Sec. 4S3 



WIND-WAVE AND SHIP-WAVE DATA 



163 



(10) At a depth h below the surface, the radius 

 R of the orbit centers at that depth is given by 



By substituting Lw = 2vRrc this becomes 



72 =Rse-'""'"' 



When h = L^,R = R^e'^' = 0.0019i?s . Thus 

 the orbital motion is virtually zero in water as 

 deep as the wave is long, and for practical pur- 

 poses the assumed unlimited depth is not neces- 

 sary; see Table 48.f of Sec. 48.7. 



(11) The total energy in the wave per unit 

 breadth is approximately 0.125w{hw)^L,r . By this 

 formula, a salt-water wave 600 ft long and 30 ft 

 high has about 2,000 ft-tons of energy per ft of 

 breadth. 



(12) Of the total energy in the wave, half of it 

 is potential energy and half kinetic energy. 



(13) From the relationship 



Steepness ratio 



hw 



2R.S 

 2TrR, 



^■kRrc 



the rolUng-circle radius Rrc = /iiF/[27r (steepness 

 ratio)]. For a limiting steepness ratio of 1/7, as 

 depicted in the diagram of Fig. 48.B, 



Rrc = /iB^/0.8976 = \.\Uhw = 2.228Rs. 



(14) The value of the virtual acceleration of 

 gravity is. 



At the crest, {Rrc — Rs)g/RRc 

 At any intermediate point, Rig /Rrc 

 At the trough, {Rrc + Rs)g/RRc ■ 



For a limiting steepness ratio hw/L^ = 1/7, the 

 value of the first is 1 .228^/2.228 = 0.55^; of the 

 last, 3.228£?/2.228 = lA5g. 



(15) The maximum slope of the wave surface 

 is sin"' {Rs/Rrc) or, in radians, irhw/L^ . It 

 occurs at the point where the orbit radius Rg is 

 normal to the radius Ri . For the same limiting 

 steepness ratio of 1/7, where Rrc = 2.228fls , 

 the sine in question is 1/2.228 = 0.4488, whence 

 the slope angle f (zeta) is about 26.7 deg, as com- 

 pared to 30 deg for the highest possible Stokes 

 irrotational wave of approximately the same 

 steepness ratio. 



As an example of the use of the data in the 

 foregoing, an estimate is made of the charac- 

 teristics of a trochoidal wave Avithin the range of 

 size and proportions listed for the operation of the 

 ABC ship in item (24) of Table 64.d. 



Assume that the ship would give its least com- 



fortable performance when steaming nearly ahead 

 into a regular train of waves having a length Lw 

 of 1.2 times the length of the ship. Then L^ = 

 1.2(L) = 1.2(510) = 612 ft; \/Z^ = 24.75 ft. 



The celerity c of this wave, reckoned with 

 res pect to the undisturbed water, is equal to 

 VgL^/2T = 2.263 a/L^ = 2.263(24.75) = 56.01 

 ft per sec. This is equivalent to 33.16 kt. For an 

 angle of encounter a (alpha) of about 180 deg, 

 representing a head sea, the speed of encountering 

 the waves is this wave speed plus the speed of the 

 ship, reckoned with respect to the undisturbed 

 water. 



The period T^ of this wave is ■w2-KLwlg = 

 0.4419 VLr. or 0.4419(24.75) = 10.94 sec. The 

 frequency /, corresponding to the number of 

 wave crests which would pass a given point in 

 space in 1 sec, is the reciprocal of the period or 

 0.0914 wave per sec. 



The height hw oi the wave is fixed by the 

 assumption made in item (24) of Table 64. d, 

 which stated that the height would not exceed 

 0.55 vL;^ . For the wave in question this is 

 0.55(24.75) = 13.61 ft. The steepness ratio hw/Lw 

 is then 13.61/612 = 0.0222 or 1: 45. 



The velocity and period of this wave, as taken 

 from the tables of Sec. 48.6, are listed in that 

 section. Its maximum slope and orbital velocity 

 are found in Sees. 48.5 and 48.7, respectively. 



48.5 Elevations and Slopes of the Trochoidal 

 Wave. Table 48. a gives the ordinates of a tro- 

 choidal wave surface in terms of the wave height 

 hw • The ordinates are spaced at equidistant 



TABLE 48.a — Ordinates for Construction of a 

 Trochoidal Wave Profile 



The data listed here are from PNA, 1939, Vol. I, p. 207. 

 The stations are spaced equally along the horizontal plane 

 from crest to trough. The base line for ordinates ia at the 

 bottom of the wave trough. 



