DML Phased Array Acoustic Demonstrator

System Architecture & Mathematical Formulation

This formulation models a 16-channel ($4 \times 4$) beamforming audio array utilizing Distributed Mode Loudspeakers (DML). Accurate acoustic steering requires solving for both the spatial interference patterns and the structural modality of the individual radiating planes.

I. Analytical Beamforming Equations

Assuming the array lies in the $xy$-plane, radiating forward along the $z$-axis. With elements at coordinates $(x_n, y_m)$, the far-field complex Array Factor (AF) at elevation angle $\theta$ (from the $z$-axis) and azimuth angle $\phi$ (in the $xy$-plane) is computed as a phasor summation:

Where $k = \frac{2\pi}{\lambda}$ is the wavenumber, $w_{nm}$ is the amplitude weight (coherence) of the specific element, and $\Delta\Phi_{nm}$ is the programmatic phase shift required to steer the main lobe to target coordinates $(\theta_0, \phi_0)$:

II. Modal Suppression via Mass Asymmetry

The transverse out-of-plane displacement $W(x,y,t)$ of a compliant DML plate is governed by the biharmonic equation:

Symmetrical plates generate degenerate eigenmodes (Chladni patterns) that severely degrade phase coherence. We suppress these by breaking geometric symmetry: applying off-center excitation and distributing localized backing weights in a Fibonacci spiral to disrupt nodal formation:

Numerical Optimization & Computational Stack

To transition from continuous analytical boundaries to a real-time visualization and design tool, the continuous surface integrals must be discretized and optimized algorithmically.

• Spatial Discretization (Three.js): The far-field pressure is evaluated over a discrete spherical grid. The acoustic intensity scalar field is mapped directly onto the vertex displacements of a high-resolution spherical manifold in real-time. Floating-point boundary clamping ensures mathematically stable coordinate transformations from Cartesian to Spherical space.
• Stochastic Annealing (Optimization): Optimizing the 50-dimensional parameter space (physical sizes, 16 transducer offsets, 16 weight counts) relies on a Monte Carlo random-walk with adaptive simulated annealing. The fitness function evaluates the ratio of coherent main-lobe energy to scattered secondary energy, driving an integrated Web Audio synthesizer to provide real-time acoustic feedback.