Practical Ship Hydrodynamics

256 Practical Ship Hydrodynamics
and the diffraction problem are solved separately. The total solution is a linear
combination of the solutions for each independent problem.
The harmonic potential ⊲1⊳ is divided into the potential of the incident wave
w , the diffraction potential d , and 6 radiation potentials:
⊲1⊳ D d C w C
6�
iD1
i ui
It is convenient to decompose w and d into symmetrical and antisymmetrical
parts to take advantage of the (usual) geometrical symmetry:
Thus:
w ⊲x, y, z⊳ D
w w
⊲x, y, z⊳ C ⊲x,
� ��
2
w
y, z⊳ ⊲x, y, z⊳
C
� �
w
⊲x,
��
2
y, z⊳
�
w,s
w,a
d D d,s C d,a D 7 C 8
⊲1⊳ D w,s C w,a C
6�
iD1
i ui C 7 C 8
The conditions satisfied by the steady flow potential
from section 7.3 without further comment.
⊲0⊳ are repeated here
The particle acceleration in the steady flow is: Ea ⊲0⊳ D ⊲r ⊲0⊳r⊳r ⊲0⊳
We define an acceleration vector Ea g Ea g DEa ⊲0⊳ Cf0, 0,ggT For convenience I introduce an abbreviation: B D 1
a g
3
∂
∂z ⊲r ⊲0⊳ Ea g ⊳
In the whole fluid domain: 1 ⊲0⊳ D 0
At the steady free surface: r ⊲0⊳ Ea g D 0
1
2 ⊲r ⊲0⊳ ⊳ 2 C g ⊲0⊳ D 1
2 V2
On the body surface: En⊲Ex⊳r ⊲0⊳ ⊲Ex⊳ D 0
Also suitable radiation and decay conditions are observed.
The linearized potential of the incident wave on water of infinite depth is
expressed in the inertial system:
w D Re
� igh
ω e ik⊲x cos y sin ⊳ kz e iωet
�
D Re⊲ O w e iωet
⊳
ω D p gk is the frequency of the incident wave, ωe Djω kV cos j the
frequency of encounter. k is the wave number. The derivation of the expression
for w assumes a linearization around z D 0. The same formula will be
used now in the seakeeping computations, although the average boundary is
at the steady wave elevation, i.e. different near the ship. This may be an
inconsistency, but the diffraction potential should compensate this ‘error’.
We write the complex amplitude of the incident wave as:
O w D igh
ω eExEd with Ed Df ik cos , ik sin , kzg T