RESUMEM

We present a mathematical framework for the active control of time-harmonic acoustic disturbances. Unlike many existing methodologies, our approach provides for the exact volumetric cancellation of unwanted noise in a given predetermined region of space while leaving unaltered those components of the total acoustic field that are deemed friendly. Our key finding is that for eliminating the unwanted component of the acoustic field in a given area, one needs to know relatively little; in particular, neither the locations nor structure nor strength of the exterior noise sources need to be known. Likewise, there is no need to know the volumetric properties of the supporting medium across which the acoustic signals propagate, except, perhaps, in the narrow area of space near the boundary (perimeter) of the domain to be shielded. The controls are built based solely on the measurements performed on the perimeter of the region to be shielded; moreover, the controls themselves (i.e., additional sources) are also concentrated only near this perimeter. Perhaps as important, the measured quantities can refer to the total acoustic field rather than only to its unwanted component, and the methodology can automatically distinguish between the two.

In the paper, we construct a general solution to the aforementioned noise control problem. The apparatus used for deriving the general solution is closely connected to the concepts of generalized potentials and boundary projections of Calderon's type. For a given total wave field, the application of Calderon's projections allows us to definitively split its incoming and outgoing components with respect to a particular domain of interest, which may have arbitrary shape. Then the controls are designed so that they suppress the incoming component for the domain to be shielded or alternatively, the outgoing component for the domain, which is complementary to the one to be shielded. To demonstrate that the new noise control technique is appropriate, we thoroughly work out a two-dimensional model example that allows full analytical consideration.

To conclude, we very briefly discuss the numerical (finite-difference) framework for active noise control that has, in fact, already been worked out, as well as some forthcoming extensions of the current work: optimization of the solution according to different criteria that would fit different practical requirements, applicability of the new technique to quasi-stationary problems, and active shielding in the case of the broad-band spectra of disturbances. In the future, the aforementioned finite-difference framework for active noise control is going to be used for analyzing complex configurations that originate from practical designs.