Environmental Engineering Reference
In-Depth Information
Rearranging gives us the phase velocity of the primary wave:
r
g
k tanh k d
c ¼ T ¼ k ¼
ð
Þ
Consequently, a harmonic primary wave in water with a limited depth can be unambigu-
ously defined by one of the groups of three values [H,
l
,d]or[H,T,d].The
approximation tanh (k d) 1 is valid for d
>l
/2, that is k d
>p
. Therefore, for
deep water we get
r
g l
2 p
r
g
k
¼ g kandc ¼ T ¼ k ¼
2
v
¼
or
g T 2
2 p
c ¼ T ¼
g T
2 p
l ¼
and
as well.
The Airy wave in deep water is unambiguously defined by one of the pairs of values
[H, l ] or [H, T].
For longer waves in very shallow water, that is d l or k d ! 0, the approximation
tanh (k d) k d gives us
p
g d
c ¼
that is the phase velocity only depends on the depth of water and no longer on the
wavelength.
All the variables of the wave field can be derived from the velocity potential of the
primary wave. Taking into account the dispersion equation, we get the following for the
components of the velocity field in the wave:
u ¼ @F
ð
x
;
z
;
t
Þ
H
2
cosh k
½
ð
z þ d
Þ
¼ v
cos k x v t
ð
Þ
@
x
sinh k d
ð
Þ
w ¼ @F
ð
x
;
z
;
t
Þ
H
2
sinh k
½
ð
z þ d
Þ
¼ v
sin k x v t
ð
Þ
@
z
sinh k d
ð
Þ
Using these velocity components it is easy to show that the water particles pursue closed
orbital trajectories during the motion of the wave (Figure 2.20: ellipses in a limited depth of
water which gradually change to circles in deep water). There is no transportation of mass;
the Airy wave transports energy only.
When calculating the acceleration field in the Airy wave, only the local component of
the substantial acceleration is taken into account because assuming a low wave
steepness means that the convective acceleration is negligible with respect to the
local acceleration. The following then applies:
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