23 
jV-waves 
Fig. 5.1.5: N-wave (Bryant 2001, Fig. 2.4). 
However, this too does not always approximate a tsunami correctly. During slope failures the 
period of seabed motion is relatively long, and the initial phase of the tsunami is described 
adequately by hydrostatic theory. Model computations have shown that in this case a wave 
crest comes first, followed by a seaward wave trough (Ward 2002). Solitary waves may then 
form in the closer far field (Rubino et al. 1998). 
Summarising the above, it may at best be possible in limited evolution phases of a tsunami to 
describe it analytically as a single wave or wave train. In the following, different wave 
theories, in conjunction with other model concepts and simulation results, will continue to be 
used to discuss the propagation and modification of a tsunami travelling into shallow-water 
areas like the North Sea. 
5.2 Propagation and modification in the deep ocean 
Especially the impulse-type excursion of the water surface following an earthquake does not 
constitute a solution in analytical wave theory. If it is interpreted as a linear superposition of 
simple waves, the individual waves propagate in all directions with their specific phase 
velocities. In the deep ocean, part of the spectrum will be short waves. Short waves 
(h/L> 0.25) have a period-dependent phase velocity gT short /In. Short partial waves with 
small periods thus lag behind waves with greater periods. This process is called frequency 
dispersion. It weakens the primary signal of a tsunami. According to this theory, the dominant 
long-wave signal (h/L < 0.05) propagates in a dispersion-free way, i.e. with a velocity ^fgh 
that is only depth dependent. 
Fig. 5.2.1 shows a tsunami as the solution to linear Boussinesq equations. It is part of a two- 
dimensional, frictionless computation by Pedersen (10.5: PEDpT), which will be used in the 
following to demonstrate important evolution phases of an exemplary tsunami. 
DISPERSION DISPERSION 
Fig. 5.2.1 Initial phase of the propagation (right) of an impulse-type signal (left) in the deep 
ocean (linear Boussinesq equations, ;/ surface elevation, h undisturbed water depth, 
10.5: PEDpT).