Spectral gaps for the linear surface wave model in periodic channels
F. L. Bakharev, K. Ruotsalainen, Jari Taskinen · The Quarterly Journal of Mechanics and Applied Mathematics · 2014
We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width ɛ. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as ɛ → 0: the endpoints are determined within corrections of order ɛ 3/2 . The width of the first bands is shown to be O (ɛ). Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.