{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T14:55:30Z","timestamp":1763564130514},"reference-count":25,"publisher":"Oxford University Press (OUP)","issue":"1","license":[{"start":{"date-parts":[[2021,2,1]],"date-time":"2021-02-01T00:00:00Z","timestamp":1612137600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/academic.oup.com\/journals\/pages\/open_access\/funder_policies\/chorus\/standard_publication_model"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,3,2]]},"abstract":"<jats:title>Summary<\/jats:title>\n               <jats:p>In this article, we study the existence of solutions for the problem of interaction of linear water waves with an array of three-dimensional fixed structures in a density-stratified multi-layer fluid, where in each layer the density is assumed to be constant. Considering time-harmonic small-amplitude motion, we present recursive formulae for the coefficients of the eigenfunctions of the spectral problem associated with the water-wave problem in the absence of obstacles and for the corresponding dispersion relation. We derive a variational and operator formulation for the problem with obstacles and introduce a sufficient condition for the existence of propagating waves trapped in the vicinity of the array of obstacles. We present several (arrays of) structures supporting trapped waves and discuss the possibility of approximating the continuously stratified fluid by a multi-layer model.<\/jats:p>","DOI":"10.1093\/qjmam\/hbaa019","type":"journal-article","created":{"date-parts":[[2020,12,10]],"date-time":"2020-12-10T03:40:01Z","timestamp":1607571601000},"page":"34-54","source":"Crossref","is-referenced-by-count":7,"title":["Trapped modes in a multi-layer fluid"],"prefix":"10.1093","volume":"74","author":[{"given":"F S","family":"Cal","sequence":"first","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00c1tica, Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro, 1959-007 Lisboa, Portugal"}]},{"given":"G A S","family":"Dias","sequence":"additional","affiliation":[{"name":"Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"given":"B M M","family":"Pereira","sequence":"additional","affiliation":[{"name":"\u00c1rea Departamental de Matem\u00c1tica, Instituto Superior de Engenharia de Lisboa, Instituto Polit\u00e9cnico de Lisboa, Rua Conselheiro Em\u00eddio Navarro, 1959-007 Lisboa, Portugal"}]},{"given":"J H","family":"Videman","sequence":"additional","affiliation":[{"name":"Center for Mathematical Analysis, Geometry and Dynamical Systems, Department of Mathematics, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]}],"member":"286","published-online":{"date-parts":[[2021,2,2]]},"reference":[{"key":"2021030711123213500_B1","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":"Proc. Camb. Phil. 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