T. von Karman 
Froude divided the total resistance into “frictional” and “residual” resistance. The 
computation of frictional resistance was based on experiments with towed flat planks and 
was assumed clearly dependent on viscosity and the roughness of the surface. Thus the 
frictional coefficient for a surface of given roughness was assumed to be a given function 
of the Reynolds number. 
On the other hand the residual-resistance coefficient was assumed to be unaffected by 
viscosity and to satisfy the similarity law for an incompressible, nonviscous fluid with a 
free surface. From this similarity law it follows that the coefficient of the residual resist- 
ance is a function of the Froude number F only. 
We have to note that in fact the residual resistance includes not only the wave-making 
and eddy-making resistances, but all interaction effects and especially the difference 
between the true frictional resistance of the ship and the frictional resistance of a flat 
plank of the same surface. Thus the residual resistance cannot be expected to be actually 
free of viscosity effects as was assumed by Froude and likewise in the conventional 
practice in naval architecture. 
Sir Thomas Havelock is one of those rare combinations of an extremely astute math- 
ematician with a feeling and understanding for the application of mathematics to practical 
ends. He is the major contributor to the theoretical hydrodynamics involved in the calcula- 
tion of the resistance due to the generation of waves by moving bodies, either on the 
surface or submerged. Over the years, since 1908, he has published in excess of 40 
papers in the field, and it is only conjecture how many publications in the field were insti- 
gated by his findings and contributions. 
One of his major contributions, and, interestingly, the first one that he made to the 
field, was in 1908 in which he extended the work of Lord Kelvin concerning the waves pro- 
duced by traveling point disturbances. Essentially, he considered the waves generated by 
a disturbance as a simple group or aggregate of wave trains expressed by a Fourier integral. 
He was able to obtain the solution for the wave group for any depth of the fluid rather than 
the infinite depth of Kelvin’s theory. He also found that the significance of finite depth 
was to introduce a critical velocity above which the wave crests emanating from the source 
change their character from convex to concave. At the critical velocity, the transverse and 
divergent waves coincide and the resultant wave is normal to the path of the disturbance. 
This finding is completely analogous to a situation which exists in high-speed aerodynam- 
ics. As early as then, although he didn’t realize it, he had developed an insight into the 
analogy between the shallow depth water waves generated by bodies and their shock wave 
patterns at supersonic speeds, which underlies modern day water table experiments. 
Evidently motivated by his solution and its application to ship resistance calculations, 
Sir Thomas Havelock embarked on concentrated efforts in this field. Many of his contribu- 
tions over the next 10 years (1908 to 1918), oddly enough, were somewhat semiempirical in 
nature. This is unusual for a theoretical mathematician and points up his interest in the 
practical application of mathematics. In the series of papers during that period, published 
in the Proceedings of the Royal Society, he attempted to obtain an analytical formulation 
for calculating the family of curves which are indicative of the “residual resistance” of 
ships as postulated by Froude. Initially, he considered a transverse linear pressure dis- 
turbance traveling uniformly over a water surface of infinite depth and arrived at a relation 
involving three universal constants (determined from experiment) and three parameters 
which depend upon the ship form. A major problem involved with the semiempirical 
vill 
