284 Dr. T. H. Havelock. [ Apr. 1, 
§ 4. Interference of Bow and Stern Wave-trains. 
The cause of the undulations in the resistance curves was shown by 
W. Froude to be interference of the wave system produced by the bow (or 
entrance) with that arising at the stern (or run). His experiments on the 
effect of introducing a parallel middle body between entrance and run 
confirmed his theory, which may be stated briefly. Let the wave-making 
features of the bow produce transverse waves which would have at 
a breadth 6 an amplitude a; owing to the spreading out of the transverse 
waves they will be equivalent to simple waves at the stern of smaller 
amplitude ka, at the same breadth b. Let a’ be the amplitude there of the 
waves produced by the stern. Then in the rear of the ship we suppose there 
are simple waves of amplitude ka superposed upon others of equal wave- 
length of amplitude a’. At certain velocities the crests of the two systems 
coincide in position, giving rise to a hump on the resistance curve; and at 
intermediate velocities there are hollows on the curve owing to the crests of 
one system coinciding with the troughs of the other. 
In developing a form for the resistance, subsequent writers have generally 
taken R proportional to an expression of the form a?+a7?-+ 2kaa’ cos (mgL/v°), 
where L is the length of the ship. This means that the bow is supposed to 
initiate a system of waves with a first crest at a short distance behind the 
bow, and that similarly the stern waves have their first crest shortly after 
the stern; the length mL is the distance between these two crests, and is 
called the wave-making length of the ship. The determination of a value 
for m appears to be doubtful, but from interference effects it is said to vary 
for different ships between the values 1 and 1:2. 
It has seemed desirable here to follow more closely the point of view in 
W. Froude’s original paper already quoted.* We regard the entrance of the 
ship as forming transverse waves with their first crest shortly aft of the bow, 
and the run of the ship as forming waves with their first trough in the 
vicinity of the middle of the run. It is suggested that this distance between 
first crest and first trough, in practice found to be about 0-9L, should be 
taken as the “wave-making distance”; the cosine term in the formula 
is then prefixed by a minus sign instead of a positive sign. We return to 
this point later; we first work out a definite simple illustration in “two- 
dimensional waves,’ and then build up a more complete formula for 
comparison with experiment. With the same notation as in § 1, let the 
pressure system be given by 
1 Pa? P22? ie 
a (aC ee) 
* W. Froude, loc. cit. ante, p. 83. 
42 
