CALCULATION OF HYDRODYNAMIC FORCES BY STRIP THEORY 



339 



Fig. 14 Sketch illustrating notation used in Appendix 



iiated </)(,„, 



= -.-..OS a 



(13) 



The sccoikI term of t-quatioii (12) is tlic p<itciitial due to 

 body-wave iiiteraetion, (ptw 



r- 



= - — ' '■'"' r-'«/^ fos a C(js — (,r - ct) (14) 

 The potential due to \va\'e motion alone is <f>^ 



4>u 



hce-"'"'^ eos — (x — ct) 



A 



(15 



The total veloeity potential is the sum of eciuations do), 

 (14) and (15) 



4> ^ <i>ti'ii + <t>t>u' + <Pii' (II') 



Exciting Forces 



Attention will now be eoncentrated on the second and 

 third terms of e(|uation (Ki), the two component parts of 

 the ^-eloeity potential which '^We the exciting forces due 

 to waves. 



The pressure due to 0,,„, is, from eciuations (11) and 

 (14) 



Pt,-u. = P - 



dt 



4 w'-c'-h 



p c 



( — 2ttR cos cx)/\ 



X COS a sin ~ (.(■ — ct) 



A 



R 



rp 



X 



IttIk 



A-'-i^It cos a)/X 



cos a cos — (,(■ — (■/) 



X 



Nomenclature 



A, B, C,\ _ coefficients of miscellaneims 

 D,E,Gj ~ terms of the differential 

 equations of motions, 

 equations (2) 

 a, b, c,\ coefficients of miscellaneous 

 </, e, (I I terms of the differential 



equations of motions, 

 equations (2) 

 A = ratio of amplitude of waves 

 made b\' ship to amijlilude 

 of heaving motion 

 Oo = amplitude of vertical ac- 

 celeration at bow 

 B = beam (local) 

 /(, .s = instantaneous distances of 

 ship bow and stern from 

 nodal of wave as defined in 

 Fig. 14 

 r = wave celcritj- 

 E = area under spectrum 

 /■' = hydrodynamic heaving force 

 F — heaving force imposed on a 



ship by waves (= Foe") 

 (I = acceleration of gravity 

 Hj = force due to water pressures 

 generated by waves and 

 ship's oscillations 

 H,„ = moment about CG due to 

 water pressures generated 

 b}' waves and ship's oscil- 

 lations 

 h = wave amplitude 



K, 



J = longitudinal moment of in- 

 ertia of a ship in mass 

 units 

 /vj = coefficients of equation (25) 

 h> = atlded mass coefficient in two- 

 dimensional vertical flow 

 about a ship section 

 ^4 = correction coefficient for ef- 

 fect of free water surface 

 L = ship length 

 i1/ = hydrodynamic moment 

 M = pitching moment imposed on 

 a ship by waves (= Moe'') 

 m = mass of a ship or of a ship 

 section 

 A'(t) = vertical damping force per 

 unit of body length per 

 foot per second 

 groupings of coefficients of 

 differential ecjuations of 

 motions defined bj' equa- 

 tions (7) 

 pressure 



radial distance to a point i] 

 in fluid 

 r = local radius of semi-cylindri- 

 cal bod}- 

 S = sectional area 

 So = amplitude of vertical tlis- 



placement at bow 

 ( = time 

 u — horizontal component of orbi- 



R,&\ 



V 

 R 



tal velocity of water in 

 waves 



V = ship speed 



V = vertical velocity 



V = vertical component of wave 



orbital velocity 



X = longitudinal co-ordinate with 

 respect to wave nodal 

 point 



y = vertical co-ordinate or local 

 half-breadth of LWL plane 



Z = complex amplitude of heav- 

 ing motion (= Zoe'*) 



z = vertical co-ordinate or heav- 

 ing displacement 



a = polar co-ordinate 



/3 = angle between longitudinal 

 tangent to body surface 

 and j-axis 

 b, t = phase angles of ship motions 



■q = local wave-height co-ordinate 



d = angle of jiitch 



d = complex amplitude of pitch- 

 ing motion (= Sof") 



X = wave length 



^ = longitudinal co-ordinate with 

 respect to CG 



p = water density 

 (7, T = phase angles of exciting 

 forces 



4> = velocity potential 

 CO or oj, = fre(|uency of wave encounter 



