{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T08:18:24Z","timestamp":1760170704317,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T00:00:00Z","timestamp":1636416000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Modelling"],"abstract":"<jats:p>Numerical models are useful tools for studying complex wave\u2013wave and wave\u2013current interactions in coastal areas. They are also very useful for assessing the potential risks of flooding, hydrodynamic actions on coastal protection structures, bathymetric changes along the coast, and scour phenomena on structures\u2019 foundations. In the coastal zone, there are shallow-water conditions where several nonlinear processes occur. These processes change the flow patterns and interact with the moving bottom. Only fully nonlinear models with the addition of dispersive terms have the potential to reproduce all phenomena with sufficient accuracy. The Boussinesq and Serre models have such characteristics. However, both standard versions of these models are weakly dispersive, being restricted to shallow-water conditions. The need to extend them to deeper waters has given rise to several works that, essentially, add more or fewer terms of dispersive origin. This approach is followed here, giving rise to a set of extended Serre equations up to kh \u2248 \u03c0. Based on the wavemaker theory, it is also shown that for kh &gt; \u03c0\/10, the input boundary condition obtained for shallow-waters within the Airy wave theory for 2D waves is not valid. A better estimate for the input wave that satisfies a desired value of kh can be obtained considering a geometrical modification of the conventional shape of the classic piston wavemaker by a limited depth \u03b8h, with \u03b8\u2264 1.0.<\/jats:p>","DOI":"10.3390\/modelling2040033","type":"journal-article","created":{"date-parts":[[2021,11,9]],"date-time":"2021-11-09T21:39:07Z","timestamp":1636493947000},"page":"626-640","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Remarks on the Boundary Conditions for a Serre-Type Model Extended to Intermediate-Waters"],"prefix":"10.3390","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5527-3116","authenticated-orcid":false,"given":"Jos\u00e9 Sim\u00e3o","family":"Antunes Do Carmo","sequence":"first","affiliation":[{"name":"Department of Civil Engineering, FCTUC, University of Coimbra, 3030-788 Coimbra, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2021,11,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1016\/0378-3839(82)90022-9","article-title":"Verification of numerical wave propagation models for simple harmonic linear water waves","volume":"6","author":"Berkhoff","year":"1982","journal-title":"Coast. Eng."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"435","DOI":"10.1017\/S0022112083002232","article-title":"A parabolic equation for the combined refraction\u2013diffraction of Stokes waves by mildly varying topography","volume":"136","author":"Kirby","year":"1983","journal-title":"J. Fluid Mech."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1016\/0378-3839(83)90017-0","article-title":"A note on the accuracy of the mild-slope equation","volume":"7","author":"Booij","year":"1983","journal-title":"Coast. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1029\/JC089iC01p00745","article-title":"A note on linear surface wave-current interaction over slowly varying topography","volume":"89","author":"Kirby","year":"1984","journal-title":"J. Geophys. Res. Space Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1061\/(ASCE)0733-950X(1988)114:4(423)","article-title":"Model for Refraction of Water Waves","volume":"114","author":"Dalrymple","year":"1988","journal-title":"J. Waterw. Port Coastal Ocean Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1016\/j.advengsoft.2009.07.007","article-title":"Wave\u2013current interactions over bottom with appreciable variations in both space and time","volume":"41","author":"Carmo","year":"2010","journal-title":"Adv. Eng. Softw."},{"key":"ref_7","first-page":"237","article-title":"Theory of unsteady water flow, with application to river floods and to propagation of tides in river channels","volume":"73","year":"1871","journal-title":"Computes Rendus. Acad. Sci."},{"key":"ref_8","unstructured":"Seabra-Santos, F.J. (1985). Contribution a L\u2019\u00e8tude des Ondes de gravit\u00e9 Bidimensionnelles en eau Peu Profonde. [Ph.D. Thesis, Universit\u00e9 Scientifique et M\u00e9dicale et Institut National Polutechnique de Grenoble]. (In French)."},{"key":"ref_9","first-page":"55","article-title":"Th\u00e9orie des ondes et des remous qui se propagent le long d\u2019un canal rectangulaire horizontal","volume":"17","author":"Boussinesq","year":"1872","journal-title":"J. Math\u00e9matiques Pures Appliqu\u00e9es"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"374","DOI":"10.1051\/lhb\/1953034","article-title":"Contribution \u00e0 l\u2019\u00e9tude des \u00e9coulements permanents et variables dans les canaux","volume":"39","author":"Serre","year":"1953","journal-title":"Houille Blanche"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1017\/S0022112076002425","article-title":"A derivation of equations for wave propagation in water of variable depth","volume":"78","author":"Green","year":"1976","journal-title":"J. Fluid Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1002\/(SICI)1097-0363(19960315)22:5<429::AID-FLD388>3.0.CO;2-8","article-title":"On breaking waves and wave-current interaction on shallow water: A 2DH finite element model","volume":"22","year":"1996","journal-title":"Int. J. Num. Meth. Fluids"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"589","DOI":"10.1016\/j.euromechflu.2011.02.005","article-title":"Recent advances in Serre\u2013Green Naghdi modelling for wave transformation, breaking and runup processes","volume":"30","author":"Bonneton","year":"2011","journal-title":"Eur. J. Mech. B Fluids"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1016\/j.jcp.2016.01.027","article-title":"A flexible genuinely nonlinear approach for nonlinear wave propagation, breaking and run-up","volume":"310","author":"Filippini","year":"2016","journal-title":"J. Comput. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1016\/S0965-9978(01)00052-7","article-title":"Sudden bed changes and wave\u2013current interactions in coastal regions","volume":"33","author":"Carmo","year":"2002","journal-title":"Adv. Eng. Softw."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1016\/j.wavemoti.2017.10.007","article-title":"Behaviour of the Serre equations in the presence of steep gradients revisited","volume":"76","author":"Pitt","year":"2018","journal-title":"Wave Motion"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1016\/j.apm.2017.03.059","article-title":"Numerical solution of the fully non-linear weakly dispersive serre equations for steep gradient flows","volume":"48","author":"Zoppou","year":"2017","journal-title":"Appl. Math. Model."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"54","DOI":"10.1016\/j.coastaleng.2012.04.004","article-title":"A new approach to handle wave breaking in fully non-linear Boussinesq models","volume":"67","author":"Tissier","year":"2012","journal-title":"Coast. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"16","DOI":"10.1016\/j.ocemod.2018.01.003","article-title":"On wave breaking for Boussinesq-type models","volume":"123","author":"Kazolea","year":"2018","journal-title":"Ocean Model."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/j.wavemoti.2018.11.008","article-title":"On the accurate simulation of nearshore and dam break problems involving dispersive breaking waves","volume":"85","author":"Carmo","year":"2019","journal-title":"Wave Motion"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"618","DOI":"10.1061\/(ASCE)0733-950X(1993)119:6(618)","article-title":"Alternative Form of Boussinesq Equations for Nearshore Wave Propagation","volume":"119","author":"Nwogu","year":"1993","journal-title":"J. Waterw. Port. Coastal. Ocean Eng."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1017\/S0022112099007247","article-title":"A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4","volume":"405","author":"Gobbi","year":"2000","journal-title":"J. Fluid Mech."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1080\/00221686.2013.814090","article-title":"Boussinesq and Serre type models with improved linear dispersion characteristics: Applications","volume":"51","author":"Carmo","year":"2013","journal-title":"J. Hydraul. Res."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/j.jcp.2014.11.016","article-title":"A new class of fully nonlinear and weakly dispersive Green\u2013Naghdi models for efficient 2D simulations","volume":"282","author":"Lannes","year":"2015","journal-title":"J. Comput. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1016\/j.cnsns.2016.10.009","article-title":"Conservative modified Serre\u2013Green\u2013Naghdi equations with improved dispersion characteristics","volume":"45","author":"Clamond","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.apm.2017.12.005","article-title":"An improved Serre model: Efficient simulation and comparative evaluation","volume":"56","author":"Ferreira","year":"2018","journal-title":"Appl. Math. Model."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"447","DOI":"10.1002\/fld.1650160602","article-title":"Surface waves propagation in shallow water: A finite element model","volume":"16","year":"1993","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_28","unstructured":"Lynett, L., and Liu, P.L.-F. (2002). Modeling Wave Generation, Evolution, and Interaction with Depth Integrated, Dispersive Wave Equations COULWAVE Code Manual, Cornell University. Cornell University Long and Intermediate Wave Modeling Package."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1016\/j.coastaleng.2009.01.001","article-title":"Hybrid finite volume\u2013finite difference scheme for 2DH improved Boussinesq equations","volume":"56","author":"Tonelli","year":"2009","journal-title":"Coast. Eng."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"105","DOI":"10.1007\/s10915-010-9395-9","article-title":"Numerical Simulation of Strongly Nonlinear and Dispersive Waves Using a Green\u2013Naghdi Model","volume":"48","author":"Chazel","year":"2011","journal-title":"J. Sci. Comput."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1002\/fld.4293","article-title":"A modified Galerkin\/finite element method for the numerical solution of the Serre-Green-Naghdi system","volume":"83","author":"Mitsotakis","year":"2017","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"ref_32","unstructured":"Seabra-Santos, F.J. (1989, January 14\u201316). As aproxima\u00e7\u00f5es de Wu e de Green & Naghdi no quadro geral da teoria da \u00e1gua pouco profunda. Proceedings of the Simp\u00f3sio Luso-Brasileiro de Hidr\u00e1ulica e Recursos H\u00eddricos (4\u00b0 SILUSBA) 209\u2013219, Lisbon, Portugal. (In Portuguese)."},{"key":"ref_33","first-page":"343","article-title":"Nonlinear and dispersive wave effects in coastal processes","volume":"16","year":"2016","journal-title":"Management"},{"key":"ref_34","unstructured":"Chaoqun, L. (2015). Modeling of wave propagation from arbitrary depths to shallow waters-A review. New Perspectives in Fluid Dynamics, InTech-Open Access Publisher."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0378-3839(92)90019-Q","article-title":"A new form of the Boussinesq equations with improved linear dispersion characteristics. Part A slowly-varying bathymetry","volume":"18","author":"Madsen","year":"1992","journal-title":"Coast. Eng."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1016\/0029-8018(96)84408-8","article-title":"A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth","volume":"23","author":"Beji","year":"1996","journal-title":"Ocean Eng."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"1296","DOI":"10.1016\/j.oceaneng.2004.12.004","article-title":"Two sets of higher-order Boussinesq-type equations for water waves","volume":"32","author":"Liu","year":"2005","journal-title":"Ocean Eng."},{"key":"ref_38","unstructured":"Dean, R.G., and Dalrymple, R.A. (1984). Water Wave Mechanics for Engineers and Scientists 1984, Prentice-Hall, Inc."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/0378-3839(93)90022-Z","article-title":"Experimental investigation of wave propagation over a bar","volume":"19","author":"Beji","year":"1993","journal-title":"Coast. Eng."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Carmo, J.S.A.D. (2016). Processos F\u00edsicos e Modelos Computacionais em Engenharia Costeira, Imprensa da Universidade de Coimbra\/Coimbra University Press.","DOI":"10.14195\/978-989-26-1153-2"}],"container-title":["Modelling"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2673-3951\/2\/4\/33\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T07:28:16Z","timestamp":1760167696000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2673-3951\/2\/4\/33"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,11,9]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2021,12]]}},"alternative-id":["modelling2040033"],"URL":"https:\/\/doi.org\/10.3390\/modelling2040033","relation":{},"ISSN":["2673-3951"],"issn-type":[{"type":"electronic","value":"2673-3951"}],"subject":[],"published":{"date-parts":[[2021,11,9]]}}}