On uniqueness and trapped modes in the water-wave problem for vertical barriers

Abstract

Uniqueness in the linearised water-wave problem is considered for a fluid layer of constant depth containing two, three or four vertical barriers. The barriers are parallel, of infinite length in a horizontal direction, and may be surface-piercing and/or bottom mounted and may have gaps. The case of oblique wave incidence is included in the theory. A solution for a particular geometry is unique if there are no trapped modes, that is no free oscillations of finite energy. Thus, uniqueness is established by showing that an appropriate homogeneous problem has only the trivial solution. Under the assumption that at least one barrier does not occupy the entire fluid depth, the following results have been proven: for any configuration of two barriers the homogeneous problem has only the trivial solution for any frequency within the continuous spectrum; for an arbitrary configuration of three barriers the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum; for three-barrier configurations symmetric about a vertical line, it is shown that there are no correspondingly symmetric trapped modes for any frequency within the continuous spectrum; for four-barrier configurations symmetric about a vertical line, the homogeneous problem has only the trivial solution for certain ranges of frequency within the continuous spectrum. The symmetric four-barrier problem is investigated numerically and strong evidence is presented for the existence of trapped modes in both finite and infinite depth. The trapped mode frequencies are found for particular geometries that are in agreement with the uniqueness results listed above.