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

…2† !
DI ˆÿi
2g
‡ i 2!
g
…2† …2† …2†
D ˆ DI ‡ DD and ˆ !2
…1†
D
!
…1†
@ I ÿ 2 @z @z
2 …1†
@ I
…1† …1†
r I r D
…2† !
DD ˆÿi
2g
…1†
D
2 …1†
@ D
ÿ i !
2g
!
…1†
@ D ÿ
@z2 @z
…1†
I
g
!
…1†
@ D ÿ 2 @z @z
2 …1†
@ D
‡ i 2! …1†
…r D
g †2
…8:49†
…8:50†
…8:51†
The second-order boundary condition can be thought of as identical to the ®rst-order
problem, but with a speci®ed pressure applied over the entire free surface, of value .
Now there is no a priori reason why such a pressure distribution should give rise to
outgoing waves as in the ®rst-order problem, and so the usual radiation condition
is not applicable. The conventional procedure is to split the second-order wave into
two parts, one the `locked' wave, in phase with the ®rst-order wave, and the other
the `free' wave, which is like the ®rst-order wave but at twice the frequency, and
with an appropriate wavenumber obtained from the dispersion relation. For further
details of the theory, see Clark et al. 69 Figure 8.12 shows results for the second-order
wave elevation around a circular cylinder, obtained by Clark et al. Although not
shown, good agreement has been obtained with predictions made by boundary
Incident
wave
Three-dimensional effects in surface waves 269
1.0
0.5
0.0
–1.0
–1.5
0 0.5 0.0 0.5
Real {ηD /(H
0.0
–0.5
1.0
–0.5
1.0
0.0
–0.5
1.0
–1.0
1.5
–0.5
2.0
2.0
–0.5
–1.5
2.0
0.5
2.5
1.0
0.0
–1.0
0.0
1.0
1.5
0.0
–0.5
–1.5
0.5
0.0
–1.0 –0.5
–1.0
0.5
0.0
0.5
1.0
–1.0 0.0
0.5
0.0
2 (2)
/4a)}
Imaginary {ηD /(H 2 (2)
/4a)}
Fig. 8.12 Second order wave elevations around cylinder ± real and imaginary parts Clark et al. 69