Concentration of trapped modes in problems of the linearized theory of water waves

S. A. Nazarov

Sbornik. Mathematics, Tome 199 (2008) no. 12, pp. 1783-1807

Résumé

Problems of the linearized theory of waves on the surface of an ideal fluid filling a half-space or an infinite 3D-canyon are considered. Families of submerged or surface-piercing bodies parametrized by a characteristic linear size $h>0$ are found that have the following property: for each $d>0$ and each positive integer $N$ there exists $h(d,N)>0$ such that for $h\in(0,h(d,N)]$ the interval $[0,d]$ of the continuous spectrum of the corresponding problem contains at least $N$ eigenvalues corresponding to trapped modes, that is, to solutions of the homogeneous problem that decay exponentially at infinity and possess finite energy. Bibliography: 38 titles.