20 
h 2 /l} «0.5H/h (U»1) allows a hydrostatic approximation, and h 2 /L 2 »0.5H/h 
(U «1) allows linear equations. h 2 /l} ~0.5H/h (U ~ 1) is the prerequisite to the validity of 
simple Boussinesq equations (Ursell 1953). In Fig. 5.1.3, the plotted red line represents 
0 5 H Lr 
— = 1, with values >1 to the left of it and values <1 to the right. It is also common to 
h h~ 
interpret U as the relation between wave steepness H/L and relative water depth h/L, i.e. 
Fig. 5.1.3: Flange of validity of different wave theories as a function of H/h and h/L (Komar 
1978, Fig. 3.17) 
Of the above frictionless and irrotational wave theories, the theory of cnoidal waves has the 
highest generality. It applies to non-breaking waves, i.e. in the case of long waves up to 
H/h- 0.78 (Miche 1944). The other waves referred to are limiting cases of this theory 
based on simplifying assumptions. „Airy deep water“ in Fig. 5.1.3 marks the validity of linear 
non-hydrostatic wave theory, „Airy shallow water“ the region of linear hydrostatic wave 
theory. Komar assumed a wider range of validity of linear theory than Ursell, i.e. U ~ 50 
(U = 16;r 2 /3). Frohle et al. (2002) set the limit at U =13. Solitary waves are located in the 
left part of Fig. 5.1.3. They are single waves, cnoidal waves in the limit L toward infinity. In 
Fig. 5.1.3, an idea of Munk (1949) is taken up according to which steep-crested waves 
separated by long, flat troughs may be considered to constitute single solitary waves. Fig. 
5.1.3, the boundary line for the validity of the theory of solitary waves has been established 
at H/h = 1600(L//z)“ 25 (Housley und Taylor 1957). 
Let tsunami be simple long waves with typical periods of 10 to 30 minutes. Then the 
parameter h/L (estimating L by T --Jgh) for depths of less than 8,000 m is smaller than 
0.05 (Table 5.1.1). In Fig. 5.1.3, this limits possible theories for the description of tsunami to 
„Airy shallow water“, „cnoidal wave“, and „solitary wave“. With regard to solitary waves, it 
should be noted that h/L theoretically declines to zero due to L—>oo, while for tsunami 
h/L declines to zero with h, while L decreases.