Research output: Contribution to journal › Article › peer-review
We construct asymptotic expansions as ϵ→+0 for an eigenvalue embedded into the continuous spectrum of water-wave problem in a cylindrical three-dimensional channel with a thin screen of thickness O(ϵ). The screen may be either submerged or surface-piercing, and its wetted part has a sharp edge. The channel and the screen are mirror symmetric so that imposing the Dirichlet condition in the middle plane creates an artificial positive cut-off-value Λ† of the modified spectrum. Depending on a certain integral characteristic I of the screen profiles, we find two types of asymptotics of eigenvalues, λϵ = Λ† - O(ϵ2) and λϵ = Λ† - O(ϵ4) in the cases I > 0 and I = 0, respectively. We prove that in the case I < 0 there are no embedded eigenvalues in the interval [0,Λ†], while this interval contains exactly one eigenvalue, if I ≥ 0. For the justification of these result, the main tools are a reduction to an abstract spectral equation and the use of the max-min principle.
Original language | English |
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Pages (from-to) | 187-220 |
Number of pages | 34 |
Journal | Quarterly Journal of Mechanics and Applied Mathematics |
Volume | 71 |
Issue number | 2 |
DOIs | |
State | Published - 1 May 2018 |
ID: 40973916