{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:32:55Z","timestamp":1760059975763,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T00:00:00Z","timestamp":1753142400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>In this study, we study the trapping of linear water waves by infinite arrays of three-dimensional fixed periodic structures in a three-layer fluid. Each layer has an independent uniform velocity field with respect to the fixed ground in addition to the internal modes along the interfaces between layers. Dynamical stability between velocity shear and gravitational pull constrains the layer velocities to a neighbourhood of the diagonal U1=U2=U3 in velocity space. A non-linear spectral problem results from the variational formulation. This problem can be linearized, resulting in a geometric condition (from energy minimization) that ensures the existence of trapped modes within the limits set by stability. These modes are solutions living the discrete spectrum that do not radiate energy to infinity. Symmetries reduce the global problem to solutions in the first octant of the three-dimensional velocity space. Examples are shown of configurations of obstacles which satisfy the stability and geometric conditions, depending on the values of the layer velocities. The robustness of the result of the vertical column from previous studies is confirmed in the new configurations. This allows for comparison principles (Cavalieri\u2019s principle, etc.) to be used in determining whether trapped modes are generated.<\/jats:p>","DOI":"10.3390\/computation13080176","type":"journal-article","created":{"date-parts":[[2025,7,22]],"date-time":"2025-07-22T15:00:42Z","timestamp":1753196442000},"page":"176","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 Three-Layer Fluid with a Background Steady Flow"],"prefix":"10.3390","volume":"13","author":[{"given":"Gon\u00e7alo A. S.","family":"Dias","sequence":"first","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00e1tica, Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro 1, 1959-007 Lisbon, Portugal"}]},{"given":"Bruno M. M.","family":"Pereira","sequence":"additional","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00e1tica, Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro 1, 1959-007 Lisbon, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1017\/S0022112006009803","article-title":"Trapped modes in the water-wave problem for a freely floating structure","volume":"558","author":"McIver","year":"2006","journal-title":"J. Fluid Mech."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s10665-006-9103-9","article-title":"Motion trapping structures in the three-dimensional water-wave problem","volume":"58","author":"McIver","year":"2007","journal-title":"J. Eng. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"283","DOI":"10.1017\/S0022112003004397","article-title":"Trapping structures in the three-dimensional water-wave problem","volume":"484","author":"McIver","year":"2003","journal-title":"J. Fluid Mech."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1017\/S0022112010001503","article-title":"Passive trapped modes in the water-wave problem for a floating structure","volume":"657","author":"Fitzgerald","year":"2010","journal-title":"J. Fluid Mech."},{"key":"ref_5","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_6","unstructured":"Stokes, G.G. (1846). Report on Recent Researches in Hydrodynamics, British Association for the Advancement of Science. Report of the British Association for the Advancement of Science."},{"key":"ref_7","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_8","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_9","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_10","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_11","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_12","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_13","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_14","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_15","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_16","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_17","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_18","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_19","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_20","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_21","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_22","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_23","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_24","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_25","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_26","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_27","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_28","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_29","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_30","doi-asserted-by":"crossref","unstructured":"Vallis, G.K. (2017). Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation, Cambridge University Press. [2nd ed.].","DOI":"10.1017\/9781107588417"},{"key":"ref_31","unstructured":"Lamb, H. (1932). Hydrodynamics, Cambridge University Press. [6th ed.]."},{"key":"ref_32","unstructured":"Kundu, P., and Cohen, I. (2010). Fluid Mechanics, Elsevier Science."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Paterson, A.R. (1983). A First Course in Fluid Dynamics, Cambridge University Press.","DOI":"10.1017\/CBO9781139171717"},{"key":"ref_34","unstructured":"Reed, M., and Simon, B. (1978). IV: Analysis of Operators, Elsevier Science. Methods of Modern Mathematical Physics."},{"key":"ref_35","unstructured":"Drazin, P.G., and Reid, W.H. (2004). Hydrodynamic Stability, Cambridge University Press. [2nd ed.]. Cambridge Mathematical Library."},{"key":"ref_36","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_37","unstructured":"Irving, R.S. (2004). Integers, Polynomials, and Rings: A Course in Algebra, Springer."},{"key":"ref_38","first-page":"309","article-title":"Die Greensche Funktion der Schwingungsgleichung","volume":"21","author":"Sommerfeld","year":"1912","journal-title":"Jahresber. Dtsch. Math.-Ver."},{"key":"ref_39","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_40","unstructured":"Evans, L.C., and Gariepy, R.F. (1992). Measure Theory and Fine Properties of Functions, CRC Press."},{"key":"ref_41","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_42","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_43","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":["Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/8\/176\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:14:10Z","timestamp":1760033650000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-3197\/13\/8\/176"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,22]]},"references-count":43,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["computation13080176"],"URL":"https:\/\/doi.org\/10.3390\/computation13080176","relation":{},"ISSN":["2079-3197"],"issn-type":[{"type":"electronic","value":"2079-3197"}],"subject":[],"published":{"date-parts":[[2025,7,22]]}}}