Wave Resistance Theory and Its 
Application to Ship Problems 
By T. H. Havetock, Visiror? 
It is now just over fifty years since the first 
mathematical analysis was made of the wave re- 
sistance of a ship form, and during the latter half 
of that period there has been a considerable out- 
put of work, both theoretical and experimental. 
It is impossible to give any adequate survey of 
this work here, and fortunately it is unnecessary 
to make the attempt; there are excellent sum- 
maries which have been published from time to 
time, and in particular I would refer, for a com- 
prehensive account with references, to Wigley’s 
recent paper, ‘“The Present Position of the Cal- 
culation of Wave Resistance’ (L’Association 
Technique Maritime et Aeronautique, Paris, 
1949). 
In the following notes I deal first with a solid 
body which is completely submerged; a short 
descriptive account of one method of developing 
the mathematical theory is followed by some re- 
cent results on motion in a curved path and on ac- 
celerated motion. The second section deals with 
floating bodies, or surface ships. Reference is 
made to the need for improving the approximate 
theory for models of fine form and extending its 
range of application; and a short account is 
given of some attempts, dealing in particular with 
(1) models of fuller form, (2) models of non- 
mathematical form and methods of approximate 
calculation, (3) the inclusion of the effects of vis- 
cosity and the possible interaction between fric- 
tional resistance and wave resistance. 
Submerged Bodies. Consider a solid body 
wholly submerged in water and moving in a hori- 
zontal line with given velocity. Assuming the 
water to be frictionless, the fluid motion is speci- 
fied by a velocity potential ¢ satisfying given 
boundary conditions: (1) the normal fluid veloc- 
ity on the solid is equal to the normal velocity of 
the solid at each point, (2) the pressure is con- 
1 Paper presented at meetings of the New England Section, August 
28, 1950, and of the Chesapeake Section, September 7, 1950. 
2 Kings College, Durham University, Newcastle-upon-Tyne. 
stant at the free surface of the water, (3) for deep 
water the velocity diminishes to zero with in- 
creasing depth. We may also impose a condition 
for the motion far in advance of the solid, such 
as, for instance, to insure that in the usual phrase 
the solid is advancing into still water. In gen- 
eral, this problem has only been attacked by some 
method of continued approximation. We may 
suppose that the wave motion at the surface is a 
relatively small effect, and we take ¢) for the 
velocity potential as if the solid were moving in 
an infinite liquid, and satisfying condition (1). 
We then add a correcting potential ¢, so that ¢) + 
¢; satisfies condition (2) at the free surface; and 
then a potential ¢. to maintain condition (1), and 
soon. Thus we may picture the solution ¢ as an 
infinite series do + ¢1 + ¢2 + .... We may as- 
sume this process to be convergent; but the ex- 
pression of it in any particular mathematical form 
would involve consideration of convergence and 
of the uniqueness of the solution so obtained. It 
has only been possible to carry out this process in 
any detail for solids of simple form, such as a cir- 
cular cylinder, sphere, or spheroid. In fact, for 
most cases it has not been carried further than 
the first three terms; while for bodies of ship- 
shape form nearly all the results have meantime 
been built up on the first two terms—denoted 
here by ¢@) + ¢;. Assume now that we know the 
first function ¢o, giving the solution if we neglect 
the wave motion completely, and consider the 
determination of the next function ¢,. There 
are various methods available; the one I wish to 
outline may not be the best from a mathematical 
point of view, but it has some advantages for dc- 
scriptive purposes. The method is one which was 
used long ago by Kelvin for the waves produced 
by a pressure disturbance traveling over the sur- 
face of the water. Consider for a moment the 
classical problem of the traveling pressure point. 
Instead of treating this directly as a continuous 
process, we may regard the motion as the limit of 
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