Analytical solutions for oscillations in a harbor with a hyperbolic-cosine squared bottom Gang Wang a,b,n , Jinhai Zheng a,b,n , Qiuhua Liang b,c , Yanna Zheng d a State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China b College of Harbor Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China c School of Civil Engineering & Geosciences, Newcastle University, Newcastle upon Tyne NE1 7RU, UK d School of Ocean and Civil Engineering, Dalian Ocean University, Dalian 116034, China article info Article history: Received 12 November 2013 Accepted 15 March 2014 Available online 3 April 2014 Keywords: Harbor resonance Wave oscillation Seiche Analytical solutions Shallow water equations abstract Based on the linear shallow water approximation, longitudinal oscillations in a rectangular harbor with a hyperbolic-cosine squared bottom induced by incident perpendicular waves are analytically investi- gated, which could be described by combining the associated Legendre functions of the first and second kinds. The effects of topographic parameters on the resonant spectrum and response are examined in detail. When the width of the harbor is of the same order magnitude as wavelengths, transverse oscillations may exist due to the wave refraction. Analytic solutions for transverse oscillations within a harbor of hyperbolic-cosine squared bottom are derived. These oscillations are typically standing edge waves. The transverse eigenfrequency is found to be related not only to the width of the harbor, but also to the varying water depth parameters. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction Better understanding of wave trapping and oscillations in bays and harbors is of importance to many practical applications. These oscillations may be induced by tsunamis, infragravity waves, atmo- spheric fluctuations and variable currents traveling into a semi- enclosed domain, e.g. bays and harbors (Bellotti et al., 2012; De Jong and Battjes, 2004; Fabrikant, 1995; Okihiro and Guza, 1996). The oscillations may cause unacceptable vessel movements, affect normal operation of docks, and generate excessive mooring forces that may break the mooring lines. Rabinovich (2010) reviewed recent advances in understanding and modeling of seiches and harbor oscillations. In order to reduce the disturbance to normal harbor operation and minimize the possible damages, a further research effort is necessary to enhance our current knowledge for this type of wave amplifica- tion and its excitation mechanisms and thus improve predictive capability. Numerical modeling has provided an effective way to repro- duce such a phenomenon and identify the eigenvalues of oscilla- tions, through simulating waves propagating from offshore and subsequently being amplified inside a bay/harbor. Researchers have developed a variety of numerical models based on the mild-slope equations to predict wave oscillations in harbors induced by different offshore wave conditions (Bellotti, 2007; Maa et al., 2011; Panchang et al., 2000). These linear models are useful for predicting short wave disturbances in harbors and identifying harbor resonance periods and long wave amplifica- tion factors. Furthermore, this type of models is generally compu- tationally efficient and thus suitable for applications to large computational domains; this facilitates explicit account of possible effects of ambient wave transformation outside the harbor in a larger region. However, these models cannot reliably predict long- period oscillations induced by incident short waves or higher harmonics generated from nonlinear interactions. Boussinesq-type models provide a more reliable tool for simulating such nonlinear hydrodynamic problems including the nonlinear generation of long waves by groups of short waves propagating from deep to shallow water, diffraction of both short- and long-period waves into a harbor, and resonant amplification of long waves inside a harbor. They represent a group of models based on extended and higher-order Boussinesq equations that are solved using the finite difference method on structured Cartesian (Shi et al., 2012; Zou and Fang, 2008) or curvilinear meshes (Shi et al., 2001) or the finite element method on triangular or quadrilateral grids (Losada et al., 2008; Walkley and Berzins, 2002). Most of these numerical models are capable of capturing the major characteristics of the resonant response from a given geometry and hence are useful for designing plan shape of harbors. However, designing a new geometry generally involves a large number of variables, including the size of the basin and the width of the entrance, among others. Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/oceaneng Ocean Engineering http://dx.doi.org/10.1016/j.oceaneng.2014.03.027 0029-8018/& 2014 Elsevier Ltd. All rights reserved. n Corresponding authors at: State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, Hohai University, Nanjing 210098, China. Tel.: þ86 83787706. E-mail addresses: gangwang@hhu.edu.cn, wanggang1015@mail.dlut.edu.cn (G. Wang), jhzheng@hhu.edu.cn (J. Zheng). Ocean Engineering 83 (2014) 16–23