Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

268 Waves
wave elevations in shallow water and large wave elevations in intermediate to deep
water. We have discussed a similar problem in Chapter 5.
8.16.2 Cnoidal and solitary waves
The equations modelled in Chapter 7 can deal with large-amplitude waves in
shallow water. These are called cnoidal waves when periodic, and solitary waves
when the period is in®nite. For more details see references 1±4. The ®nite element
methodology of Chapter 7, can be used to model the propagation of such waves.
It is also possible to reduce the equations of momentum balance and mass conservation
to corresponding wave equations in one variable, of which there are several
di€erent forms. One famous equation is the Korteweg±de Vries equation, which in
physical variables is
@ p
‡ gH
@t
1 ‡ 3
2h
@
@x
‡ h2
6
p @
gH
3
ˆ 0 …8:46†
3 @x
This equation has been given a great deal of attention by mathematicians. It can be
solved directly using ®nite element methods, and a general introduction to this ®eld
is given by Mitchell and Schoombie. 68
8.16.3 Stokes waves
When the water is deep, a di€erent asymptotic expansion can be used in which the
velocity potential, , and the surface elevation, , are expanded in terms of a small
parameter, ", which can be identi®ed with the slope of the water surface. When
these expressions are substituted into the free surface condition, and terms with the
same order in " are collected, a series of free surface conditions is obtained. The
equations were solved by Stokes initially, and then by other workers, to very high
orders, to give solutions for large-amplitude progressive waves in deep water.
There is an extensive literature on these solutions, and they are used in the o€shore
industry for calculating loads on o€shore structures. In recent years, attempts have
been made to model the second-order wave di€raction problem, using ®nite elements,
and similar techniques. The ®rst-order di€raction problem is as described in Sec. 8.9.
In the second-order problem, the free surface condition now involves the ®rst-order
potential.
First order:
Second order:
@ …1†
@z
@ …2†
@z
ÿ !2
g
ÿ !2
g
…1† ˆ 0 …8:47†
…2† ˆ …2†
D
…8:48†