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

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 $ε>0$ while the distance of the body to the water surface is also of order $ε$. 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 $ε(d,N)>0$ such that the problem has at least $N$ eigenvalues in the interval $(0,d)$ of the continuous spectrum in the case $ε\in(0,ε(d,N)) $. The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.