The point spectrum of water-wave problem in intersecting canals

Trapped modes are examined on the water surface in two canals which intersect each other at the right angle and have the same symmetric cross-section. These trapped modes correspond to eigenvalues embedded into the continuous spectrum of the Steklov boundary value problem, decay exponentially at infinity, i.e., are localized near the crossing of the canals. A sufficient condition is presented for the existence of such trapped waves. The effect is discussed of the concentration of eigenvalues under a perturbation in the vicinity of the canals crossing by means of the formation of a shoal, a thin water layer. A condensed review of known results on curved, cranked and branched waveguides is given and open questions are formulated. Bibl. 24 titles.