498
Appendix C Useful Theoretical Approaches and Formulas
C.5.2 Boundary Conditions
We start with the condition at the sea bottom (plane z = -h in Fig. 4.2).
If the sea bottom is not permeable, there is no motion through the boundary
(z = - h), i. e. velocity, normal to the boundary, has to be equal to zero. Hence,
we have:
w = o¢ = 0 at
oz
z = -h.
(C.91)
This boundary condition is consistent with the particle motion in waves, explained in Sect. 3.4.2 (see also Fig. 3.4). At the sea bottom, the horizontal
velocity component, u, is not necessarily zero while the vertical component vanishes completely as the path of the particle motion deteriorates into horizontal
motion only.
At the sea surface, the boundary conditions are more complicated. First of
all, let us consider a water particle Pl(X, z, t) located at the sea surface (see
Fig. 4.2). Existence of the sea surface means that there is no transfer of matter
across the surface. As this surface moves, the velocity of a point P1(x, z, t)
on the surface, in the direction normal to the surface, is equal to the normal
component of velocity of the fluid particle at the same point on the surface.
This condition, which is known as the kinematic boundary condition, can
be expressed in terms of the velocity potential, ¢(x, z, t), as follows (Massel,
1989) :
o( o¢ o(
ot ox ox
o¢
oz
at z = (,
(C.92)
in which ((x, t) is the surface displacement from the still water level (see
Fig. 4.2). Physically, the term o(/ot denotes the velocity of vertical movement of the sea surface. The second term on the left-hand side of Eq. (C.92)
expresses the fact that water particles subjected to wave motion move along
the surface. Therefore, a particular point P(x, z, t) not only moves vertically
but also possesses a small back- and forward motion in the horizontal direction.
The summation of the two terms on the left-hand side of Eq. (C.92) provides
the total vertical velocity o¢ / 0 z of the water particle. A detailed derivation of
Eq. (C.92) will not be given here, as it can be found in many papers and books
(see, for example, Massel, 1989; Dean and Dalrymple, 1992).
As we assumed that wave steepness was very small, it follows that 0(/ ox
term is very small and can be neglected. Therefore, Eq. (C.92) simplifies to:
o(
ot
o¢
oz
at z = O.
(C.93)
Note that the kinematic condition is now applied on the plane z = 0, instead
of the ~urface z = (, due to the assumption of low wave steepness (see also
Fig. 4.1).
Appendix C Useful Theoretical Approaches and Formulas
C.5.2 Boundary Conditions
We start with the condition at the sea bottom (plane z = -h in Fig. 4.2).
If the sea bottom is not permeable, there is no motion through the boundary
(z = - h), i. e. velocity, normal to the boundary, has to be equal to zero. Hence,
we have:
w = o¢ = 0 at
oz
z = -h.
(C.91)
This boundary condition is consistent with the particle motion in waves, explained in Sect. 3.4.2 (see also Fig. 3.4). At the sea bottom, the horizontal
velocity component, u, is not necessarily zero while the vertical component vanishes completely as the path of the particle motion deteriorates into horizontal
motion only.
At the sea surface, the boundary conditions are more complicated. First of
all, let us consider a water particle Pl(X, z, t) located at the sea surface (see
Fig. 4.2). Existence of the sea surface means that there is no transfer of matter
across the surface. As this surface moves, the velocity of a point P1(x, z, t)
on the surface, in the direction normal to the surface, is equal to the normal
component of velocity of the fluid particle at the same point on the surface.
This condition, which is known as the kinematic boundary condition, can
be expressed in terms of the velocity potential, ¢(x, z, t), as follows (Massel,
1989) :
o( o¢ o(
ot ox ox
o¢
oz
at z = (,
(C.92)
in which ((x, t) is the surface displacement from the still water level (see
Fig. 4.2). Physically, the term o(/ot denotes the velocity of vertical movement of the sea surface. The second term on the left-hand side of Eq. (C.92)
expresses the fact that water particles subjected to wave motion move along
the surface. Therefore, a particular point P(x, z, t) not only moves vertically
but also possesses a small back- and forward motion in the horizontal direction.
The summation of the two terms on the left-hand side of Eq. (C.92) provides
the total vertical velocity o¢ / 0 z of the water particle. A detailed derivation of
Eq. (C.92) will not be given here, as it can be found in many papers and books
(see, for example, Massel, 1989; Dean and Dalrymple, 1992).
As we assumed that wave steepness was very small, it follows that 0(/ ox
term is very small and can be neglected. Therefore, Eq. (C.92) simplifies to:
o(
ot
o¢
oz
at z = O.
(C.93)
Note that the kinematic condition is now applied on the plane z = 0, instead
of the ~urface z = (, due to the assumption of low wave steepness (see also
Fig. 4.1).