26 
2 - 
1 - 
0 v 
AMPLIFICAI ION 
AMPLIFICATION 
Fig. 5.4.1: Transformation on the shelf (linear Boussinesq equations, ij surface elevation, 
h undisturbed water depth, 10.5:PEDpT) 
With respect to the transition from 540 m to 60 m water depth, the result of the simple, linear, 
dispersion-free estimation in section 5.1 is that the wave height increases further, by the 
factor 3.0, to 4.3H deep , that the wavelength decreases by an additional factor of 0.3, and 
steepness increases by an additional factor of 5.2. 
On the shelf, especially on a wide shelf like that of the North Sea, these values are not valid, 
however, because linear equations lose their validity with increasing steepness and, in 
particular, because it is no longer justifiable to neglect bottom friction. Also frictionless 
simulation (Fig. 5.4.1) thus gives only a general picture. 
5.4.2 Attenuation 
The depth at which sea and swell (T ~10 s) are strongly affected by the ocean bottom 
(h/L> 0.05) is about 25 m (Holthijsen 1998). For the long-wave proportion of a tsunami 
(with T > 10 minutes), the condition h/L> 0.05 is met on the entire shelf (cf. Table 5.1.1). 
If only large-scale processes are considered, the destructive energy of a tsunami may be 
assessed without considering the details of wave mechanics. The width of the North 
European Shelf in the area of the North Sea is 800 km. The average slope angle in this 
region is 0.014°. Therefore, in the following assessment of energy (Kleine 2005), a tsunami is 
approximated as a portion of energy travelling at ^fgh . 
The starting point is the balance equation for the total spectral energy of a wave train: 
- + V-(ce) = Q e 
a g 
(c^the effective propagation velocity of total energy, Q E net effect of energy sources, non 
linear interaction and dissipation, used for deep water waves, Reistad et al. 1998). 
Neglecting non-linear effects and concentrating on the long-wave part of the spectrum, the 
energy balance equation in advective form applies: 
^ + i^hWE + (y(i^h))E = -D (3) 
at 
(Tthe unit vector in the direction of propagation, -Jgh scalar propagation velocity, D> 0 
energy dissipation due to bottom friction). 
Equation 3 thus describes the temporal change of energy due to advection, shoaling, and 
dissipation at a particular shelf location. On a flat bottom, only the mechanical energy arriving