Practical Ship Hydrodynamics

Numerical example for BEM 255
Let Ev DEv⊲Ex⊳ be any velocity relative to the Oxyz system and Ev DEv⊲x⊳ the
velocity relative to the Oxyz system where Ex and Ex describe the same point.
Then the velocities transform:
Ev DEv CE˛ ðEv C ⊲E˛t ðEx CEut⊳
Ev DEv E˛ ðEv ⊲E˛t ðEx CEut⊳
The differential operators rx and rx transform:
rx Df∂/∂x, ∂/∂y, ∂/∂zg T Drx CE˛ ðrx
rx Df∂/∂x,∂/∂y,∂/∂zg T Drx E˛ ðrx
Using a three-dimensional truncated Taylor expansion, a scalar function transforms
from one coordinate system into the other:
f⊲Ex⊳ D f⊲Ex⊳ C ⊲E˛ ðEx CEu⊳rxf⊲Ex⊳
f⊲Ex⊳ D f⊲Ex⊳ ⊲E˛ ðEx CEu⊳rxf⊲Ex⊳
Correspondingly we write:
rxf⊲Ex⊳ Drxf⊲Ex⊳ C ⊲⊲E˛ ðEx CEu⊳rx⊳rxf⊲Ex⊳
rxf⊲Ex⊳ Drxf⊲Ex⊳ ⊲⊲E˛ ðEx CEu⊳rx⊳rxf⊲Ex⊳
A perturbation formulation for the potential is used:
total D ⊲0⊳ C ⊲1⊳ C ⊲2⊳ CÐÐÐ
⊲0⊳ is the part of the potential which is independent of the wave amplitude
h. It is the solution of the steady wave resistance problem described in the
previous section (where it was denoted by just ). ⊲1⊳ is proportional to h,
⊲2⊳ proportional to h 2 etc. Within a theory of first order (linearized theory),
terms proportional to h 2 or higher powers of h are neglected. For reasons of
simplicity, the equality sign is used here to denote equality of low-order terms
only, i.e. A D B means A D B C O⊲h 2 ⊳.
We describe both the z-component of the free surface and the potential in a
first-order formulation. ⊲1⊳ and ⊲1⊳ are time harmonic with ωe, the frequency
of encounter:
total ⊲x, y, z; t⊳ D ⊲0⊳ ⊲x, y, z⊳ C ⊲1⊳ ⊲x,y,z; t⊳
D ⊲0⊳ ⊲x, y, z⊳ C Re⊲ O ⊲1⊳ ⊲x, y, z⊳e iωet
⊳
total ⊲0⊳ ⊲1⊳
⊲x, y; t⊳ D ⊲x, y⊳ C ⊲x, y; t⊳
D ⊲0⊳ ⊲x, y⊳ C Re⊲O ⊲1⊳ ⊲x, y⊳e iωet ⊳
Correspondingly the symbol O is used for the complex amplitudes of all other
first-order quantities, such as motions, forces, pressures etc.
The superposition principle can be used within a linearized theory. Therefore
the radiation problems for all 6 degrees of freedom of the rigid-body motions