Anisotropic Cylindrical Waves in a Square Lattice of Acoustic Waveguides

Technical Summary: Anisotropic Cylindrical Waves in a Square Lattice of Acoustic Waveguides

Problem Statement
While nonlinear wave dynamics in one-dimensional (1D) airborne acoustic metamaterials (specifically waveguides loaded with Helmholtz resonators) have been extensively studied, the exploration of such phenomena in two-dimensional (2D) settings remains limited. Previous 2D studies often utilized resonant elements to suppress the inherent anisotropy of square lattices, thereby enforcing radial symmetry. This work investigates the opposite scenario: a square network of acoustic waveguides without Helmholtz resonators. The central problem is to characterize how the intrinsic anisotropy of the square lattice geometry influences the propagation of cylindrical waves, particularly in the interplay between direction-dependent dispersion and nonlinearity.

Methodology
The study employs a multi-faceted approach combining theoretical derivation, experimental validation, and numerical simulation:

1. Theoretical Framework:

   • Dispersion Analysis: The authors derive the dispersion relation for the square lattice using the Transfer Matrix Method (TMM) under a monomodal approximation. This establishes the band structure and identifies high-symmetry directions ($\Gamma X$, $\Gamma M$, and $XM$).
   • Effective Modeling: To model the long-wavelength, weakly nonlinear dynamics, the authors develop an improved electroacoustic (EA) analogy. They introduce a refined "supercell" discretization scheme to capture anisotropic dispersion accurately. This leads to a discrete differential-difference equation (DDE) for the pressure field.
   • Continuum Limit: By applying a continuum approximation to the DDE, the authors derive an effective 2D Boussinesq equation. Crucially, this model includes fourth-order spatial derivatives to account for the square symmetry, resulting in explicit angle-dependent dispersive coefficients.
   • Reduction to cKdV: In the limit of large radii and small amplitudes, the 2D Boussinesq equation is reduced to a cylindrical Korteweg–de Vries (cKdV) equation. The dispersive coefficient in this equation is shown to be a function of the propagation angle $\theta$.

2. Experimental Validation:

   • A physical 2D square acoustic network ($15 \times 15$ unit cells) was constructed using rigid square obstacles arranged on a lattice with spacing $d = 12.5$ cm.
   • Acoustic waves were generated using a central compression chamber source.
   • Pressure fields were measured at various nodes using a portable microphone to capture temporal signals.
   • Dispersion relations were extracted via 2D Fast Fourier Transform (FFT) of the spatiotemporal data.

3. Numerical Simulation:

   • Direct numerical simulations of the full lattice dynamics were performed.
   • Additionally, the authors utilized the 2D Westervelt equation for nonlinear simulations to corroborate theoretical predictions.

Key Results

• Anisotropic Dispersion: The study confirms that the square lattice exhibits strong anisotropic dispersion. Propagation along the diagonal direction ($\theta = \pi/4$, or $\Gamma M$) is dispersionless, whereas propagation along the principal axes ($\theta = 0$, or $\Gamma X$) is highly dispersive. This behavior is attributed to in-phase multiple scattering along the diagonals due to square symmetry.
• Model Accuracy: The derived improved 2D Boussinesq equation, and subsequently the angle-dependent cKdV equation, show excellent agreement with both the TMM analytical predictions and experimental measurements across the first propagating band. The model successfully captures the transition from dispersive to dispersionless behavior as the propagation angle changes.
• Waveform Evolution:
  • Linear Regime: The network supports cylindrical waves with waveforms that vary significantly with direction. Along dispersive directions ($\theta \neq \pi/4$), the waves evolve into Airy-like packets. Along the dispersionless diagonal ($\theta = \pi/4$), the waves maintain a nearly dispersionless pulse shape.
  • Nonlinear Regime: The interplay between anisotropy and nonlinearity yields distinct propagation regimes. For $\theta \neq \pi/4$, the system supports smooth solitary waves (cylindrical solitons). However, as $\theta \to 0$ (approaching the dispersionless limit along the diagonal is not the case here; rather, the reduction to the inviscid radial Burgers equation occurs when dispersion vanishes in specific limits, though the text notes that for $\theta \to 0$ in the cKdV context, the equation reduces to support shock-like structures), the waveforms transition to sharp, shock-like fronts. Conversely, along the dispersionless diagonal, the dynamics are governed by the generalized Burgers equation, supporting cylindrical shock waves.
• Numerical Corroboration: Simulations of the full lattice and the 2D Westervelt equation confirm the coexistence of these distinct nonlinear regimes depending on the propagation direction.

Significance and Claims
The paper claims to establish a general framework for predicting direction-dependent wave phenomena in 2D acoustic metamaterials. Its primary significance lies in demonstrating that the intrinsic anisotropy of a simple square lattice, even without resonant elements, profoundly influences the evolution of radially symmetric waves. The work reveals a rich family of anisotropic cylindrical waves where characteristics such as waveform shape (from Airy-like to shock-like) and dispersion properties vary continuously with the propagation angle. By validating these findings experimentally and theoretically, the authors provide a robust model for understanding and manipulating wave propagation in structured 2D acoustic networks, extending the understanding of nonlinear wave dynamics beyond 1D settings.