Heaving and Pitching Motion of a Ship. 66) 
4. Consider a long prism, of rectangular cross-section and of breadth 
2b, immersed in water to a depth f and made to perform small vertical 
oscillations asin pt. For an approximate solution we suppose the 
motion to be two-dimensional and to be that due to a uniform distribution 
of sources, of density (pa/27) cos pt, over the immersed base of the prism 
at its mean depth f. The regular waves on the side x>0 are given, 
from (5), by 
C= (pale | cos {pt—x,(v—h)}dh 
—b 
=2ae-" sin (gb) cos (pi—Koz). . . - . . - (7) 
Hence, for the mean rate of radiation of energy per unit length of the 
prism, we have 
DE=(Hrypyaerips a= eae (es 9 2 2 0 « (&) 
If the wave-length 27/«, is large compared with the breadth of the prism, 
we have the simpler forms 
Ce Ge a ee Pe) 
ESA aMe awe No ee be ee LD) 
In the experiments by Schuler (1936) a rectangular prism was used 
and the amplitude of the waves and other quantities measured directly. 
Schuler obtained the expression (7) by an indirect energy method sug- 
gested by Prandtl, and it was contrasted with the source theory of the 
effect ; however, we have seen that it follows from assuming a uniform 
distribution of sources over the base of the prism. ‘The interesting point 
is that the experimental results agree reasonably well with the expression 
(7) for periods such that the wave-length 27/«, is greater than the breadth 
26 of the prism. 
5. We now apply these results to the heaving of a ship in still water. 
We may, as in similar cases, treat the motion as two-dimensional in the 
first instance, an approximation which may be supported by the experi- 
ments of Browne, Moullin and Perkins. These authors also give an 
approximate formula for the added mass of a ship of normal form in 
vertical heaving motion ; this is given as 0-95pb7/, where p is the density 
of water, 2b the maximum beam and / the total length of the ship. 
We take an example from recent work by Kent and Cutland (1941), 
carried out on models at the National Physical Laboratory. The data 
for the corresponding ship are: length 400 ft., beam 55 ft., draught 
24 ft., displacement 11,332 tons, natural heaving period 7-42 sec. From 
the formula just given the added mass comes out as 8200 tons; thus 
the total effective mass M’ in equation (2) is about 20,000 tons. It is 
of interest to check this result in a different way. If 27/o is the period 
when damping is neglected, we have the relation given in (3). The 
change in period due to damping is relatively small, so we may use the 
recorded period ; further, estimating the water plane area S as 17,600 
sq. ft., we obtain from (3) a value for M’ of about 20,000 tons. 
495 
