LINEAR PLANE WAVES
67
w = â7 at z = 0
(3.12)
w = 0 at z = — d
(3.13)
P — Pa at z=0
(3.14)
Equation (3.11a) is the zéro vorticity or irrotational condition, which follows
from Assumption 4. Equation (3.11b) is the continuity condition. Equations
(3.11c) and (3.11d) are the momentum conservation équations, otherwise known
as the Eulerian équations of motion. Equations (3.12) and (3.14) are the boundary conditions at the surface, and équation (3.13) is the boundary condition at
the sea floor. The free surface is defined by z — p. However, in view of Assumption 1, the conditions on w and p of équations (3.12) and (3.14) are applied at
the mean water level, z = 0.
A particular solution that satisfies the linear équations (3.11) is the plane
wave form for surface élévation p = p(x,t), already presented as équation
(3.10b). With this resuit, together with équations (3.11 )-(3.14), the water particle velocities u and w, their respective accélérations du/dt (= u) and dw/dt
(= w), and the dynamic pressure p can be deduced. Ail of these quantifies are
summarized in Table 3.1. It is noted that the total or absolute water pressure is
the sum of three pressure terms: the atmosphère pressure, pa; the hydrostatic
pressure, p3 = —pgz-, and the dynamic pressure, or
Ptotai = Pa + Ps + P
(3-15)
In calculating the forces on offshore structures, the atmospheric pressure is of
no conséquence.
The wave number and frequency relation that is compatable with the solutions just presented is
cv2 = gk tanh kd
(3.16)
From équation (3.6), the wave phase velocity or celerity is given by c = w/fc.
With this and équation (3.16), the celerity becomes
/ ff
\ 1 ' ■
c — y — tanh kd |
(3-17)
This équation for celerity reduces to rather simple limiting forms for cases in
which k is either very large (short wave lengths) or very small (long wave
lengths). Such examples are included as problems at the end of this chapter.
67
w = â7 at z = 0
(3.12)
w = 0 at z = — d
(3.13)
P — Pa at z=0
(3.14)
Equation (3.11a) is the zéro vorticity or irrotational condition, which follows
from Assumption 4. Equation (3.11b) is the continuity condition. Equations
(3.11c) and (3.11d) are the momentum conservation équations, otherwise known
as the Eulerian équations of motion. Equations (3.12) and (3.14) are the boundary conditions at the surface, and équation (3.13) is the boundary condition at
the sea floor. The free surface is defined by z — p. However, in view of Assumption 1, the conditions on w and p of équations (3.12) and (3.14) are applied at
the mean water level, z = 0.
A particular solution that satisfies the linear équations (3.11) is the plane
wave form for surface élévation p = p(x,t), already presented as équation
(3.10b). With this resuit, together with équations (3.11 )-(3.14), the water particle velocities u and w, their respective accélérations du/dt (= u) and dw/dt
(= w), and the dynamic pressure p can be deduced. Ail of these quantifies are
summarized in Table 3.1. It is noted that the total or absolute water pressure is
the sum of three pressure terms: the atmosphère pressure, pa; the hydrostatic
pressure, p3 = —pgz-, and the dynamic pressure, or
Ptotai = Pa + Ps + P
(3-15)
In calculating the forces on offshore structures, the atmospheric pressure is of
no conséquence.
The wave number and frequency relation that is compatable with the solutions just presented is
cv2 = gk tanh kd
(3.16)
From équation (3.6), the wave phase velocity or celerity is given by c = w/fc.
With this and équation (3.16), the celerity becomes
/ ff
\ 1 ' ■
c — y — tanh kd |
(3-17)
This équation for celerity reduces to rather simple limiting forms for cases in
which k is either very large (short wave lengths) or very small (long wave
lengths). Such examples are included as problems at the end of this chapter.