, air density $\Im \omega > 0$2, areal mass $\Im \omega > 0$3:

$\Im \omega > 0$4, $\Im \omega > 0$5

• Shunted Loudspeakers: The passive shunt cannot alter $\Im \omega > 0$6 or $\Im \omega > 0$7, only damping $\Im \omega > 0$8, providing leverage only on finite-band absorption bounds.

3. Waterbed Effect and Trade-off Inequalities

The sum rules guarantee that the “area under the absorption curve” in wavelength space is strictly fixed by the system's static stiffness:

$\Im \omega > 0$9

Partitioning the absorption spectrum over $H_1(-\omega^*) = -H_1(\omega)^*$0 sub-bands, any increase in the mean absorption $H_1(-\omega^*) = -H_1(\omega)^*$1 over $H_1(-\omega^*) = -H_1(\omega)^*$2 above its statistical share enforces a compensating decrease elsewhere:

$H_1(-\omega^*) = -H_1(\omega)^*$3

This is the waterbed effect: gain in absorption at some frequencies requires automatic loss at others. Expressed for a finite frequency band $H_1(-\omega^*) = -H_1(\omega)^*$4 of width $H_1(-\omega^*) = -H_1(\omega)^*$5 and central frequency $H_1(-\omega^*) = -H_1(\omega)^*$6: $H_1(-\omega^*) = -H_1(\omega)^*$7

$H_1(-\omega^*) = -H_1(\omega)^*$8

Recasting in terms of thickness-to-wavelength ratio $H_1(-\omega^*) = -H_1(\omega)^*$9: $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$0 This enforces that, at fixed $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$1, broadening the absorption band mandates lower average absorption, and vice versa (Mo et al., 2024).

4. Practical Implications for Absorber and Isolator Design

The sum rules yield concrete guidelines for passive designs:

• Average absorption-limited by effective mass and damping: To raise average absorption, reduce $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$2 or increase $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$3.
• Thickness-bandwidth trade-off: Achieving high absorption over a broad band requires proportionally greater normalized thickness $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$4.
• Deep-subwavelength absorbers: For $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$5, broadband and high-efficiency absorption are fundamentally in conflict; the bound $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$6 is a tight predictor.
• Resonator arrays: For parallel arrays, select minimal $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$7 and maximal $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$8; layering increases $$\Im H_1(\omega + i0) = \Re[1 - R(\omega)] = \frac{1 + \theta(\omega)}{2\theta(\omega)} a(\omega)$$9, counter-productively constricting the absorption area.
• Membranes and shunts: Manipulate $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$0, $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$1, and $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$2 for membranes; recognize shunted loudspeakers cannot modify stiffness or mass bounds.

Case studies verify:

• Increasing number of resonators ($$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$3) elevates the absorption spectrum while the total area under $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$4 remains fixed.
• Increasing bandwidth $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$5 forces a reduced mean height $$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$6, demonstrating the waterbed trade-off effect (Mo et al., 2024).

5. Summary and Significance in Acoustic Limits

The energy-centric sum rules

$$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$7

and their bounded forms

$$\frac{2}{\pi} \int_0^\infty \omega^{2q} \Im H(\omega + i0)\, d\omega = a_{2q-1} - b_{2q-1}, \quad q=0,1$$8

provide sharp, physically transparent constraints for passive, LTI, one-dimensional waveguides with Lorentz-type loading. These identities supplant earlier bounds by avoiding logarithmic and frequency-weighted integrals, directly constraining absorption and transmission in terms of dynamic mass, static stiffness, and input impedance properties. The resulting waterbed effect encapsulates the universal trade-off inherent to any passive absorber design: improvements in one aspect (absorption, bandwidth, or thickness) incur mandatory performance reductions elsewhere. These principles establish a rigorous baseline for evaluating and designing optimal subwavelength absorbers, guiding future developments and indicating the inescapable limitations of passive acoustic noise control (Mo et al., 2024).