In this work we construct and discuss special solutions of a homogeneous problem for the Laplace
equation in a domain with cone-shaped boundaries. The problem at hand is interpreted as that describing
oscillatory linear wave movement of a fluid under gravity in such a domain. These solutions
are found in terms of the Mellin transform and by means of the reduction to some new functionaldifference
equations solved in an explicit form (by quadrature). The behavior of the solutions at
large distances is studied by use of the saddle point technique. The corresponding eigenoscillations
of a fluid are then interpreted as generalized eigenfunctions of the continuous spectrum.