Spc. ^SA 



WIND-WAVE AND SHIP-WAVE DATA 



lf)l 



of 19.47 deg for a traveling pressure point, as 

 described in Sec. 10.6 on page 174 of Volume I 

 [Hovgaard, W., INA, 1909, Vol. 51, table facing 

 p. 260 and PL XXIV; Taylor, D. W., S and P, 

 1943, pp. 27-28]. 



48.3 Hogner's Contribution to the Kelvin 

 Wave System. Owing to an apparent misunder- 

 standing of the exact nature of a wave-pattern 

 diagram which was published by Ijord Kelvin 

 in the papers referenced in Sec. 10.5 on page 170 

 of Volume I, tlie planform illustrated in Fig. 10. B 

 of that section has for the past half-century been 

 described in many text and reference books as 

 the exact crest pattern developed by him. E. Hog- 

 ner brought out, in the 1920's, the fact that the 

 figure given by Kelvin showed only "the envelope 

 curves of the crests of the two-dimensional waves 

 from which he has constructed his three-dimen- 

 sional waves" ["A Contribution to the Theory of 

 Ship Waves," Arkiv for Matematik, Astronomi 

 och Fysik, Stockholm, 1922-1923, Vol. 17, paper 

 12, footnote on p. 42]. Kelvin's mathematical 

 formulas actually showed a phase difference at 

 the boundary planes (marked "Line of Crest 

 Intersections" in Fig. 10. B of Volume I) but 

 this was apparently overlooked by most of those 

 who worked in this field in the 1900's and 1910's, 

 until Hogner brought it to fight. 



Depending upon the assumptions made and 

 the approximations employed, Kelvin's earlier 

 theory gives a phase difference, at the boundary 

 planes lying at angles of 19.47 deg to the direction 

 of motion of the traveling pressure point, of 



^Line of Crest Interaections Qs in Fiq lO.B 



Broken Lines Depict Pattern of Fio. lO.B; 



Full Lines Depict Norninol Crest Pattern 

 Qccordinq to E. Hoqn 



Fig. 48.A Modification of Kelvin Wave System 

 According to E. Hogneb 



The Kelvin system, corresponding to that in Fig. lO.B of 

 Volume I, is shown in broken lines. The modification 

 according to Hogner, with the transverse waves ahead of 

 the diverging waves, is indicated in solid lines. 



L„,/4 between the crests of the transverse waves 

 and those of the divergent waves. The crests of 

 the former system lead the others, just as if there 

 were a first transverse crest formed at a position 

 Lh^/4 ahead of the pressure point. A planform 

 diagram of such a pattern, showing the variations 

 from the Kelvin wave system of Fig. 10. B of this 

 book, is given by Hogner, from which Fig. 48. A 

 is adapted [Proc. First Int. Congr. Appl. Mech., 

 Delft, 1924, Fig. 5, p. 149]. In a reworking of the 

 entire analytic procedure, described by Hogner 

 in the paper listed as the first reference in this 

 section, he shows that the phase difference at the 

 boundary planes is actually Lwfi, indicated in 

 detail by his diagram in Fig. 17 on page 46 of 

 that reference. 



These same phase differences, although ex- 

 pressed as 27r/4 and 27r/3, are derived by J. K. 

 Lunde [SNAME, 1951, p. 71]. In Fig. 6 of that 

 reference he gives a diagram corresponding to 

 the solid lines of Fig. 48.A. 



E. Hogner carries his analytic procedure to 

 the point where he predicts the nature of waves 

 formed in the area beyond the boundary planes. 

 He also substantiates his analytic derivation by 

 photographs which reveal the patterns actually 

 formed on the surface of the water. 



48.4 Summary of the Trochoidal-Wave The- 

 ory. The notes which follow, adapted from G. C. 

 Manning [PNA, 1939, Vol. II, pp. 6-7], summarize 

 the relationships described and presented in 

 Chap. 9. They are illustrated, as far as practica- 

 ble, in the definition sketch of Fig. 48.B, which 

 is the elevation of a wave having a steepness 

 ratio hw/L^ of 1:7. 



The principal assumptions of the trochoidal- 

 wave theory, satisfying the requirements of 

 equilibrium, continuity, and uniformity of pres- 

 sure, are: 



(a) The motion is 2-diml, around circular orbits 

 in a vertical plane 



(b) The liquid particles revolve in circular orbits 

 with uniform angular velocity u (omega) 



(c) The liquid particles at the crest move in the 

 same direction as the wave is advancing 



(d) There are equidifferent phase angles d6{theta,) 

 = u dt between successive particles whose orbit 

 centers lie at equidifferent distances dL^r along 

 a given horizontal line 



(e) Liquid particles whose orbit centers lie in 

 the same vertical line rotate about those centers 

 in the same phase 



