  Exact solutions of the linear water-wave problem describing oblique waves
over a submerged horizontal cylinder of small (but otherwise fairly arbitrary)
cross-section in a two-layer fluid are constructed in the form of convergent
series in powers of the small parameter characterizing the "thinness" of the
cylinder. The terms of these series are expressed through the solution of the
exterior Neumann problem for the Laplace equation describing the flow of
unbounded fluid past the cylinder. The solutions obtained describe trapped
modes corresponding to discrete eigenvalues of the problem (lying close to the
cut-off frequency of the continuous spectrum) and resonances lying close to the
embedded cut-off. We present certain conditions for the submergence of the
cylinder in the upper layer when these resonances convert into previously
unobserved embedded trapped modes.
