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4 How to Determine Wave Parameters
or
and
(4.9)
For very shallow water w 2 h/g « 1, the tanh(kh) - t kh. Thus, Eq. (4.3)
yields now:
ff2 h
kh= - 9 ,
(4.10)
or
k = ~ and L = f;h T.
(4.11)
The dispersion relation (4.3) is shown in Fig. 4.4 in the form of the function
kh = f (w 2 h / g). The asymptotes for both extreme cases, deep and shallow
water, are also marked. A computer program is provided to solve Eq. (4.3) for
both cases (see Appendix D, program D.41).
Surface Displacement ((x, t). Surface displacement ((x, t) can be calculated
from the dynamic boundary condition at the sea surface (see Eq. C.96), e.g.:
1 (8¢) H
((x, t) = -- at
= "2 cos(kx - wt),
9
z=o
(4.12)
or
H
(271"X 271"t)
((x,t)="2cos T-T·
(4.13)
Equation (4.13) provides the oscillation of surface displacement in time and
space. The phase (kx-wt) = 0 (or 271") corresponds to the wave crest, while the
phase (kx - wt = 71") is associated with the wave trough. When (kx - wt) = 71" /2
or 371"/2, the wave surface intersects the still water level.
The highest point of the wave surface (wave crest) is at the vertical distance
H /2 from the still water level, while the lowest point (wave trough) is located
H /2 below the still water level. Therefore, the wave profile is symmetric with
respect to the still water level. Equation (4.13) can be rewritten in a slightly
different form:
H [271" ( L)] H [271"
]
((x, t) = 2" cos L x - t T = 2" cos L (x - Ct) .
(4.14)