Mechanics of
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Part A | 4.1
4. Mechanics of Ocean Waves
James M. Kaihatu, Palaniswamy Ananthakrishnan
This chapter reviews mechanics of water waves and
wave–body interactions pertaining to ocean and
coastal engineering based on linear and weakly
nonlinear wave theories. Numerical methods
based on Green’s theorem and mixed Eulerian–
Lagrangian formulation for fully nonlinear wave
and wave–body interaction problems are also
discussed. The discussion also covers methods to
determine the wave forces on fixed and floating
structures, including the viscous drag force.
4.1 Ocean Surface Waves ............................... 77
4.2 Wave Theories ........................................ 78
4.2.1 Potential Flow Formulation ........... 78
4.3 Properties of Small Amplitude
Gravity Waves ......................................... 80
4.3.1 Linear Dispersion Relation ............. 80
4.3.2 Phase Speed ................................ 80
4.3.3 Group Speed ................................ 80
4.3.4 Amplitude Modulation
of Water Waves............................. 81
4.3.5 Average Wave Energy Density ......... 81
4.3.6 Propagation of Wave Energy .......... 81
4.3.7 Water Particle Trajectory ................ 82
4.3.8 Spatio-Temporal Evolution
of Waves ...................................... 82
4.3.9 Shoaling and Refraction of Waves .. 83
4.3.10 Closing Remarks to the Section ...... 83
4.4 Weakly Nonlinear Deep Water
Wave Theories ........................................ 83
4.4.1 Properties of Weakly Nonlinear
Deep Water Waves ........................ 84
4.4.2 Evolution of Weakly Nonlinear
Deep Water Waves ........................ 85
4.5 Shallow Water Wave Theories .................. 87
4.5.1 Properties of Weakly Nonlinear
Shallow Water Waves .................... 88
4.5.2 Evolution of Weakly Nonlinear
Shallow Water Waves .................... 89
4.6 Transformation of Waves
Approaching Land................................... 90
4.7 Computational Method
for Fully Nonlinear Waves........................ 93
4.8 Wave Forces
on Fixed and Floating Structures ............. 94
4.8.1 Incident Wave Force:
Froude–Krylov Force ..................... 94
4.8.2 Morison Force
on a Stationary Body .................... 95
4.8.3 Wave Diffraction over a Body ......... 96
4.8.4 Wave Radiation Force
on an Oscillating Body .................. 96
4.9 Concluding Remarks................................ 97
References..................................................... 98
4.1 Ocean Surface Waves
The interface between the atmosphere and water, when
disturbed, results in the generation of surface waves.
In an open domain, the disturbance has to continue
for the waves to persist as the propagating waves radiate energy. There is a limit to the energy contained
by waves; once exceeded, wave breaking occurs, at
which point energy is dissipated by turbulence during
the breaking process. The limit depends on parameters
such as wave height to wave length (or steepness) and
wave height to water depth ratios. Wave instabilities or
damping are also governed by air flow and separation,
surface tension and surfactant effects, and instability of
the free surface boundary layers. The physics of water waves is unique in many respects compared to other
wave motions in fluids because of the dispersive nature
of the water waves. Water wave problems are among
the earliest topics attempted in applied mathematics,
as illustrated, for example, by the classical Cauchy–
Poisson problem on transient waves, tackled as early as
in 1815 [4.1] and Michell’s theory of ship wave resistance developed in 1898 [4.2, 3]. A historical account
of the development of classical water wave theory is
given in [4.1]. Today, from an engineering viewpoint,
understanding the properties of waves and wave–body