Skip to main content

Full text: 41: Tsunami - a study regarding the North Sea coast

17 
V v = -v,yh at the bottom (h undisturbed water depth). 
For analytical progress it is frequently assumed that the water initially is at rest and the 
velocity field is irrotational. In an inviscid fluid, motion then remains irrotational. 
Each particular wave theory neglects certain processes. By neglecting advection (non-linear 
term) in the equations and in their boundary conditions, changes in wave shape are 
suppressed. Solutions to linear equations (Airy waves) thus preserve the shape of the 
surface elevation. A simple wave whose propagation speed is a function of its period is 
called dispersive (more precisely: frequency dispersive). Whether an approximation leads to 
dispersive waves can only be seen when the solution has been obtained (Whitham 1999). By 
neglecting local vertical acceleration, dw/dt, linear equations become hydrostatic linear 
equations whose solution is non-dispersive. More complex approximations such as the 
Boussinesq equations (e.g. Boussinesq 1871), and their special forms Korteweg de Vries 
(Korteweg et al. 1895, 10.6: WEI05) and KP equations (Kadomtsev et al. 1970), are both 
dispersive and shape changing but, for both properties, only approximately (cnoidal and 
solitary waves). Other non-linear non-hydrostatic theories (e.g. Stokes 1847) apply only to 
very small surface elevations and hence are not of interest in this context. Linear Boussinesq 
equations are shape preserving and, to first order, dispersive. 
In linear non-hydrostatic and non-linear hydrostatic equations, the manifold influences of 
variable bottom topography, reflection, refraction, diffraction, and energy concentration 
(shoaling) are taken into account through the bottom boundary condition. Of the Boussinesq 
type, only more complex equations (Peregine 1972, Madsen et al. 1991, Madsen et al. 1992, 
Liu et al. 2002) can be applied to variable bottom topography. 
Fig. 5.1.1 shows several solutions of analytical wave theories for shallow water (Komar 1976, 
Table 3.1). The first three wave types are simple periodic waves with L,T,H and the 
undisturbed depth h as parameters, while the fourth solution, a limit of the cnoidal wave, is a 
solitary wave valid only for L —»°o. 
AIRY WAVES Itlnusoldol) 
Applicotion: Wovtt of ynoll arrpfih.de ir deep water. 
References: Lep'oce 0776), Airy (1845). 
STOKES AND GERSTNER WAVES itrccholciol) 
Application; Wo«*» of finite (rnplitv.de In deep. Interred lore, and iho'low water. 
Reference»: Gerstner <1807), Stele es 0647), Freude Ronllr* «'1863', 
RayleigK 0877). 
CNOIDAL WAVES 
Application: Wo.es of finite otiDlifv.de in intermediate to »Hollow water. 
Reference»: Korteweg erd DeVries 118951, Keller 0949). 
SOLITARY WAVES 
Application: Solitary or i»olc*ee crests of finite or-plit.ee moving In »hollow water. 
References: S<oft-Ru»sell (1844), taussinesq 11871!, Rayleigh 118761, McCowar '1891«. 
Fig. 5.1.1: Wave theories (Komar 1976, Table 3.1)
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.