27 
at this location due to advection would be available for partial dissipation by bottom friction. 
On a sloped bottom, more mechanical energy per unit area is available for dissipation 
because of shoaling. The simple balance equation is limited to gently sloping shelves 
(A h/L h <h/L). 
With the approach D = x{E/hf 5 for dissipation by bottom friction (k dissipation coefficient) 
and a bottom slope assumed to be constant, -iVh = Ah/L h =: a > 0, a closed solution can 
be found for equation 3: 
1 
Í 1 2 k 1 ^ 
1 
1 2 k 1 
h 0.25 
77O..5 ^ 0.5 r 
1E So g hj 
_ h 0 ' 25 
™deep 
E°' 5 5 ct j? 05 /z 
^ ^ deep J u <5 n deep J 
or, expressed by wave height H of a representative shallow water wave, E = gH 2 /8 : 
H/H 
deep 
■0.25 
/ h 
h + 
.0.25 
v ^deep 
K JJ 
5a/2£7 dmP 
f 
1— 
h 
'deep J j 
i.e. for k - 0 the expression in section 5.3 is regained. In near-shore waters of /2 —> 0 the 
result shows, as an important property, an asymptotic behaviour of energy, or wave height, 
according to — —> 5^2 —. Table 5.4.2, with /r = 6 10“ 3 , gives the asymptotic behaviour at 
h k 
different bottom slopes. 
Bottom slope 
H (asymptotic) 
H (asymptotic, h=20 m) 
North Sea 
1:4,000 
0.27 h 
5.4 m 
German Bight 
1:2,500 
0.43 h 
8.6 m 
Thailand 
1:1,000 
1.08 h 
21.7 m 
[Lisbon 
1:100 
10.83 h 
216.6 ml 
Table 5.4.2: Asymptotic wave height with different bottom slopes 
The above energy balance ceases to be valid at bottom slopes of 1:1,000. However, it 
provides a good estimate for the North Sea. In addition, all results are limited to relative wave 
heights of Hlh< 1. 
There are two important conclusions: firstly, clearly less energy is available for the as yet 
insufficiently understood near-shore processes on a shallow shelf than on a steep one and, 
secondly, the asymptotic value for wave height is independent of the initial wave height. 
Fig. 5.4.3 shows the amplitude profile for several initial amplitudes on an idealised north- 
south profile across the North Sea. Fig. 5.4.2 shows profiles on an equally idealised, 
somewhat steeper section in the German Bight. The amplitude development in Fig. 5.4.2-3 is 
comparable to the dotted lines in the frictionless simulation (Figs. 5.3.1 and 5.4.1) which 
mark the amplitude change of the primary signal.