161 



IIVHRonVX XMICS I\ SHIP DESIGN 



Sec. 48.5 



Fig. 4S.C Expijvnatory Sketch for Surface Slopes of a Trochoidai, Wave 



intervals along the horizontal plane between the 

 crest and the trough, in a direction normal to 

 the crest and trough lines. 



The expression for the wave slope j'(zeta) is 

 based upon the fact that a normal to the wave 

 surface at any surface-particle point passes 

 through a point at a distance Rrc above the 

 instantaneous orbit center of the particle. In 

 diagram 1 of Fig. O.G on page 163 of Volume I, 

 the line P,Ci is normal to the wave surface at P, , 

 whence the wave slope f equals the angle PiCiOi . 

 In the diagram of Fig. 48. C the line Vid is normal 

 to the wave surface at Vt , and the wa\e slope ^^ 

 is represented by either the angle VidOi or the 

 angle MiPiN* . By simple geometry, indicated 

 in Fig. 48.C, 



P4M. 



Ki, sin 04 



(48.i) 



_ J4'>'< _ /l.s a 111 P4 



lan u f^Q^ _,_ Q^^j^ i^^^ _ ^^ ^^^ g 



where 6 is measured from the top center of the 

 orbit and cos in the lower quadrants is negative. 

 The maximum wave slope occurs when the 

 Hurfacc-parlicle orbit radius P3O3 or lis in the 

 diagram of Fig. 48. C lies normal to the line PaC'a , 

 whence 



fii.. = sin 



■(c^) = »-'Ct) <--> 



Since lia is /»«-/2 and Rhc = Lw/2-k, 



f M.. = sin' 



ha 

 2_ 



2ir 



/.„ 



M8.iia) 



as indicated at the right on Figs. 48. B and 48. C. 



It is to be noted particularly that the slope 

 indicated by Eq. (48. i) is that at the position of a 

 surface particle lying at the extremity of a radius 

 72s which lies at the angular orbit position 6. 

 For a w-ave traveling to the left, as in Figs. 48. B 

 and 48. C, this angle is reckoned counter-clockwise 

 from the top center, starting at a crest at the left 

 or advancing end of a wave. The horizontal 

 distance from the crest to the position of tlie 

 particle is therefore the offset distance of the 

 orbit center from the crest, namely (3()0 — 

 0/36O)(Li,), modified by the offset position in 

 the orbit, namelj' Rs sin 9. 



Thus for Of = 2ir/3 or 120 (l(•^^ with the surface 

 particle Ij'ing at Pi , 



VM* Rs sin e« 



tan f = 



C^O, -F O^M, Rrc - Rs cos fl« 



J2s(0.8(iC)) 



0.86C.«, 



Rrc - /es(-0.5) Rhc + 0.5«a 



The value of (360 - ej:im)Lw is L,r/3 and 

 that of Rs sin 0, = O.SGC)/?., . The latter is meas- 

 ured toward the crest, so the particle position 

 desired, reckoned from the crest in the direction 

 of advance of the wave, is 



To determine the niaxininn) slope of the t)12-ft 

 wave of Sec. 48.4, for the .\B(" ship of Part 4, 

 it is nece.s.sary to know the orbital radius Rs of 

 a surfaic particle and the radius of the rolling 

 circle A';,,- by which the trochoidai surface is 



