19 
P = Po +P =-PogZ + Pog i l 
cosh L/27t(z + h 0 ) 
cosh L/2^ 0 
p' differs from p' h by a factor approaching 1 for small L/2^ 0 . 
Boussinesq equations: 
T7 - r, du du dp dw dp 
V-v = 0, p Q — + p 0 u— = —— and p 0 — = ——~p 0 g with boundary conditions 
dt dx dx dt dz 
dr/ dr/ , 
w = —- + u—-, p = 0 at z = r/ and w = 0 at z = -h 0 . 
dt dx 
1* 1* 1* O u 
It follows from V ■ v = 0 that w--\ —dz , from irrotationality u = u(z = 0) + —-/dz"dz'. 
o z' -\2 
d H j // j / 
~K 
dx 
z -An 
dx 2 
Expressed by the approximation for w, the pressure is: 
0 z 02^ 
P = Po+Pb =-PogZ + PogP-Po$ j ~^f Z " dz ' ■ 
z -h, 
dw 
p' B differs from p' h by a summand vanishing with > 0. A solution to the equations is 
dt 
possible by assuming = anc | ¡t ¡ s determined by the choice of u(z- 0). The 
dx dx 
additional terms in this simple version of the Boussinesq equations as compared to non 
linear hydrostatic equations can be deduced from the local temporal derivative of u and 
from the horizontal pressure gradient (cf. Peregine 1972). 
5.1.3 Applicability of the equations to tsunami events 
The various approximations of equations 2 apply only to a certain region of the parameters 
wave height, wave length, and undisturbed water depth. In section 5.1.1, a solitary wave was 
stated to be a possible way of describing a tsunami. It is the solution to simple Boussinesq 
equations. 
Let a tsunami be a simple wave characterised by the parameters H,L, and h. With suitable 
scaling of the equations (Voit 1978), the orders of magnitude of the individual terms in (2) 
can be estimated through these parameters, and simplifications evaluated. The non-linear 
terms in the horizontal momentum equations and in the surface boundary conditions are on 
the order of 0.5H/h, in the vertical momentum equation 0.5H/h ■ h 2 /1} and in the bottom 
boundary conditions 1. The order of magnitude of the local temporal change of horizontal 
motion with this scaling is 1, that of vertical motion h 2 /L 2 . Komar (1976) accordingly 
determined the range of validity of different wave theories in dependence on the two 
important parameters h/L and H/h (Fig. 5.1.3). Compare also Mader (2004, Table 1.1); 
Frohle et al. (2002, Table A.3.1); and Peregine (1972). 
Consequently, /z 2 /l 2 toward zero (i.e. h/L <0,05) marks the hydrostatic limit case of 
equations 2, 0.5H/h toward zero (i.e. H/h« 1) the linear limit case. However, both 
parameters alone are not sufficient to determine the range of validity of individual wave 
theories. According to Ursell (1953), it is the relative order of magnitude of non-hydrostatics 
(local temporal change of vertical velocity) h 2 /l} and non-linearity 0.5H/h, the so-called 
Ursell parameter U, which determines the necessary generality of the equations.