Water Wave and Sea State Models for Ship Design

Prof. F. Bonnefoy

Code = EMSHIP S3-1 (Semester S3)

Workload: lectures (20h), practical training (10h)

Number of credits: 5



The objective of this series of lectures is to introduce the students to the most important environmental source of loading structure loading: sea waves. The lectures are divided into two main parts.
First we give an overview of some of the numerous mathematical models used to represent free surface gravity waves, and the associated underlying flow. The scope is voluntarily restricted to the most useful models generally used by naval engineers and researchers. In a few cases, a deeper theoretical insight is presented in order to allow the students to understand the subtleties of water wave theory. In the second part, the use of the statistical approach is presented, both for the representation of sea states and for the ship’s response.


a) Waves modelling:

Derivation of governing non-linear equations; introduction of multiple scales method to generate particular subset of equations.


Dispersive waves:

    • Airy Potential; derivation of the solution by separation of variables. Expression of all the related physical quantities: group velocity, energy density, energy flux, limits of the linear model.
    • Higher order Stokes solutions (3rd order, 5th order). Sequential construction of the Stokes higher order solutions. Specific nonlinear features of Stokes waves.
    • Stream function model. Explanation of the method – numerical application

Shallow water (non dispersive) waves:

    • Derivation of Boussinesq equation.
    • The solitary wave as a particular solution of Boussinesq equation.
    • KdV equations: cnoidal waves.
    • Introduction to wave refraction & diffraction in coastal areas.

b) Statistical models:

    • Random sea state modelling.
    • Usual wave spectra models .
    • Wave generation.
    • Random responses of a linear system.
    • Review of the results for ship responses by a deterministic theory.
    • Motions on a real sea state.
    • Extreme responses, design factors.

Recommended reading :

“Water Wave Mechanics for Engineers & Scientists (advanced series on ocean engineering)” by R.G. Dean and R.A. Dalrymple

Form of exams:

Written exam (1h)+ report of practical training