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parameter U (Table 5.1.2), it is found that non-linear influences prevail on the shelf and, with 
/i/L <0.01, hydrostatic non-linear theory is adequate. 
All wave theories referred to in this section are severely limited by a breaking criterion, which 
is h/L= 0.78 for long waves. At 10 m water depth, it limits wave heights to maximally 7.8 m, 
and to 0.78 m in 1 m of water. Therefore, the description of tsunami by the analytical theory 
of long waves in frictionless media, in particular as solitary waves, is no longer valid in 
shallow water. 
Also the interpretation of tsunami as simple waves with a given period, determined by the 
process that generated them, ceases to be valid in coastal regions, where wave period rather 
is a function of location (Munk 1962), because frictional effects prevail (Sabatier 1986). In 
other respects, too, the interpretation of tsunami as simple waves is simplistic. Despite a 
relatively narrow spectrum, a tsunami is a superposition of simple waves having different 
periods. As such, it has been described as a soliton. Such waves preserve their overall 
shape in spite of major non-linearities, whose influence is compensated by dispersion i.e. by 
the change in shape of the composed signal due to different propagation speeds of single 
waves (Fig. 5.1.4). A soliton thus is described well by Boussinesq equations. In real tsunami, 
however, such equilibrium is rarely present. In addition, a soliton is not a good model for a 
tsunami because it is a positive signal and thus fails to reproduce the initial receding of water 
(wave trough) which is often observed on the coast. 
Fig. 5.1.4: Equilibrium in a soliton (Brunelli 2000, Fig. 3, Lomdahl 1984, Fig. 1). 
Earthquake triggered tsunami often are generated in deep water, with an impulse-type initial 
elevation of the water surface. In incompressible, frictionless media, an initial surface 
elevation represented as a delta function develops a shape that can be described by an Airy 
function. The first excursion of this function is negative, as desired (Gill 1982). Another way 
of generating an initial wave trough is to consider the viscosity of water. In a viscous fluid, 
the dissipation and non-linear propagation of an impulse-type signal, in the most simple, one 
dimensional case, is described by a non-linear diffusion equation (Burgers equation, Burgers 
1974) for the scalar velocity potential (Whitham 1999). With a suitable initial distribution, its 
solution gradually develops a surface elevation termed ,,N-wave“ based on its shape, which 
also has a leading wave trough (Fig. 5.1.5).