502
Appendix C Useful Theoretical Approaches and Formulas
C.5.4 Wave Parameters Based on Small Amplitude Wave Theory
Orbital Velocity Components. The velocity components of the water particle subjected to wave motion can be found through the differentiation of the
velocity potential (C.114). Thus, we have:
• horizontal velocity components:
(
) _ a _ gkH cosh [k(z + h)] (k _ )
uX,z,t -
-
h( )
cos x wt,
ax 2w cos kh
(C.115)
• vertical velocity components:
(
) _ a _ gkH sinh[k(z+h)] . (k _ )
w x, z, t -
-
( )
sm x wt.
oz 2w cosh kh
(C.116)
In deep water, the dependence of the velocity potential on z can be presented
in a simpler form. We note that (see Sect. C.6):
eX + e- X
cosh(x) =
2
Therefore,
cosh [k(z + h)] ek(z+h) + e-k(z+h)
cosh(kh)
e kh + e- kh
e kz + e- 2kh e- kz
1 + e- 2kh
(C.117)
(C.US)
For very deep water, h --> 00, and relation (C.11S) simplifies as follows:
cosh [k(z + h)]1
cosh(kh)
limh ...... oo
(C.U9)
After substituting Eq. (C.119) into (C.115), the velocity u in deep water becomes:
u(z) = gkH e kz cos(kx _ wt) = gTH e(27fz/L) cos(kx - wt).
2w
2L
(C.120)
The rate of attenuation of velocity, u, with submergence is now controlled by the
term e(27fz/L). In particular, if submergence, z, is equal to half the wavelength
L, i.e. z = -L/2, Eq. (C.120) yields that velocity, u, at level z is only about
4% of its value at the sea surface. Therefore, the submergence z = -L/2 is
usually considered as a limiting depth, below which the influence of surface
waves can be neglected.