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.