Environmental Engineering Reference
In-Depth Information
Fig. 2.22 Wave pro
les
-
Airy and third-order Stokes (H¼10m, d¼30m,
l¼150m)
H
2
1
2
p
H
z
x
ðÞ¼
;
cos k
x
v
t
ð
Þ þ
cos 2
½
ð
k
x
v
t
Þ
l
)
2
3
8
p
H
l
þ
cos 3
½
ð
k
x
v
t
Þ
This expression includes the influence of the finite wave steepness, the maximum value of
which for deep water is (H/
)
max
¼
1/7.
With Stokes waves it is important to note that - as with all waves of finite steepness - the
substantial acceleration (Du/dt) must be calculated and - as with the Airy wave - not
just its local component (
@
u
=@
t):
l
Du
dt
¼
@
u
t
þ
u
@
u
x
þ
w
@
u
@
@
@
z
Another non-linear theory with practical significance is the stream function wave theory of
Dean [19], which covers a wide range of applications. The boundary conditions at the free
water surface are completely satisfied by this theory but only partly by Stokes waves [18].
Elliptical waves are used in areas of shallow water where Stokes waves become
inaccurate. As a global estimate, this is the case when d/
l
0.1.
The Stokes and elliptical waves, which are oscillatory, are joined by the solitary wave
for the surf zone. This purely translational wave consists of a single infinitely long
wave crest that lies entirely above the still water level and whose two flanks slope
asymptotically towards this. The profile of the solitary wave is defined as
(
"
#
)
1
cosh
2
r
3
H
4
d
3
z
ðÞ¼
H
;
ð
x
c
t
Þ
x
with the phase velocity
s
H
d
c
¼
g
d
1
þ
