  We consider the eigenproblem for the Laplacian inside a three-dimensional
revolutional domain diffeomorphic to a solid torus and construct asymptotic
eigenvalues and eigenfunctions (quasimodes) of the whispering gallery-type. The
whispering gallery-type asymptotics are localized near the boundary of the
domain, and an explicit analytic representations in terms of Airy functions is
constructed for such asymptotics. There are several different scales in the
problem, which makes it possible to apply the procedure of adiabatic
approximation in the form of operator separation of variables to reduce the
initial problem to one-dimensional problems up to the small correction. We also
discuss the relation between the constructed whispering gallery-type
asymptotics and classical billiards in the corresponding domain, in
particularly, such asymptotics correspond to almost integrable billiards with
proper degeneracy. We illustrate the results in the case when a revolutional
domain is obtained by rotation of the triangle with rounded wedges.
