The far field of the moving and oscillating source: The resonance case

Abstract

We study the field W of an oscillating source moving in a 2D dispersive medium. The source starts moving at t = 0 with a velocity equal to the group velocity corresponding to its oscillation frequency (calculated by taking into account the Doppler effect). Then the dispersive curve \\ ̂ gw(λ, μ) = 0 has a self-intersection point ( λ 0, μ 0) and the resonance takes place. It is shown in the paper that in this case as t → ∞ the field increases as lgt, and for 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 special function, the Ff -integral. This allows us to describe the large-time far-field asymptotics of W both qualitatively and quantitatively. In particular, there arises in the vicinity of the source the resonance zone where the field is of order unity or larger. In the directions which 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 directions it increases more slowly, as √ t. The degenerate singular point 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.