THE FAR-FIELD OF THE MOVING AND OSCILLATING SOURCE - THE RESONANCE CASE

We study the field W of an oscillating source moving in a 2D dispersiv e medium. The source starts moving at t = 0 with a velocity equal to t he group velocity corresponding to its oscillation frequency (calculat ed by taking into account the Doppler effect). Then the dispersive cur ve <(omega)over cap>(lambda,mu) = 0 has a self-intersection point (lam bda(0),mu(0)) and the resonance takes place. It is shown in the paper that in this case as t --> infinity the field increases as lg t, and f or large t and r, that is for large time and far away from the source the field can be represented in terms of the recently introduced speci al function, the F f-integral. This allows us to describe the large ti me far-field asymptotics of W both qualitatively and quantitatively. I n particular, there arises in the vicinity of the source the resonance zone where the field is of order unity or larger. In the directions w hich are normal to the dispersion curve at the self-intersection point the size of this zone is of order t(2/3) and far away from these dire ctions it increases more slowly, as root t. The degenerate singular po int of the dispersion curve is considered as well. There a more sharp resonance arises and the field increases as t(1/6). At the end of the paper we briefly discuss the 3D case.