Proceedings of the Thirty-second (2022) International Ocean and Polar Engineering Conference
Shanghai, China, June 5–10, 2022
Copyright © 2022 by the International Society of Offshore and Polar Engineers (ISOPE)
ISBN 978-1-880653-81-4; ISSN 1098-6189

www.isope.org

Boussinesq-type equations for flexural-gravity waves
in shallow water
Yingfen Xiong, Liangsheng Zhu, Shanran Tang
School of Civil Engineering & Transportation, South China University of Technology,
Guangzhou, Guangdong, China

depth, 𝜖, and the ratio of water depth to wavelength, 𝜇, are introducedin the derivation. They represent the nonlinear and dispersion
characteristics of shallow water waves respectively. These twoparameters are assumed to be small in the classical Boussinesq
equations, such that the equations are applicable to weakly nonlinear
long waves. McCowan (1987) found that the ratio of water-depth towavelength must less than 1/5 to ensure an accurate calculation.
Madsen et al. (1991) introduced a third-order term obtained from longwave equations to the momentum equation, whose coefficient
determines the frequency dispersion. These equations are only
applicable to constant water depth by the constraint of the long waveequations. Madsen and Sørensen (1992) extended their equations to themild-slope bottoms, and improved the accuracy of shoaling effectproperty up to 𝑂(𝜇4 ). Nwogu (1993) used a velocity variable at anarbitrary depth for deriving a new set of Boussinesq equations. Its
momentum equation also includes a third-order term, and thedispersion depends on this term similar to Madsen’s study. The
dispersion of Nwogu’s model is improved significantly, while itremains weakly nonlinear. Based on Nwogu’s work, Schäffer andMadsen (1995), Chondros and Memos (2014) and Choi et al. (2015)
continued to improve the nonlinearity of the classical Boussinesqequations.

ABSTRACT
Many previous studies on hydroelasticity of floating plates assume
uniform stiffness and deep waters. In this paper, we study a floating
elastic plate with varying stiffness in shallow water over varying
bathymetry. Boussinesq-type equations for flexural-gravity waves inshallow water are derived up to second-order accuracy. Using themethod of separation of variables, the analytical solution and thedispersion relationship to the linear, first-order governing equation offlexural-gravity waves are obtained for constant water depth anduniform plate stiffness with periodic conditions. The results implies
that wave celerity decreases with decreasing flexural stiffness or waterdepth.

KEY WORDS: Boussinesq-type equation; Flexural-gravity wave;
Shallow water wave; Dispersion relationship.

INTRODUCTION
Sea-ice and very large floating structure (VLFS) have been research
hotspots in hydroelasticity for decades. An elastic floating plate is often
employed as the model for simulating the interaction between thefloating structure and water waves. Many floating structures locate incoastal regions, and their hydroelastic waves are affected by water
depth remarkably. In this paper, a new mathematical model for
flexural-gravity waves in shallow water is derived with the use of theexpansion method that has been applied in the derivation of theclassical Boussinesq equation.

Different from the classical Boussinesq equations that are based on
Euler equations and the continuity equation, researches of flexuralgravity waves in floating plates are mainly under the framework ofpotential flow theory. Elastic beam and plate models are often
employed to study the interaction between floating plates and waterwaves. Both sea-ice and VLFS can be modeled as a floating elastic
plate, while early studies of hydroelasticity mainly are focused on thesea-ice. Wadhams (1973, 1986) studied the propagation properties offlexural-gravity waves in sea-ice. Due to insufficient computing power,
they only got an approximate numerical solution. It was not until Foxand Squire (1994) applied the eigenfunction expansion method to study
the scattering of waves on a semi-infinite sea-ice that the challenge innumerical computation was overcame. Compared with other solutions,the eigenfunction expansion method is the most direct and effective
method to solve this kind of problem. The eigenfunctions in Fox andSquire (1994) are not orthogonal, which is improved by other scholars(Barrett and Squire, 1996; Sahoo et al., 2001; Teng et al., 2001). The
Wiener–Hopf method (Evans and Davies, 1968), the residue calculus
approach (Linton and Chung, 2003) and the integro-differential
equation (Hermans, 2000) are alternative methods that can be used forAmong the existing wave mathematical models, Boussinesq equations
is the most suitable for simulating the motion of long waves in coastalwater. Boussinesq (1872) obtained a set of one-dimensional nonlinear
equations for wave motion under the assumption that the horizontal
velocity is constant and the vertical velocity varies linearly with water
depth. The appearance of these equations promoted the development ofBoussinesq-type equations. In shallow water, the depth of water has
significant effects on the propagation of flexural-gravity waves. Basedon the perturbation expansion of Euler equations and the continuity
equation, Peregrine (1967) derived the classical Boussinesq equations
for describing the wave motion in water of varying depth. The classicalBoussinesq equations enable a complicated three-dimensional problem
to be reduced to a set of two-dimensional equations that does notinclude the vertical velocity. The ratio of wave amplitude to water

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