Actions on wind turbines - Concrete Structures for Wind Turbines - page 37

2

u ¼ @

u
@ t ¼ @

F H

2

cosh k ½ ð z þ d Þ

2

_

t ¼ v

sin k x v t ð Þ

@ x @ sinh k d ð Þ

w ¼ @

w
@ t ¼ @

2

F ð x ; z ; t Þ H

2

sinh k ½ ð z þ d Þ

2

_

t ¼v

cos k x v t ð Þ

@ z @ sinh k d ð Þ

The pressure field in the Airy wave is obtained from the linearised Bernoulli equation for
unsteady flows:

p þ r @F
@

p ¼ p p 0 ¼r g z r @F
@ t þ r g z ¼ p 0

! D

t
Accordingly, the difference between this and atmospheric pressure (p 0 ) consists of a
hydrostatic ( r g z) and a hydrodynamic (unsteady) component:

p inst ¼r @F

@ t

Substituting the equation for the velocity potential in this relationship gives us

p inst ¼r @F

ð ; ; Þ

cosh k

½ ð

z þ d

Þ

x z t H

2

¼ r g cos k x v t ð Þ

@ t cosh k d ð Þ

Easy to recognise here is the fact that the dynamic pressure is proportional to the wave
profile and subsides with the depth ordinate (z < 0):

cosh k ½ ð z þ d Þ

p inst ¼ r g z ðÞ x ;

cosh k d ð Þ

When designing marine structures it is often important to consider the processes involved
with the superposition of a current (e.g. tidal or river current) and a primary wave. This
results in a change to the wavelength and the wave height. Some important physical
relationships can be derived from the simplest case of a situation in deep water.
Let U C be the flow velocity (U C >

0 in the direction of wave propagation) and index 0
indicate the undisturbed waves. Using a fixed system of coordinates, the condition that
the frequency remains constant upon superposition then gives us the following for the
change in wavelength from l 0 to l :

2

l
l 0 ¼

k 0
k ¼

ð 1 þ x Þ

4

where

r

1 þ 4

U C
c 0

g T
2 p

x ¼ and c 0 ¼

We obtain

r

g l
2 p

2

ð 1 þ x Þ

c ¼ T ¼

1 þ x
2

2

v ¼ g k 0 ¼ g k and ¼ c 0

4

for the modified dispersion equation and the phase velocity.