Horizontal Sound Field Reproduction with Point Sources using Distributed Constraints

A preliminary investigation is presented of horizontal sound field control using a rectangular array of spherical wave driving sources, with focus on wide area plane wave reproduction. Comparison is made between Wave Field Synthesis and a method based on distributed pressure and velocity constraints. Advantages of the second approach are discussed, and modifications of the both methods are proposed. Introduction Horizontal sound field reproduction with point driving sources using either Wave Field Synthesis (WFS) [1, 2, 3] or High Order Ambisonics (HOA) [4, 5] suffers from reproduction errors that do not fall to zero as the number of drivers is increased. The error is unsurprising because a 3D source restricted to the plane is not a solution of the 2D wave equation, most obviously because the 2D field divergence is not zero around the 3D source. A line array with 3D drivers can be analyzed fully because of its symmetry, however it is not immediately clear what is possible with more general arrays. In an attempt to find better solutions a distributed constraint (DC) method is investigated here following on from related work using 2D and 3D drivers for 2D and 3D control. Simulations are used to compare the fields generated using WFS and DC. The test fields used are plane waves, which are harder to reproduce accurately than virtual point source fields due to the use of more drivers. The arrays used are rectangular which is a common practical configuration, particularly in concert halls. From the scattering view in the Simple Source formulation [6], driving functions are expected to be more complex for boundaries with sharper corners. Reproduction methods Distributed constraints Previously the control of 2D sound fields was considered by applying modal constraints on a grid of points in the target region [7]. This is effective because the constraints at each point can accurately describe a sound field region of any radius. In 3D with a horizontal array the contribution from each driver at a point in the array plane cannot be expressed in a 2D modal expansion. Of course it is possible to apply derivative constraints of any order, but the non-attainability of the solution implies that higher-order constraints will dominate the error and the problem will be over-constrained. For this reason only pressure and gradient constraints are given at each point in the target region. This is equivalent to constraint by the horizontal components of the first order Fourier-Bessel expansion. The set of all constraints can be written at each frequency as