Water-waves modes trapped in a canal by a body with the rough surface

Details Water-waves modes trapped in a canal by a body with the rough surface Water-waves modes trapped in a canal by a body with the rough surface

2009-10-06

arxiv.org/abs/0910.1065v1

Zeithschrift fur Angewandte Mathematik und Mechanik, vol. 90 (2010), p. 983-1004

Mathematics

Spectral Theory

25 pages, 8 figures

Scientific paper

10.1002/zamm.201000042

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter $\epsilon>0$ while the distance of the body to the water surface is also of order $\epsilon$. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely, it is proved that, for any given $d>0$ and integer $N>0$, there exists $\epsilon(d,N)>0$ such that the problem has at least $N$ eigenvalues in the interval $(0,d)$ of the continuous spectrum in the case $\epsilon\in(0,\epsilon(d,N)) $. The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.

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