12 WAVE PATTERNS AND WAVE RESISTANCE. 
We might even carry this calculation a step further and divide the integration into two 
parts: (i) from @ = 0 to 6 = 35° 16’, (ii) from 8 = 35° 16’ to 6 = 90°; and we might associate 
the first part of R so calculated with the transverse waves of the pattern, and the second 
part with the diverging waves. On that basis we easily find that for (23) the transverse 
waves account for about 77 per cent. of the wave resistance, and the diverging waves for 
the remaining 23 per cent. 
The formula for R given in (22) was for a sine pattern (19), but the same expression 
holds for a similar cosine pattern. For instance, to compare with (23) we may take the case 
7 
2 
C= h| es PCS . 5 o .o 6 o (4a) 
7 
— 
Z 
which is like a term of (13) or (15) giving the effect of the curved sides of the model. For 
this pattern the corresponding wave resistance is 
= 
Ra npctit| co Odd = tempeh. ut oly! selec wao) 
0 
If we make a similar division into transverse waves and diverging waves we find that 
the former now account for a greater proportion of the total resistance, about 86 per cent. 
However, this is, no doubt, carrying the dissection too far; the wave pattern as a whole 
should be treated as a single system. 
13. As an example of (22) we may consider the model with parabolic lines for which 
the wave pattern was given in the expression (12). We have at once the wave resistance 
given by 
64 b? p c? 
R= ED coy («sec 8) — cos @ sin («I sec 0)}? cos? 0d0 . (26) 
On expanding this expression we have 
48 
32 B2 pc? | * 
R= SEE ane cos? 6 + x? 1? cos (2 « Lsec 6) 
TK 
“0 
— 2x 1cos @ sin (2 «1 sec 8) + cos? 0 cos (2 « 1 sec 4)} cos? 6d (27) 
And this leads to 
q sale G a. ae a iG a [ow Dicos (ZiT see8)id0 
na =I! i 8 sin (2 kU sec 6) d8 — (Ss ) [co 6 cos (2 « L sec 6) a | (28) 
The result has been put into this form for direct comparison with the expression for 
the waves given in (13), where they are analysed into four simple patterns, one for the 
bow, one for the stern, and two for the combined effects of the curved sides of the model. 
From this, and the calculations of the previous section, we can now identify the origin of 
each of the terms in the expression (28). The first term is the resistance due to the bow 
and stern patterns as if each existed alone, while the second term is similarly due to the 
curved sides calculated separately. The last three terms of (28) have been left in the form 
388 
