Mechanics of Ocean Waves 4.4 Weakly Nonlinear Deep Water Wave Theories 83
Part A | 4.4
4.3.9 Shoaling and Refraction of Waves
One can generalize the derivations and findings of the
previous section to allowing the waves to propagate
along any arbitrary direction. In the case of slowly varying, small amplitude steady-state waves approaching an
ideal coast (straight and long shore line with all bottom
bathymetry contours parallel to the shore line) the direction of wave propagation is governed by Snell’s law
sin Â
C p
D constant ;
(4.24)
where  denotes the local angle of wave incidence measured from the shore normal, with  D 0 corresponding
to crests parallel to the shore and  D 90
ı to crests
normal to the shore. The change of direction of wave
propagation is called wave refraction. As one may recall
from classical physics, the path of waves as predicted
by the above Snell’s law corresponds to the shortest
time of travel between two points in a variable velocity
field. If the velocity is constant, then  will also be constant, meaning that the path of wave propagation will be
a straight line.
As the waves approach the shore with depth h ! 0,
C p !
p
gh ! 0, then  should also ! 0 for the ratio
sin Â
Cp
to remain constant; in other words, the wave rays
will be normal to the shoreline irrespective of the angle
of incidence of the wave while offshore.
Note that in the case of slowly varying steady-state
waves, the wave frequency will remain constant and the
wave number evolve satisfying the dispersion relation
gk tanh kh D constant :
Assuming no wave breaking (which would enhance
dissipation through turbulence) and with conservation
of mechanical energy, one can show that in the case of
a slowly varying, steady-state, nonbreaking wave approaching an ideal shore, the wave height will vary,
satisfying
H
p
cos Â
p
C g D constant ! H D
constant
p
C g cos Â
:
(4.25)
4.3.10 Closing Remarks to the Section
The linear (Airy) wave theory enables us to understand
the basic and unique properties of surface waves. There
are several books that deal with the linear theory of water waves, which the reader may refer to for details.
These include [4.5–9]. The linear potential flow theory
will hold good only for small amplitude waves. For finite and large amplitude waves, one has to use weakly
and fully nonlinear wave theories, as reviewed in the
following sections.
4.4 Weakly Nonlinear Deep Water Wave Theories
While linear theory has proved to be quite useful, it
is often insufficient to investigate many of the details of wave propagation. Bulk properties of waves
(some measure of wave height) are generally well predicted by linear theory in many instances. However,
the small amplitude assumption inherent in linear wave
theory is often invalidated (when waves approach their
breaking condition), so nonlinear wave theory becomes
essential.
The nonlinearity in water wave theory arises from
the treatment of the surface boundary conditions. The
kinematic and dynamic free surface boundary conditions are both nonlinear (they involve the products of
the dependent variables and Á) and apply at a surface whose position is unknown a priori, as mentioned
above. To move beyond simple linearization of the
free surface condition (which involves neglecting the
nonlinear terms and applying the boundary conditions
on the surface z D 0), successive approximations to
the nonlinear boundary conditions are required. This
requires the establishment of physical scales, which become the basis of these approximations.
We first discuss deep water nonlinear wave theory,
or Stokes wave theory (Stokes [4.10]). In deep water,
the relative depth kh (where k is the wave number and h
the water depth) is large. This can be used to adjust the
velocity potential toward its deep water asymptote
D
H
2
g
!
cosh k.z C h/
cosh kh
sin.kx !t/
gH
2!
O.1/ ;
(4.26)
which reflects the observation that the hyperbolic cosine
term approaches 1 as kh becomes large. The particle
velocities u and w similarly become
u D
@
@x
D
gkH
2!
cosh k.z C h/
cosh kh
cos.kx !t/
gka
!
O.1/ D
g
!
O.1/ ;
(4.27)
Part A | 4.4
4.3.9 Shoaling and Refraction of Waves
One can generalize the derivations and findings of the
previous section to allowing the waves to propagate
along any arbitrary direction. In the case of slowly varying, small amplitude steady-state waves approaching an
ideal coast (straight and long shore line with all bottom
bathymetry contours parallel to the shore line) the direction of wave propagation is governed by Snell’s law
sin Â
C p
D constant ;
(4.24)
where  denotes the local angle of wave incidence measured from the shore normal, with  D 0 corresponding
to crests parallel to the shore and  D 90
ı to crests
normal to the shore. The change of direction of wave
propagation is called wave refraction. As one may recall
from classical physics, the path of waves as predicted
by the above Snell’s law corresponds to the shortest
time of travel between two points in a variable velocity
field. If the velocity is constant, then  will also be constant, meaning that the path of wave propagation will be
a straight line.
As the waves approach the shore with depth h ! 0,
C p !
p
gh ! 0, then  should also ! 0 for the ratio
sin Â
Cp
to remain constant; in other words, the wave rays
will be normal to the shoreline irrespective of the angle
of incidence of the wave while offshore.
Note that in the case of slowly varying steady-state
waves, the wave frequency will remain constant and the
wave number evolve satisfying the dispersion relation
gk tanh kh D constant :
Assuming no wave breaking (which would enhance
dissipation through turbulence) and with conservation
of mechanical energy, one can show that in the case of
a slowly varying, steady-state, nonbreaking wave approaching an ideal shore, the wave height will vary,
satisfying
H
p
cos Â
p
C g D constant ! H D
constant
p
C g cos Â
:
(4.25)
4.3.10 Closing Remarks to the Section
The linear (Airy) wave theory enables us to understand
the basic and unique properties of surface waves. There
are several books that deal with the linear theory of water waves, which the reader may refer to for details.
These include [4.5–9]. The linear potential flow theory
will hold good only for small amplitude waves. For finite and large amplitude waves, one has to use weakly
and fully nonlinear wave theories, as reviewed in the
following sections.
4.4 Weakly Nonlinear Deep Water Wave Theories
While linear theory has proved to be quite useful, it
is often insufficient to investigate many of the details of wave propagation. Bulk properties of waves
(some measure of wave height) are generally well predicted by linear theory in many instances. However,
the small amplitude assumption inherent in linear wave
theory is often invalidated (when waves approach their
breaking condition), so nonlinear wave theory becomes
essential.
The nonlinearity in water wave theory arises from
the treatment of the surface boundary conditions. The
kinematic and dynamic free surface boundary conditions are both nonlinear (they involve the products of
the dependent variables and Á) and apply at a surface whose position is unknown a priori, as mentioned
above. To move beyond simple linearization of the
free surface condition (which involves neglecting the
nonlinear terms and applying the boundary conditions
on the surface z D 0), successive approximations to
the nonlinear boundary conditions are required. This
requires the establishment of physical scales, which become the basis of these approximations.
We first discuss deep water nonlinear wave theory,
or Stokes wave theory (Stokes [4.10]). In deep water,
the relative depth kh (where k is the wave number and h
the water depth) is large. This can be used to adjust the
velocity potential toward its deep water asymptote
D
H
2
g
!
cosh k.z C h/
cosh kh
sin.kx !t/
gH
2!
O.1/ ;
(4.26)
which reflects the observation that the hyperbolic cosine
term approaches 1 as kh becomes large. The particle
velocities u and w similarly become
u D
@
@x
D
gkH
2!
cosh k.z C h/
cosh kh
cos.kx !t/
gka
!
O.1/ D
g
!
O.1/ ;
(4.27)