282 Dr. T. H. Havelock. [Apr. 1, 
begins to fall off. This effect is general in residuary resistance curves; we 
see that it is really an interference effect, the character of the curve being 
due to the mutual interference of the wave-making elements of the pressure 
system. Superposed on the mean curve we have a further interference effect 
due to the combination of two systems, the bow and stern systems. 
From Table I we infer that the mean curve agrees well with the calculated 
values R from about 18 knots upwards, but at the lower speeds the values of 
R are much too small; this suggests the addition of a term to represent the 
effect of the diverging waves. 
§ 3. Diverging Wave System. 
In the example considered above, the calculated values of R are much too 
small at the lower velocities. This might have been expected; for we 
obtained (12) by the consideration of line-waves on the surface, that is waves 
with crests of uniform height along parallel infinite lines. But the model 
experiments correspond more to a point disturbance travelling over the 
surface, with the formation of diverging waves as well as transverse waves. 
In fact, W. Froude* infers from his experimental curves that the residuary 
resistance at the lower velocities is chiefly due to the diverging wave system, 
on account of the absence of undulations; for the latter signify interference 
of the transverse systems initiated by the bow and stern, and these become 
very important at the higher velocities. 
We have to add to (12) a term representing the diverging waves; the 
comparison in Table I suggests for this a term of the same type, e~#V’/V 
with V’” much smaller than the corresponding velocity V’ for the transverse 
waves. With the data at our disposal we might then determine the various 
constants so as to obtain the closest fit possible ; however, we can make the 
process appear less artificial by the following considerations. We know that 
the wave pattern produced by a travelling point source consists of a system 
of transverse waves and a system of diverging waves, the whole pattern 
being contained with two radial lines making angles of about 19° 28’ with 
the direction of motion; a fuller investigation of the effects produced by 
a diffused source must be left over at present. In applying energy con- 
siderations as in the previous sections, the usual method is to suppose that 
the transverse waves form on the average a regular wave-train of uniform 
amplitude and uniform breadth; using the same approximation for the 
diverging waves we suppose that these form on the average a regular wave- 
train on each side, with the crests inclined at some angle 0 to the direction 
* W. Froude, ‘Inst. Nav. Arch. Trans.,’ vol. 18, p. 86, 1877. 
40 
