GUIDE WAVE ANALYSIS AND FORECASTING - WMO

the energy flux is c gH 2 . This is a constant. Hence the
shoaling factor depends on the ratio between the deepwater
group velocity c g0 and the group velocity c g at
depth h, and is given by:
WAVE FORECASTING BY MANUAL METHODS 53
(4.10)
where c 0 is the phase velocity in deep water (√g/k 0),
k 0 is the wavenumber in deep water and H 0 is the wave
height in deep water. Since the group velocity c g is given
by:
c
g
K
H cg
= = =
H c
g
1 ⎛ kh ⎞
= β tanh kh , β = ⎜ + ⎟ ,
k
2 ⎝ sinh kh⎠
1
2
2
the shoaling factor can also be expressed as:
10
1
0.1
0.01
s
0
1 c
c
0 0
g 2 g
,
c/c0 . λ/λ0
K = 2 β tanh kh .
s
– 1/ 2
(4.11)
The wavenumber k = 2π/λ at depth h can be
approximated as follows:
k = k 0 (tanh kh) –1 ,
k = k 0 [tanh (k 0 h tanh kh) –1 ] –1 ≈ k 0 (tanh k 0 h) –1 . (4.12)
Figure 4.7(a) shows various curves involved in the
transformation of properties of a wave propagating from
deep to shallow water. The shoaling factor, K s, is also
shown in this figure. But for the convenience of the
user interested in obtaining only the shoaling factor K s
associated with a deep water wave of a given wavenumber
k 0 (= 2π/λ 0), the curve of K s vs h/λ 0 is presented
in Figure 4.7(b) in an expanded scale.
0.0001 0.001 0.01 0.1 1
h/λ0
Figure 4.7(a) — Illustration of various functions of h/λ 0 (derived from CERC, 1984)
K s
cg/c0
h/λ
2β