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)