Active manipulation of Helmholtz scalar fields: near-field synthesis with directional far-field control

Abstract

In this article, we propose a strategy for the active manipulation of scalar Helmholtz fields in bounded near-field regions of an active source while maintaining desired radiation patterns in prescribed far-field directions. This control problem is considered in two environments: free space and homogeneous ocean of constant depth, respectively. In both media, we proved the existence of and characterized the surface input, modeled as Neumann data (normal velocity) or Dirichlet data (surface pressure) such that the radiated field satisfies the control constraints. We also provide a numerical strategy to construct this predicted surface input by using a method of moments-approach with a Morozov discrepancy principle-based Tikhonov regularization. Several numerical simulations are presented to demonstrate the proposed scheme in scenarios relevant to practical applications.

Feasibility analysis for active near/far field acoustic pattern synthesis in free space and shallow water environments

“…Meanwhile, our scheme uses slightly larger mutually disjoint regions W 1 and W 2 such that [20], an accurate control in the sense of the L 2 -norm on oW 1 and oW 2 implies, via regularity and uniqueness results for the solution of the interior Helmholtz equation, the smooth control required in (2). As pointed out in [21,35], within the framework mentioned above, the boundary input data, either normal velocity v n or pressure p on the surface of the source can be characterized using a smooth density function w 2 L 2 ðoD 0 a Þ such that,…”

“…In this section, we present a general description of the active manipulation scheme for Helmholtz fields proposed in our previous works. The unified functional and numerical framework have already been discussed in [20,21,35]. We shall briefly recall several essential theoretical results and describe some geometric configurations of interest.…”

“…The problem geometry in these two environments are sketched in Figures 1 and 2. Although the theoretical discussion in [20,21,35] indicates that an arbitrary number of source regions, exterior control regions, and far-field directions can be considered in the active control scheme, here we shall only consider a single source D a , two control regions D 1 , D 2 , and two far-field directions x 1 , x 2 for illustrative purposes. A single source D a b R 3 is modeled as a compact region in both free space and homogeneous ocean.…”

“…In general, it is fairly difficult to find the analytical fundamental solution *Corresponding author: jchen82@central.uh.edu and thus, in this paper we follow the paradigm proposed in [22] and consider a simpler marine environment modeled as a shallow water or a homogeneous finite-depth ocean. Our sensitivity analysis builds up on the numerical framework developed in [20,21,35]. We use the associated Green's function to represent the solution of the Helmholtz equation and employ the integral equation (IE) method to formulate the forward propagator.…”

“…The rest of this paper is organized as follows. In Section 2, we formally describe the problem and provide relevant theoretical results obtained in [35]. Section 3 shows the numerical results and sensitivity analysis in free space.…”