110
4 How to Determine Wave Parameters
or,
(4.4)
Only the combinations of the wave frequency, w, and the wave number, k, which
satisfy the dispersion relation represent possible wave motion. In practice, the
dispersion relation has to be solved for two cases:
• Wavelength, L, and water depth, h, are known; wave period, T, has to
be found.
The wave period, T, or wave frequency, w, can be calculated from the
above equations in a straightforward manner:
211"
T = --;:::=====
211"g h (211"h)
-tan - -
L
L
(4.5)
or
w = vgk tanh(kh).
(4.6)
• Wave period, T, or wave frequency, w, and water depth, h, are known;
wavelength, L, has to be found.
The determination of wave number, k, or wavelength, L, is not as straightforward as in the first case. In order to clarify the nature of the solution,
let us rewrite the second equation in (4.3) in the form:
w
2 h ( 1 )
h = -
-
= tanh(kh) = h
9
kh
(4.7)
If the functions h(kh) = (w 2 h/g) (l/kh) and h(kh) = tanh(kh) are
plotted, as illustrated in Fig. 4.3, then the inter-section of these curves
at point P denotes the value of (kh) which is a solution of the dispersion
relation (4.7).
The solution of Eq. (4.3) can be obtained much more easily for extreme
water depths, i.e. for very deep water and very shallow water. For very
large water depth w 2 h/g » 1, the tanh(kh) --+ 1.0. Using this value in
Eq. (4.3), we obtain:
(4.8)