{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T13:14:12Z","timestamp":1763558052237,"version":"3.45.0"},"reference-count":35,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,11,18]],"date-time":"2025-11-18T00:00:00Z","timestamp":1763424000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This study examines the trapping of linear water waves by an endless structure of stationary, three-dimensional periodic obstacles within a two-layer fluid system. The setup features a lower layer of either limited or unlimited depth, overlaid by an upper layer of finite thickness bounded by a free surface, with each layer exhibiting its own constant background speed relative to the fixed reference frame. For real roots to emerge in the dispersion relation, an additional stability condition on the layer velocities is necessary. By selecting adequate choices for the background flow, a non-linear eigenvalue problem is derived from the variational formulation; its reasonable approximation yields a geometric criterion that guarantees the presence of trapped modes (subject to the aforementioned stability bounds). The selection of the eigenvalue is influenced by velocity owing to the presence of an interface and free surface. Due to inherent symmetries, the overall analysis can be confined to the positive quadrant of the velocity domain. Illustrations are provided for various obstacle setups that produce trapped modes in diverse ways.<\/jats:p>","DOI":"10.3390\/axioms14110846","type":"journal-article","created":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T12:55:31Z","timestamp":1763556931000},"page":"846","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Trapped Modes Along Periodic Structures Submerged in a Two-Layer Fluid with Free Surface and a Background Steady Flow"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0524-1239","authenticated-orcid":false,"given":"Gon\u00e7alo","family":"Dias","sequence":"first","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00e1tica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em\u00eddio Navarro 1, 1959-007 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9269-8335","authenticated-orcid":false,"given":"Bruno","family":"Pereira","sequence":"additional","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00e1tica, Instituto Superior de Engenharia de Lisboa, Rua Conselheiro Em\u00eddio Navarro 1, 1959-007 Lisboa, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,11,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Kuznetsov, N., Maz\u2019ya, V., and Vainberg, B. (2002). Linear Water Waves: A Mathematical Approach, Cambridge University Press.","DOI":"10.1017\/CBO9780511546778"},{"key":"ref_2","first-page":"1","article-title":"Report on recent researches in hydrodynamics","volume":"16","author":"Stokes","year":"1846","journal-title":"Rep. Br. Assoc. Adv. Sci."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1017\/S0305004100026700","article-title":"Trapping modes in the theory of surface waves","volume":"47","author":"Ursell","year":"1951","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_4","first-page":"79","article-title":"Edge waves on a sloping beach","volume":"214","author":"Ursell","year":"1952","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1002\/cpa.3160030106","article-title":"On the motion of floating bodies II. Simple harmonic motions","volume":"3","author":"John","year":"1950","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1017\/S0022112084002287","article-title":"Uniqueness in linearized two-dimensional water-wave problems","volume":"148","author":"Simon","year":"1984","journal-title":"J. Fluid Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0022112094000236","article-title":"Existence theorems for trapped modes","volume":"261","author":"Evans","year":"1994","journal-title":"J. Fluid Mech."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1017\/S0022112096002418","article-title":"An example of non-uniqueness in the two-dimensional linear water wave problem","volume":"315","author":"McIver","year":"1996","journal-title":"J. Fluid Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.wavemoti.2007.04.009","article-title":"Embedded trapped modes in water waves and acoustics","volume":"45","author":"Linton","year":"2007","journal-title":"Wave Motion"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1017\/S0022112093002058","article-title":"Trapped modes of internal waves in a channel spanned by a submerged cylinder","volume":"254","author":"Kuznetsov","year":"1993","journal-title":"J. Fluid Mech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1017\/S002211200300377X","article-title":"Trapped modes in a two-layer fluid","volume":"481","author":"Linton","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1017\/S0022112003005354","article-title":"Wave interaction with two-dimensional bodies floating in a two-layer fluid: Uniqueness and trapped modes","volume":"490","author":"Kuznetsov","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1007\/s00033-014-0423-8","article-title":"Linearised theory for surface and interfacial waves interacting with freely floating bodies in a two-layer fluid","volume":"66","author":"Cal","year":"2015","journal-title":"Z. Angew. Math. Phys."},{"key":"ref_14","first-page":"3799","article-title":"A sufficient condition for the existence of trapped modes for oblique waves in a two-layer fluid","volume":"465","author":"Nazarov","year":"2009","journal-title":"Proc. R. Soc. A Math. Phys. Eng. Sci."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1023\/A:1022186320373","article-title":"Exponentially decreasing solutions of the problem of diffraction by a rigid periodic boundary","volume":"73","author":"Kamotskii","year":"2003","journal-title":"Math. Notes"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"914","DOI":"10.1134\/S1064562409060325","article-title":"A simple method for finding trapped modes in problems of the linear theory of surface waves","volume":"80","author":"Nazarov","year":"2009","journal-title":"Dokl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"713","DOI":"10.1007\/s10958-010-9956-3","article-title":"Sufficient conditions for the existence of trapped modes in problems of the linear theory of surface waves","volume":"167","author":"Nazarov","year":"2010","journal-title":"J. Math. Sci."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1017\/S0022112010002429","article-title":"Existence of edge waves along three-dimensional periodic structures","volume":"659","author":"Nazarov","year":"2010","journal-title":"J. Fluid Mech."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1093\/qjmam\/hbs001","article-title":"Existence of trapped modes along periodic structures in a two-layer fluid","volume":"65","author":"Cal","year":"2012","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s10665-013-9641-x","article-title":"Edge waves propagating in a two-layer fluid along a periodic coastline","volume":"85","author":"Cal","year":"2014","journal-title":"J. Eng. Math."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1093\/qjmam\/hbaa019","article-title":"Trapped modes in a multi-layer fluid","volume":"74","author":"Cal","year":"2021","journal-title":"Q. J. Mech. Appl. Math."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Wilcox, C.H. (1984). Scattering Theory for Diffraction Gratings, Springer.","DOI":"10.1007\/978-1-4612-1130-3"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"235","DOI":"10.1016\/j.jappmathmech.2011.05.013","article-title":"Trapped surface waves in a periodic layer of a heavy liquid","volume":"75","author":"Nazarov","year":"2011","journal-title":"J. Appl. Math. Mech."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1017\/jfm.2012.364","article-title":"The surface signature of internal waves","volume":"710","author":"Craig","year":"2012","journal-title":"J. Fluid Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"18274","DOI":"10.1002\/mma.9557","article-title":"Trapped modes along periodic structures submerged in a two-layer fluid with background steady flow","volume":"46","author":"Dias","year":"2023","journal-title":"Math. Methods Appl. Sci."},{"key":"ref_26","unstructured":"Lamb, H. (1932). Hydrodynamics, Cambridge University Press. [6th ed.]."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Paterson, A.R. (1983). A First Course in Fluid Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139171717"},{"key":"ref_28","unstructured":"Kundu, P., and Cohen, I. (2010). Fluid Mechanics, Elsevier Science."},{"key":"ref_29","unstructured":"North, G.R., Pyle, J., and Zhang, F. (2015). DYNAMICAL METEOROLOGY|Kelvin\u2013Helmholtz Instability. Encyclopedia of Atmospheric Sciences, Academic Press. [2nd ed.]."},{"key":"ref_30","unstructured":"Irving, R.S. (2004). Integers, Polynomials, and Rings: A Course in Algebra, Springer."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Evans, L. (2010). Partial Differential Equations, American Mathematical Society. Graduate studies in mathematics.","DOI":"10.1090\/gsm\/019"},{"key":"ref_32","unstructured":"Evans, L.C., and Gariepy, R.F. (1992). Measure Theory and Fine Properties of Functions, CRC Press."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Birman, M.S., and Solomjak, M.Z. (1987). Spectral Theory of Self-Adjoint Operators in Hilbert Space, D. Reidel Publishing Company.","DOI":"10.1007\/978-94-009-4586-9"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Dean, R., and Dalrymple, R. (1991). Water Wave Mechanics for Engineers and Scientists, World Scientific.","DOI":"10.1142\/9789812385512"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1017\/S0022112002002227","article-title":"The existence of Rayleigh\u2013Bloch surface waves","volume":"470","author":"Linton","year":"2002","journal-title":"J. Fluid Mech."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/846\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T13:09:36Z","timestamp":1763557776000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/11\/846"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,11,18]]},"references-count":35,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2025,11]]}},"alternative-id":["axioms14110846"],"URL":"https:\/\/doi.org\/10.3390\/axioms14110846","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,11,18]]}}}