Active wave control and generalized surface potentials

In active wave control, an arbitrary bounded domain with the smooth boundary is shielded from the outside field (noise) using additional sources. Unlike passive control, there is no any mechanical insulation in the system. The general solution of the problem is obtained in unsteady linear formulation. For this purpose, the theory of potentials introduced by Ryaben'kii is extended to initial–boundary value problems and the theory of distributions. Both first- and second-order spatial differentiation operators are considered. The obtained results can immediately be applied to active control problems in electromagnetics and acoustics. Two classical problems, on a bounded conductor in an electrostatic field and superconductor in a magnetostatic field, are interpreted as active control problems. The control sources for aeroacoustics are then obtained in the form of a linear combination of single- and double-layer sources. The constructed solution of the problem requires only the knowledge of the total field on the perimeter of the shielded domain.