114
4 How to Determine Wave Parameters
1..3' " 1.4
" C.,)
~ 1.2
C.,)'"
" C.,)'" 1.0
'" 0
'p
'u 0.8
0
Q)
>
'@ 0.6
c
.9
'" c 0.4
0
S
;.a
I
0.2
c
0
Z
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Non-dimensional wave number kh
Fig. 4.5: Phase and group velocities, normalized against their values in deep water
Cg = m(kh) C,
(4.21)
in which:
m kh = - 1 +
.
1 (
2kh)
( ) 2
sinh(2kh)
(4.22)
Hence, the group velocity is proportional to the phase velocity, but the proportionality function, m, depends on the non-dimensional wave number, kh.
Again, there are two limiting water depths when group velocity takes simpler
forms, i.e.:
2kh
• deep water when
(
) -+ 0; then
sinh 2kh
1
m-+2
1
gT
C -+-C=9
2
41r '
(4.23)
and
• shallow water when sinh(2kh) -+ 2kh; then
m -+ 1 and Cg -+ C = j;h.
(4.24)