WAVE PATTERNS AND WAVE: RESISTANCE. 13 
of integrals; these integrals have been tabulated for numerical work, but we are only con- 
sidering here some general inferences. These three terms represent the mutual interference 
of the four simple patterns contained in (13), and it is obvious from the power of the factor 
(c?/g 1) whence they arise. The first of these represents the interference of bow and stern 
patterns, the second the interference of bow or stern with entrance or run, and the last 
term the mutual interference of the two patterns from the curved sides or, as one may say, 
the interference of entrance and run. It is these last three terms in (28) which have 
oscillating values, and so give rise to the well-known humps and hollows on the curve of 
wave resistance. 
14. We have seen that the wave pattern left behind by a ship can in general be put 
into the form given in (9); we have described this as sine and cosine patterns with known 
amplitude factors. The calculation of the quantity E — W can readily be extended to this 
general form and we obtain then the wave resistance for any general case. 
We first put (9) into the equivalent form 
7 
gs 
[=| tenn A cos 4, cos A in B+ By cbs A cos BY Wy sin A in B29 (29) 
0 
where A= x xsec 0, B=xysin@sec? 6, and the F’s are functions of 9, in general, and of 
the form of the ship and its speed. The calculation of R follows as in the simpler case 
of (19), and leads to the general result 
7 
2 
Ra bape | e+ RE + RE + Ry cos oat oo 6) ©) 
0 
The determination of the functions F is, of course, another matter. Approximate 
methods in use at present amount to replacing the ship by some equivalent distribution of 
sources and sinks; the functions F then usually appear as integrals taken over the surface 
of the ship, or over its longitudinal section for a first approximation. One of the out- 
standing problems of ship wave resistance is the improvement of methods for determining 
these functions; the line of attack open at present would seem to be by further steps of 
mathematical and numerical approximation, assisted and corrected by comparison with 
experimental results. 
The object of the present paper was to recall some of the elementary properties of 
wave patterns and their production by the mutual interference of simple plane waves, to 
illustrate these by examples from ship models, and further to emphasize the direct connection 
between the wave pattern and the wave resistance. 
389 
