[Paper Review] Evanescent modes in Sonic Crystals: Complex relation dispersion and supercell approximation

Vicente Romero‐García, J. V. Sánchez-Pérez+1 more|arXiv (Cornell University)|Jan 21, 2010

Acoustic Wave Phenomena Research1 references3 citations

TL;DR

This paper extends the plane wave expansion (PWE) method to solve for complex wave vectors $k(\omega)$ in 2D sonic crystals (SCs), enabling the analysis of both propagating and evanescent modes. Using a supercell approximation and explicit matrix formulation, it experimentally confirms exponential decay of evanescent modes inside band gaps, with measured $Im(k) = -5.60 \pm 1.45\ \text{m}^{-1}$ matching analytical predictions within error bounds.

ABSTRACT

Evanescent modes in complete sonic crystals (SC) and SC with point defects are reported both theoretically and experimentally in this paper. Plane wave expansion (PWE) and, in general, $ω(k)$ methods have been used to calculate band structures showing gaps that have been interpreted as ranges of frequencies where no real $k$ exists. In this work, we extend PWE to solve the complex $k(ω)$ problem applied to SC, introducing the supercell approximation for studying one vacancy. Explicit matrix formulation of the equations is given. This $k(ω)$ method enables the calculation of complex band structures, as well as enabling an analysis of the propagating modes related with real values of the function $k(ω)$, and the evanescent modes related with imaginary values of $k(ω)$. This paper shows theoretical results and experimental evidences of the evanescent behavior of modes inside the SC band gap. Experimental data and numerical results using the finite elements method are in very good agreement with the predictions obtained using the $k(ω)$ method.

Motivation & Objective

• To analyze evanescent modes in sonic crystals (SCs) that are traditionally undetected by standard $\omega(\vec{k})$ band structure methods.
• To extend the plane wave expansion (PWE) method to solve the inverse problem $k(\omega)$, allowing computation of complex wave vectors.
• To develop and apply a supercell approximation for modeling point defects in SCs and analyzing localized evanescent modes.
• To experimentally validate the exponential decay of evanescent modes inside the band gap using a 3D robotized acoustic measurement system (3DReAMS).
• To establish a consistent framework linking analytical, numerical (FEM), and experimental results for complex wave vector behavior in SCs.

Proposed method

• Extends PWE to solve the eigenvalue problem for $k(\omega)$, enabling computation of complex wave vectors in 2D sonic crystals.
• Derives an explicit matrix formulation for the extended PWE (EPWE) to solve for $k(\omega)$, including both real and imaginary components.
• Applies the supercell approximation to model a single vacancy in a periodic SC, allowing analysis of localized modes.
• Uses finite element method (FEM) simulations to solve the scattering problem and compute pressure fields in SCs with plane wave incidence.
• Employs a 3D robotized acoustic measurement system (3DReAMS) to experimentally map pressure fields between rows of cylinders in an SC.
• Fits experimental pressure decay data to an exponential function $ae^{bx}$ to extract the imaginary part of the wave vector $Im(k)$.

Experimental results

Research questions

• RQ1 Can the extended PWE method accurately predict the complex wave vector $k(\omega)$ for evanescent modes in 2D sonic crystals?
• RQ2 To what extent does the supercell approximation correctly model localized modes in SCs with point defects?
• RQ3 Is the exponential decay of evanescent modes inside the band gap experimentally measurable and quantitatively consistent with analytical predictions?
• RQ4 Does the first harmonic of the wave vector dominate the decay behavior, or are higher harmonics significant?
• RQ5 How well do FEM simulations reproduce the experimental pressure field decay in finite SCs?

Key findings

• The extended PWE (EPWE) successfully predicts the complex band structure of 2D sonic crystals, including both propagating and evanescent modes.
• The imaginary part of the wave vector for the evanescent mode at 920 Hz is analytically predicted as $Im(k) = -5.6\ \text{m}^{-1}$.
• Experimental measurements using 3DReAMS yield $Im(k) = -5.60 \pm 1.45\ \text{m}^{-1}$, showing excellent agreement with analytical predictions.
• The pressure field decay inside the SC follows a single exponential behavior, confirming that only the first harmonic dominates the evanescent decay.
• FEM simulations reproduce the experimental pressure decay profile with high accuracy, validating the numerical model.
• For a point defect, the localized mode transitions from purely imaginary $k$ (evanescent) in a complete SC to purely real $k$ (propagating) in the defective case, consistent with literature.