Practical Ship Hydrodynamics

226 Practical Ship Hydrodynamics
The derivation used � ⊲t 2 C a 2 ⊳ 3/2 dt D t/⊲a 2p t 2 C a 2 ⊳.For1x 2 C 1y 2 −
j1z1jj1z2j or 1y 2 C 1z 2 1 − 1x2 special formulae are used. Bertram (1992)
gives details and expressions for higher derivatives.
6.4 Dipole elements
6.4.1 Point dipole
The dipole (or doublet) is the limit of a source and sink of equal strength
brought together along some direction (usually x) keeping the product of
distance and source strength constant. The result is formally the same as differentiating
the source potential in the required direction. The strength of a dipole
is usually denoted by m. Again,rdenotes the distance between field point Ex
and the dipole at Exd. We consider a dipole in the x direction here. We define
1Ex DEx Exd.
1. Two-dimensional case
The potential and derivatives for a dipole in the x direction are:
D m
1x 2
2 r
x D m
2 r 2
z D 2 1z
xx D
�
r 2
2 1x
r 2
6 C 8 1x2
r 2
�
Ð r 2
xz D 2 1z Ð x C 1x Ð z
r 2
The streamlines created by this dipole are circles, (Fig. 6.9).
2. Three-dimensional case
The three-dimensional point dipole in the x direction is correspondingly
given by:
D
x D
m
1x 3
4 r
m
4 r 3
y D 3 1y
r 2
z D 3 1z
r 2
3 1x
r 2