Rendiconti Lincei - Matematica e Applicazioni

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Volume 29, Issue 2, 2018, pp. 321–342
DOI: 10.4171/RLM/809

Published online: 2018-04-26

Spectral gaps and non-Bragg resonances in a water channel

Valeria Chiadò Piat[1], Sergey A. Nazarov[2] and Keijo M. Ruotsalainen[3]

(1) Politecnico di Torino, Italy
(2) St. Petersburg State University, Russia, and Institute for Problems of Mechanical Engineering RAS, St. Petersburg, Russi
(3) University of Oulu, Finland

In this paper the essential spectrum of the linear problem of water-waves on a 3d-channel with gently periodic bottom will be studied. We show that under a certain geometric condition on the bottom profile the essential spectrum has spectral gaps. In classical analysis of waveguides it is known that the Bragg resonances at the edges of the Brillouin zones create band gaps in the spectrum. Here we demonstrate that the band gaps can be opened also in the frequency range far from the Bragg resonances. The position and the length of the gaps are found out by applying an asymptotic analysis to the model problem in the periodicity cell.

Keywords: Water waves, spectral problem, asymptotic analysis, Bragg resonances

Chiadò Piat Valeria, Nazarov Sergey, Ruotsalainen Keijo: Spectral gaps and non-Bragg resonances in a water channel. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 29 (2018), 321-342. doi: 10.4171/RLM/809