ON THE LARGE-TIME ASYMPTOTICS OF GREENS-FUNCTION FOR INTERNAL GRAVITY-WAVES

We study Green's function for the internal gravity waves in a horizont ally uniform fluid. The exact representation of Green's function is fo und for the half-space z > 0 under the assumptions that the square of Brunta-Vaissala frequency N-2(z) = cont.z, and the boundary condition at z = 0 is zero. The large-time asymptotics of this function contain two terms of the form t(-1/2)A(r,z,z(0)) exp i(t omega(r,z,z(0))) This result suggests that in the general case the large-time asymptotics o f Green's function contain analogous terms. The analogs of the eikonal equation for omega and the transport equation for A are obtained and solved. This enables us to formulate an algorithm for calculating the asymptotics of Green's function in the general case. Under an addition al assumption that N-2 has a unique maximum, these asymptotics are jus tified for the case of internal waves propagating in a waveguide layer , when as \z\ --> infinity, N-2(z) --> 0. In order to prove this we em ploy the well-known representation of Green's function as a sum of nor mal modes as obtained by the method of separation of variables. The ap propriate eigenvalues and eigenfunctions are replaced by their WKB-asy mptotics, and the Poisson summation formula is used. The resulting int egrals are evaluated by applying the stationary phase method.