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\title{WAVES IN A LAYER OF A HEAVY LIQUID ON A WEAKLY UNEVEN BOTTOM, GENERATED
BY A LOCALIZED SOURCE}
\author{S.Yu. Dobrokhotov, O.L. Tolstova}


 We construct the asymptotic solution of the evolution problem mentioned
in the title. The solution is represented in the form of "asymptotical
spectral" decomposition with respect to "3D internal" and "2D surface"
waves. This representation is then analyzed using the stationary phase
method. The "internal" and "surface" waves possess many properties of
generalized eigenfunctions and don't vanish at infinity. However the
"surface" waves decrease exponentially as the vertical variable tends to
minus infinity. The "internal" waves consist of the incident
either longitudinal or normal component and the reflected component which
include necessarily both longitudinal and normal parts. Due to the
difference of velocities, the incident normal wave can generate as
oscillating as decreasing (with respect to the vertical variable)
longitudinal waves. Rather complicated from the asymptotical point of view
is the description of the "transient layer" which is generated by the
uneven bottom. In this layer the reflected longitudinal waves are described
by both oscillating and decreasing functions. We give (assuming that the
bottom is varying slowly) the uniform asymptotics for the "internal" and
"surface" waves including the functions from the layer mentioned above.



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