Page 503 - Practical Design Ships and Floating Structures

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72 in Dahlquist( 1974)). In order to study the wave systems behind the source, we study the change of
the phase of the integrand in expression (4). Due to symmetry only positive y - v needs to be studied.
The phase function k(lx - 41 cos0, + (y - v) sin 0,) increases monotonically with increasing k and will
give no wave contribution far away. If a = arctan((y - v)/~x - 51) is larger than some value p , the
phase function k(lx -<lcosO0 - (y- q)sine0) will decrease monotonically with increasing k and will
not give any wave contribution. If a is less than B, this phase function will have one local maximum
and/or minimum. Thus p determines the wave angle behind the source. For subcritical case, the phase
function will have both local minimum and maximum, which corresponds to respectively the
transverse and divergent wave systems. For supercritical case, it has only one local maximum which
corresponds to the divergent wave systems. Fig 1 shows the relationship between wave angle and Fh .
If Fh <- 0.6, the wave angle is almost the same as in deep water (19'28'), and approaches 90' when
reaching the critical speed. It then decreases with increasing speed.
Michell's thin ship theory is used to determine the densities of the source distribution on the
U
centreplane So of the ship. This gives the velocity potential 4 = - jjdtd< -
') G(x, y. r; 6.0, <)
2x so at
due to the presence of the ship. Here y = f(x, z) descirbes the hull surface for y > 0 . By expressing
the hydrodynamic pressure as p = PUI$~ , the corresponding longitudinal force 5, vertical force
4 and pitch moment 4 about the y - axis are:

Here S, is the mean wetted ship hull surface, p is the mass density of water and n3 is the z-
component of the normal vector to the body surface. Positive normal direction is into the body. We can
use the pressure at the centreplane in the integration of 4 and F5, but must integrate over s, in order
to properly account for area. By decomposing the Green's function into an odd and even function in
x - 6 , we can see that the only non-vanishing contribution to F; is coming from the odd term. This
gives:

where P + iQ = jjdxd&(x,z)cosh(k(l+ z))exp(ikxcose). This is a well known result (Lunde (1951)).
S,,
We then have two equations to calculate the wave resistance, either Eq. (5) or Eq. (8). This can be used
to check the accuracy of the results.

3 RESULTS FOR WIGLEY HULL
Wigley's (1 942) parabolic hull is used in the case studies. The hull surface for y > 0 is given by: