APPENDIX A 



Theory of Simple Waves 



1 The TrochoJdal Wave 



111 the literature on iia\'al architeeture, the "tro- 

 ehoitial" \va\'e ha.s been ino.st often used as a represeiita- 

 t ion ( )f a simple wave. This is a purely geometrie deserip- 

 tion, shown in Fig. 1. The shape of the water surface is 

 generated by a point on the smaller of two concentric 

 circles (radius r) rotating together with the larger one of 

 radius R. The latter rolls througli an angle 6 on an 

 imaginary horizontal track located at a certain height 

 above the still-water surface. By virtue of this geometry, 

 the instantaneous co-ordinates of the point P on the small 

 circle, i.e., the co-ordinates .r and y of the described tro- 

 choidal curve, are given by the parametric e(iuations 



.(■ = Re + r sill 

 ij = R -\- r cos t 



(1) 



The origin of the horizontal co-orilinutes .c is at the 

 wave hollow and the angle d is measured from the lower 

 vertical, passing through the center of the generating 

 circle. The ordinates y are measured downwards from 

 the track .r — .r. A conspicuous feature of the trochoidal 

 curve is sharper cur\'atin'e at the wave crest and a lesser 

 curvature at the wave troughs. This difference is ag- 

 gra\-ated as r increases and approaches R. At r = /?, 

 the wave crest takes the form of a sharp cusp. In this 

 case the wave height would be '2R and, since wave length 

 X = 27ri?, the minimum ratio of wa\'e length to height 

 would be 3.14. Such a wave height never occurs in na- 

 ture, and is not compatible with the theory of potential 

 waves discussed in the next section. For waves having 

 a height-to-length ratio often found in nature, such as 

 ^20 ior instance, the trochoidal cur\'e agrees well with 

 experimental tank ob.servations. 



Etiuations (1) merely define the trochoidal form, but 

 do not give any indication of its relation to tlie still-water 

 level. This relation is defined by the condition that the 

 total water volume above this level must be equal to the 

 one below. In order to satisfy this condition, the line 

 of centers of the rolling circles must lie located r-/2R 

 above the still-water le\'el. 



The dynamic characteristics of the wa\'e, such as its 

 velocity of propagation and pressure gradients, are 

 derived from the condition that a uniform atmospheric 

 pressure acts on each part of its surface. Since there is 

 no pressure gradient along the surface, there is no fluid 

 acceleration in this direction, and all fluid accelerations 

 at the free surface are normal to it. This fact will be seen 



later to play an imiioi'tant part in the theory of ship 

 rolling. In this connection, this was demonstrated ex- 

 perimentally by W. Froude (1861), who htted a pen- 

 dulum over a small float and found that the pendulum 

 remained in its still-water position normal to the float, 

 regardless of the inclinations taken by the float in waves 

 in a towing tank. He found this to hold even when the 

 float was partially in\'erted fin the forward face of a 

 breaking wa\'e. 



Certain \-alues of the circular frequency oj and of the 

 wave celerity c are needed to fulfill the foregoing con- 

 ditions. The acceleration of a water particle at the sur- 

 face must be such that, combined with the acceleration 

 of gravity g, it would gi\'e the total acceleration normal to 

 the trochoidal surface. On this basis, the following rela- 

 tionshi]3s have been established: 



Wa\'e celerity c = '\/g\;2w 



= 2.20 -x/xTt (fps) 

 Wave peri(Ki (sec) T = \/c = ■\/2ir\/g 

 Wave length (ft) X = 0.0196c- (fps) 



= O.oTTc- (knots) 



1.34 Vx ft (knots) 

 0.442 v'xlt 



5.118 r- 



A water particle, initially at rest in still water, is lifted 

 at the approach of the wave crest, mo\'ed forward at the 

 crest, lowered at the approach of the h(jllow and moved 

 aft at the hollow. It thus describes a complete circle of 

 radius /■ at the circular fre(iuency w, the complete circular 

 path being accomplished in the pa.s.sage of one wave; i.e., 

 in the period T. This is known as the oribital motion, 

 and the particle \-elocity as tiie orbital velocitJ^ The 

 whole mass of water in the immediate \'icinity of a 

 particle goes through approximately the same motion, 

 so that there is no significant rotation of the particle or 

 of the small mass of water in relation to the ambient 

 water. Howe\'er, any initial rectangular mass abed, as 

 shown cross-hatched in Fig. 2, is distorted into a' b' c' d' 

 in such a way that the motion is not "irrotational" or 

 "potential" from the point of view of hydrodynamics, and 

 cannot be represented in a simple form needed for the 

 solution of most hydrodynamic problems. This .se\'erely 

 limits the practical u.sefulness of this theory. 



A very simple and well-developed exposition of the 

 trochoidal theory — from the na\al architect's point of 

 view — is given by [Manning (1-1942, pages 1-7 



A 



' Reft'renre to sections, equations, fignres and bililiography will 

 be designated by chapter number and section, equation, figure, or 

 bibliography; refereni-e noted is to Manning, ehapter 1 (lil4'2). 



314 



