Part A | 4.2
78 Part A Fundamentals
interactions is essential for efficient design and performance prediction of surface ships, offshore platforms,
coastal structures, beach erosion mitigation measures,
and beach fill configurations.
This chapter of the Handbook deals with the mechanics of surface waves, in particular:
1. Kinematic properties of surface waves
2. Weakly nonlinear deep water waves
3. Transformation of waves as they approach land
from the deep ocean
4. Shallow water waves
5. Evolution of deep and shallow water waves
6. Breaking waves
7. Nearshore long wave generation, and
8. Wave forces on floating and fixed ocean structure.
Linear, weakly nonlinear, and fully nonlinear theories, and results on above topics are reviewed. The
formulation of nonlinear wave theories in deep and
shallow water are outlined, including the spectral evolution of these waves. Methods to solve wave–body
interaction problems and to determine wave forces are
discussed. The subject is presented in a manner so that
it can serve as a reference for practicing engineers and
researchers in ocean engineering. In each section of the
chapter, an overview of the mathematical theory and
formulations is given along with references for details.
4.2 Wave Theories
With exact equations governing water waves, wave
transformations and wave–body interactions being nonlinear and involving arbitrary boundaries which make
analysis difficult, numerous approximate theories have
been developed over the years. These include linear Airy wave theory, Stokes weakly nonlinear theory, Boussinesq weakly nonlinear long wave theory
and Korteweg–de Vries (KdV) theory for shallow
water waves. Recent developments include computational methods to solve the fully nonlinear wave
problem with the notable one by Longuet-Higgins and
Cokelet based on mixed Eulerian–Lagrangian formulation [4.4].
Table 4.1 Key wave and flow variables
A
wave amplitude
C g
wave group speed
C p
wave phase speed
g
acceleration of gravity
h
water depth
H
wave height
k
wave number
L
wave length
p
pressure field
s
surface tension
t
time variable
T
wave period
u
velocity field
.x; y; z/ Inertial earth-fixed coordinates against g with z D 0
on the calm surface
specific weight of water
Á
wave elevation
velocity potential
water density
!
wave radian frequency
˝
vorticity
The above theories are based on the potential flow
formulation, which assumes water to be inviscid and
flow irrotational. The wave boundary conditions are,
however, still nonlinear; they may be linearized for the
case of small amplitude waves as in Airy’s water wave
theory. Despite the idealization and assumptions involved, linear theory captures many properties of wave
phenomena and measures wave effects quite reasonably
in most cases. Of course, for practical engineering applications, knowledge of large amplitude waves, including transformation over rapidly changing bathymetry, is
essential; here linear theory has limited application and
one has to consider weakly and fully nonlinear wave
models. This chapter reviews these theories and the corresponding wave properties.
4.2.1 Potential Flow Formulation
We begin with a brief overview of the mathematical
formulation of the water wave problem. The notations
used in this chapter for key wave and flow variables are
shown in Table 4.1. Additional notations used locally in
the text are explained in the context.
A typical wave–body interaction and wave transformation problem encountered in ocean engineering and
considered in this chapter is illustrated in Fig. 4.1. The
body S B may either be stationary or moving with velocity U (translation) and ˝ (rotational), resulting in
a normal velocity V n D U O
n C ˝ .r O
n/ with r denoting the position vector from the axis of rotation.
Neglecting effect of viscosity and consequently assuming the flow to be irrotational (i. e., ! Á .r r u D
0)), one can define the flow in terms of velocity potential so that
u D r :
(4.1)
78 Part A Fundamentals
interactions is essential for efficient design and performance prediction of surface ships, offshore platforms,
coastal structures, beach erosion mitigation measures,
and beach fill configurations.
This chapter of the Handbook deals with the mechanics of surface waves, in particular:
1. Kinematic properties of surface waves
2. Weakly nonlinear deep water waves
3. Transformation of waves as they approach land
from the deep ocean
4. Shallow water waves
5. Evolution of deep and shallow water waves
6. Breaking waves
7. Nearshore long wave generation, and
8. Wave forces on floating and fixed ocean structure.
Linear, weakly nonlinear, and fully nonlinear theories, and results on above topics are reviewed. The
formulation of nonlinear wave theories in deep and
shallow water are outlined, including the spectral evolution of these waves. Methods to solve wave–body
interaction problems and to determine wave forces are
discussed. The subject is presented in a manner so that
it can serve as a reference for practicing engineers and
researchers in ocean engineering. In each section of the
chapter, an overview of the mathematical theory and
formulations is given along with references for details.
4.2 Wave Theories
With exact equations governing water waves, wave
transformations and wave–body interactions being nonlinear and involving arbitrary boundaries which make
analysis difficult, numerous approximate theories have
been developed over the years. These include linear Airy wave theory, Stokes weakly nonlinear theory, Boussinesq weakly nonlinear long wave theory
and Korteweg–de Vries (KdV) theory for shallow
water waves. Recent developments include computational methods to solve the fully nonlinear wave
problem with the notable one by Longuet-Higgins and
Cokelet based on mixed Eulerian–Lagrangian formulation [4.4].
Table 4.1 Key wave and flow variables
A
wave amplitude
C g
wave group speed
C p
wave phase speed
g
acceleration of gravity
h
water depth
H
wave height
k
wave number
L
wave length
p
pressure field
s
surface tension
t
time variable
T
wave period
u
velocity field
.x; y; z/ Inertial earth-fixed coordinates against g with z D 0
on the calm surface
specific weight of water
Á
wave elevation
velocity potential
water density
!
wave radian frequency
˝
vorticity
The above theories are based on the potential flow
formulation, which assumes water to be inviscid and
flow irrotational. The wave boundary conditions are,
however, still nonlinear; they may be linearized for the
case of small amplitude waves as in Airy’s water wave
theory. Despite the idealization and assumptions involved, linear theory captures many properties of wave
phenomena and measures wave effects quite reasonably
in most cases. Of course, for practical engineering applications, knowledge of large amplitude waves, including transformation over rapidly changing bathymetry, is
essential; here linear theory has limited application and
one has to consider weakly and fully nonlinear wave
models. This chapter reviews these theories and the corresponding wave properties.
4.2.1 Potential Flow Formulation
We begin with a brief overview of the mathematical
formulation of the water wave problem. The notations
used in this chapter for key wave and flow variables are
shown in Table 4.1. Additional notations used locally in
the text are explained in the context.
A typical wave–body interaction and wave transformation problem encountered in ocean engineering and
considered in this chapter is illustrated in Fig. 4.1. The
body S B may either be stationary or moving with velocity U (translation) and ˝ (rotational), resulting in
a normal velocity V n D U O
n C ˝ .r O
n/ with r denoting the position vector from the axis of rotation.
Neglecting effect of viscosity and consequently assuming the flow to be irrotational (i. e., ! Á .r r u D
0)), one can define the flow in terms of velocity potential so that
u D r :
(4.1)