4.2 Wave Parameters Based on Small Amplitude Wave Theory
115
Equation (4.24) confirms a simpler derivation of group velocity which is given in
Sect. 3.4.3. The phase and group velocities are presented in Fig. 4.5 as functions
of the non-dimensional water depth kh. The velocities have been normalized
against their values in deep water, e.g. C /Co and Cg/Cgo ' in which Co and
CgO are the phase and group velocities at infinite water depth, respectively,
i.e.: Co = Jg/k and CgO = 1/2Jg/k. Figure 4.5 shows that phase velocity
C jCo increases monotonically as water depth increases. However, this is not
the case for normalized group velocity Cg/C gO which initially increases, when
depth increases, and reaches its maximum value at about kh = 21rh/ L ~ 1.0
(L/h ~ 21r) after which it decreases to 1. The computer program D41 IS
provided to determine both velocities (see Appendix D).
Orbital Velocity Components. Because of the periodicity of wave movement, water particles move along closed orbits (see Figs. 3.3 and 314). Both
horizontal and vertical orbital velocity components take the form (see Appendix
C.5 for derivation):
• horizontal velocity component
gkH cosh k(z + h)
u(x, z, t) = -
( )
cos(kx - wt),
2w
cosh kh
( 4.25)
• vertical velocity component
9 k H sinh k (z + h) .
w(x, z, t) = -
() sm(kx - wt).
2w
cosh kh
( 4.26)
Let us now consider a vertical profile located under the wave crest, when kx -
wt = O. Hence, Eqs. (4.25) and (4.26) give:
=
gkH coshk(z+h) }
2w
cosh(kh)
.
w = O.
u
( 4.27)
Under the wave crest, the vertical velocity component vanishes, as expected,
while the horizontal component is non-zero, attenuating from the surface to the
sea bottom. At the sea surface (z = 0), the velocity, u = gkH /2w, and at the
sea bottom (z = -h), u = (gkHj2w)/coshkh. The schematic representation
of orbital velocity components for various wave phases is given in Fig. 4.6. Four
phases on the wave profile have been selected and the corresponding orbital velocity vectors at some submergence level are indicated. Both deep and shallow
water cases are considered. In deep water, particles move along circular paths.
At the wave crest phase (point B), only horizontal orbital velocity exists, while
at the inter-section of the surface with the still water level (points A and C)
water particle moves up and down, respectively. In shallow water, the picture
115
Equation (4.24) confirms a simpler derivation of group velocity which is given in
Sect. 3.4.3. The phase and group velocities are presented in Fig. 4.5 as functions
of the non-dimensional water depth kh. The velocities have been normalized
against their values in deep water, e.g. C /Co and Cg/Cgo ' in which Co and
CgO are the phase and group velocities at infinite water depth, respectively,
i.e.: Co = Jg/k and CgO = 1/2Jg/k. Figure 4.5 shows that phase velocity
C jCo increases monotonically as water depth increases. However, this is not
the case for normalized group velocity Cg/C gO which initially increases, when
depth increases, and reaches its maximum value at about kh = 21rh/ L ~ 1.0
(L/h ~ 21r) after which it decreases to 1. The computer program D41 IS
provided to determine both velocities (see Appendix D).
Orbital Velocity Components. Because of the periodicity of wave movement, water particles move along closed orbits (see Figs. 3.3 and 314). Both
horizontal and vertical orbital velocity components take the form (see Appendix
C.5 for derivation):
• horizontal velocity component
gkH cosh k(z + h)
u(x, z, t) = -
( )
cos(kx - wt),
2w
cosh kh
( 4.25)
• vertical velocity component
9 k H sinh k (z + h) .
w(x, z, t) = -
() sm(kx - wt).
2w
cosh kh
( 4.26)
Let us now consider a vertical profile located under the wave crest, when kx -
wt = O. Hence, Eqs. (4.25) and (4.26) give:
=
gkH coshk(z+h) }
2w
cosh(kh)
.
w = O.
u
( 4.27)
Under the wave crest, the vertical velocity component vanishes, as expected,
while the horizontal component is non-zero, attenuating from the surface to the
sea bottom. At the sea surface (z = 0), the velocity, u = gkH /2w, and at the
sea bottom (z = -h), u = (gkHj2w)/coshkh. The schematic representation
of orbital velocity components for various wave phases is given in Fig. 4.6. Four
phases on the wave profile have been selected and the corresponding orbital velocity vectors at some submergence level are indicated. Both deep and shallow
water cases are considered. In deep water, particles move along circular paths.
At the wave crest phase (point B), only horizontal orbital velocity exists, while
at the inter-section of the surface with the still water level (points A and C)
water particle moves up and down, respectively. In shallow water, the picture