1D Electro-Acoustic Model

• One-dimensional electro-acoustic models are reduced representations that analyze the coupling of acoustic and electrical phenomena under assumptions of plane-wave propagation and linearity.
• These models employ discretization schemes like FEM, FDM, and transfer matrix methods to simulate devices ranging from ear canal acoustics to MEMS and quantum plasmas.
• By leveraging electrical circuit analogies, these models enable efficient parameter identification and inverse analysis, facilitating rapid system optimization in various application domains.

A one-dimensional electro-acoustic model provides a reduced-dimensionality framework for analyzing the coupled propagation and interaction of acoustic and electrical (or electrostatic, electromechanical) phenomena in media or devices where lateral variations are either negligible or explicitly suppressed. Such models appear in diverse contexts: (i) inverse estimation of the acoustic transfer properties of anatomical structures (e.g., the human ear canal), (ii) optoacoustic and piezoelectric transducer modeling under near-field conditions, (iii) capacitive bulk acoustic wave generation and detection in MEMS, and (iv) propagating electro-acoustic waves in degenerate quantum plasmas. Despite application-specific differences, all one-dimensional electro-acoustic models exploit the underlying equivalence between certain mechanical and electrical variables and adopt reduced partial differential equations or circuit analogs as the basis for analysis and simulation.

1. Mathematical Foundations and Assumptions

A one-dimensional electro-acoustic model typically assumes plane-wave propagation and linear material and field responses. The governing equations for acoustics are derived from the linearized equations of continuity (mass conservation) and momentum (Newton’s second law), often under the additional constraint of spatially varying cross-sectional area or layered material properties. In elasticity-based approaches, such as for the ear canal or layered transducer stacks, the basic system (e.g., for pressure $p(x,t)$ and volume velocity $U(x,t)$ in air) is: $$\frac{\partial}{\partial x}\left(A(x) \frac{\partial p}{\partial x}\right) - \frac{A(x)}{c^2} \frac{\partial^2 p}{\partial t^2} = 0,$$ where $A(x)$ is the cross-sectional area and $c$ the sound speed. Analogous forms appear for longitudinal displacement in solids with layered density $\rho(z)$ and modulus $K(z)$ or $c(z)$.

Coupled electromechanical effects are introduced through constitutive relations: in piezoelectric transducers, the electric field $E$ evolves in response to mechanical strain and stress via

$$\partial_t E + h\,\partial_z u = 0,\quad h = \frac{d}{\varepsilon^T s^D},$$

where $d$ is the piezoelectric constant, $\varepsilon^T$ the permittivity, and $s^D$ the elastic compliance under constant electric displacement.

In plasmas, one-dimensional electro-acoustic models arise through fluid equations for density, velocity, and electrostatic potential coupled via Poisson’s equation. The closure relation involves degenerate pressure, e.g., $P_e = K_e n_e^{\gamma_e}$ for electrons, with quantum degeneracy effects captured via $K_e$ and $\gamma_e$ appropriate to the relativistic regime (Mamun, 2017).

2. Discretization Strategies

Finite element (FE) and finite difference (FD) discretization methods are standard for numerical solutions in one-dimensional electro-acoustic models.

• Finite Element Method (FEM): For the ear canal, first-order “hat” functions are used as basis functions to discretize Webster’s horn equation, leading to a semi-discrete system

$M\,\ddot{\bf q} + K\,{\bf q} = {\bf f},$

where $M$ and $K$ are mass and stiffness matrices constructed from integrals involving $A(x)$, and ${\bf f}$ encodes boundary conditions and driving sources (Wulbusch et al., 2023).

• Finite Difference Method (FDM): In nearfield transducer modeling, a staggered-grid leap-frog scheme discretizes pressure and particle velocity, with inhomogeneous material properties assigned per grid cell. Boundary and interface conditions are enforced directly in the update rules (Melchert et al., 2017).
• Impedance/Transfer Matrix Methods: For layered electromechanical structures as in MEMS, 2×2 transfer matrices propagate displacement and stress through each layer, incorporating the effect of capacitive boundary conditions at gaps (Arapan et al., 2016).

3. Circuit and Electromechanical Analogies

Many one-dimensional electro-acoustic models exploit the mathematical equivalence between mechanical/acoustic linear systems and electrical circuits.

• Acoustic–Electric Ladder Network: Each finite element (or discrete segment) is represented by a series inductance $L_a = \rho \Delta x / A(x)$ (analogous to inertia) and a parallel capacitance $C_a = A(x)\Delta x / \rho c^2$ (compressibility). Resistive losses are introduced as $R_a$ (viscothermal or structural damping) (Wulbusch et al., 2023).
• Electrostatic Coupling in MEMS: Capacitive transduction is encoded as boundary stresses that are functionals of applied voltage and local displacements. The total two-port input–output behavior under small-signal excitation is then naturally summarized by an admittance (Y-) matrix, which includes both mechanical resonance effects and electrostatic couplings (Arapan et al., 2016).
• Piezoelectric Coupling: In piezoelectric polymers (e.g., PVDF), the time evolution of the internal electric field is driven by the local mechanical strain rate, and the measured voltage across the layer is directly proportional to spatially averaged stress (Melchert et al., 2017).

4. Boundary Conditions and Parameter Identification

Boundary conditions are central to model physical fidelity and parameter recovery in inverse problems.

• Acoustics with Unknown Termination: At the source (e.g., canal entrance), a prescribed volume velocity acts as a Neumann condition; at the distal boundary (eardrum), a lumped impedance $Z_d$ is imposed. The input impedance $Z_{\rm in}(\omega)$ is a function of both the geometrical and terminations parameters (Wulbusch et al., 2023).
• Electromechanical Interfaces: Electrostatic gaps (RF-MEMS) and piezoelectric patches (transducers) are modeled as voltage- or current-driven boundary conditions, incorporating both static bias and small-signal AC excitation. Boundary stresses due to the voltage fields are linearized about the bias operating point (Arapan et al., 2016).

Inverse identification schemes rely on minimizing a cost function between measured and modeled input impedance or output signal, parameterizing unknowns such as the area function $A(x)$ (often as a Fourier series) or impedance characteristics (e.g., via resonator models). Optimization proceeds via algorithms such as bounded Nelder–Mead with multiple starts, enforcing physical constraints ($A(x)>0$) and bounds (Wulbusch et al., 2023).

5. Model Validation, Limitations, and Application Regimes

Validation against experiment or higher-dimensional simulations is essential. One-dimensional electro-acoustic models yield quantitatively accurate predictions in systems where the assumptions of plane parallel layering, negligible lateral variation, and linearity hold.

• Ear Canal Acoustics: The method recovers area and impedance with errors below $\sim$1 dB (to 7 kHz) and below 5 dB (to 10 kHz) for diverse geometries (Wulbusch et al., 2023).
• Piezoelectric Transducer Signals: Simulated and measured transients for PVDF transducers match in arrival times and relative pulse amplitudes, within the experimental instrumental bandwidth (approximated as a first-order low-pass filter) (Melchert et al., 2017).
• MEMS Resonator Admittance: The calculated frequency response (resonances, bandwidth) for a layered Si resonator with air-gap coupling reproduces the expected modal structure, with glass layers providing negligible contribution to thickness-extensional modes (Arapan et al., 2016).

Model limitations follow from (i) the prescription of 1D propagation (no diffraction, only suited for acoustical near-field or plane-layered systems), (ii) the assumption of linearity, excluding nonlinear and loss mechanisms beyond those explicitly modeled, and (iii) neglect of mode conversion or lateral inhomogeneity except via effective parameters.

6. Physical Contexts and Notable Variants

Acoustic Propagation in Anatomical Structures

Webster’s horn-based models serve in non-invasive recovery of transfer impedance for anatomical cavities (notably the ear canal), where input impedance is measured at the canal entrance and the product of interest is the pressure transfer to a distal membrane (eardrum). The area function’s high-resolution recovery enables subject-specific modeling vital to hearing research and device design (Wulbusch et al., 2023).

electromechanical Transducer Modeling

Under nearfield and layered conditions, the one-dimensional assumption yields tractable, accurate models for optoacoustic generation, piezoelectric sensing (e.g., in PVDF), and capacitive MEMS structures. Precise propagation of mechanical pulses, reflection at impedance discontinuities, and piezoelectric/electrostatic signal generation are captured, enabling both forward and inverse analysis (Melchert et al., 2017, Arapan et al., 2016).

Quantum Plasma Electro-Acoustic Waves

In degenerate quantum plasma systems, 1D electro-acoustic models provide analytic and computational access to new dispersive waves (degenerate pressure-driven electro-acoustic [DPDEA] waves), where the wave inertia is provided by heavy species (positive particles), and the restoring force arises from electron degeneracy pressure, with detailed dispersion relations derived for different quantum regimes (Mamun, 2017).

7. Significance and Scope of One-Dimensional Electro-Acoustic Models

One-dimensional electro-acoustic models constitute a versatile, computationally efficient, and physically interpretable class of reduced-order descriptions for layered and quasi-one-dimensional systems, whether biological (ear canals), piezoelectric (PVDF transducers), MEMS (capacitive Si resonators), or quantum (degenerate plasmas). By unifying mechanical and electrical phenomena within a common formalism, these models facilitate rapid prediction, parameter identification, and physical insight into complex coupling phenomena relevant in both technological and fundamental research domains (Wulbusch et al., 2023, Mamun, 2017, Melchert et al., 2017, Arapan et al., 2016). Limitations—principally, the neglect of multidimensional wave effects—must be weighed against computational tractability and accuracy in practically relevant parameter regimes.