18 
A solitary wave, which is a long single wave, seems to be well suitable to describe a tsunami 
because, in a solitary wave, the water particles move horizontally as, for example, in long 
Airy waves, but unlike the latter they move only in the direction of wave propagation (Fig. 
5.1.2). The waves are waves of translation, with a mass transport exceeding that of the 
Stokes drift (mass transport because particle trajectories do not close) of wind waves. 
Fig. 5.1.2: Trajectories of water particles during passage of a solitary wave (Komar 1976, 
Fig. 3.15). Surface elevation:;/ = sec/r 
l3Hx' 
\4 h h J 
5.1.2 Equations of wave theories 
With regard to section 6.2, the main differences among the individual wave theories are 
presented below using a very simple version of equations 2 for irrotational flow (Cartesian 
co-ordinates, two-dimensional with v = (u,w) and constant depth h 0 , and = (0,0,g)). 
Non-linear hydrostatic: 
V-v = 0, p 0 — + p 0 u—-~— and 0 = ~—~p 0 g with boundary conditions 
dt dx dx c)z 
drt drt , 
w = —- + u—-, p = 0 at z — rj and w = 0 at z = -h 0 . 
dt dx 
From V ■ v - 0 it follows that w = - 
Irrotationality Vx v - 0 in this simple example takes the form 
du 
dz 
dw 
dx 
With — = 0 (hydrostatic), it follows from irrotationality that — = 0. Pressure is 
dx dz 
P = Po+ p'hs = -Pogz + Po8V ■ 
Linear non-hydrostatic: 
V • v = 0, p 0 —— — and p 0 —— = - p 0 g with boundary conditions 
dt dx dt dz 
w = —-, p = 0 at z = p and w = 0 at z = -h 0 . 
dt 
Irrotationality allows a solution via the introduction of a scalar velocity potential (e.g. Komar 
1976). Instead, V -Vp' - 0 may be considered (Gill 1982). The solution for pressure in both 
cases thus is: