256
6 Wave Transformation in Shallow Water
So, with decreasing depth, the index of the degree of angular direction
n =no (KI Ko) 2 increases, which is also confirmed by observations (Krylov et
al., 1976). The frequency spectrum value:
n/2
S(w,H) = J S(w,{3,H)df3
-n/2
changes with the depth as:
S(w,H)"'
( (g•)'
) So (w)2n•[•-(:..)r
4 1 + 2KH
W
sh(2KH)
r 2 (~ + 1) r (n + 1)
X r 2 (~ + 1) r (no+ 1) .
(6.11)
The spectral parameter variations in shallow water are shown in Fig. 6.1.
Relative values of the wave number Kl Ko, power index nino, phase velocity
of the waves cl c0 , as well as the ratio of the spectral densities S (w, H) IS (w ),
obtained by the formula (6.11) depending on the non-dimensional depth
H w 2 I g, are shown. If the wave number and the power index are increased and
the phase velocity is decreased monotonically with decreasing depth, then at
first the ratio of spectral densities is slightly decreased and after that begins
i :
,i ~
,., ~
I, ~
,, \
-·-·- 1
----- 2
......... J
1\ ·~
2 ~, \ ••.••
.....
.
-·
\' · ..
·', ·· ..
'· ....... ..:: .......
'· -- ······.~.~~~
·-·-·---·
f
Fig. 6.1. Transformation of wave spectrum ratioS/So (1), wave number K/Ko (2)
and power index n = no (K/Ko) 2 (3) of angular direction distribution and phase
velocity c/co (4) at their normal propagation towards a coast