318 



THEORY OF SEAKEEPING 



Table 2 Properties of Harmonic Waves in Water of any 



Depth 



Surface profile t] = a cos k(x — cD 



Com])lex potential w = 4> + i^' 



sin {kz + ikh — ckl) 



= —nc 



sinh kh 



eosh k(y + h) . , 



Velocitv potential (j) = -ac . , , , sni k(x — ct) 



snih kh 



(ja eosh k(y + h) 



= — — sni k(x — cl) 



Cl) cosh kh 



, s'"h k(y -\- h) 



Stream tunction f = — "' : , , , l'os A-(x — ct) 



snih kh 



eosh kiy -\- h) 



Horiz. water velocitv " = ack . , , , cos k{x — ct) 



snili kh 



sinh k(i/ + h) . 



Vert, water velocitv '' = ack . ,' ,, sin *:(x — ct) 



sum kn 



Wave celeritv c- = '— tanli kh 



(Mish k(ij + h) 



Pressure p = apy -— — cos k(x — ct) 



eosh kh 



Horiz. pressure gradient . dp/di 



eoshUy^M) . . 



= —iipak ; ; SHI kix — ct) 



cosh kh 



Vert, pressiire gradient . t)p/d// 



, sinh kiy + h) 



= iip(ik cos k(x — rt) 



cosh kh 



For \'ery shallow water (h < X/20), llii.s becomes 



c- = gh (20) 



and, for deep water (h > X,'2), 



c= = I' (21) 



All characteristics of the deep-water gravity wa\-es are 

 summarized for ea.se of reference in Table 1. The 

 characteristics of waves in water of hmited depth are 

 summarized in Table 2. Table 3 shows the relationships 

 aniouf;- wa\'e length, period, and rcierity in deep water. 

 2.2 Velocity and Path of Fluid Particles. The total 

 velocity of a fluid particle can be obtainetl by adding 

 vectorially the u and c components: 



U = u + ir (22) 



By inserting the values of it and r from Table 1 for deep- 

 water waves and letting .r = and 'lird/X = wt, 



U = ^^ e-"-'^'' (cos o)/ - i sin a) 

 \ 



or 



(23) 



2irl//X „-!"' 



tor who.se uniform ah.solute value is 'lirac/X at the water 

 surface and which rotates at the angular \'clocity oj in 

 radians per sec, making a complete revolution in the 

 period of the passing wave. The path of the particle at 

 the surface is therefore a circle of radius a. The ab.solute 

 value of the velocity vector and the radius of the circular 

 path of the particle diminish with depth as c'-"" ^. This 

 result is identical with the result of the trochoiilal-wave 

 theory. 



When the wave is in water of limited depth /;, the path 

 of a particle is elliptical and is defined by the equation 

 (iMilne-Thom.son, 1-F, page 3oG: Coulson, 1-A, page 

 78) 



-F' 



1, 



(24) 



where ^ and -q are the horizontal and A^ertical dis- 

 placements of a particle from its initial still-water posi- 

 tion, and a and i3 are ,semi-axes of the ellip.ses: 



a cosh k(h + y) 



fi = 



sinh kh 



a sinh k(h + y) 

 sinh kh 



(25) 



The particle velocity, therefore, is represented by a vec- 



Here, k is again written for bre\-ity in place of 27r/X. 

 All the ellipses have the same distance 2a sinh kh be- 

 tween their foci, but the semi-axes a and /S dimini.sh 

 with depth and (3 vanishes at the bottom, y = —h, where 

 water particles have a rectilinear motion between foci. 



The theory as outlined in the foregoing, i.e., based on 

 the simplifying assumptions (ii) and (iii) shown by ecjua- 

 tions (8) and (9), is known as the linear theory, or theory 

 of waves of small amplitude. The wave described by 

 efiuation (18) is known as a "simple harmonic wave." 

 The derivation becomes exact for vanishingly small wave 

 amplitude a. With waves of growing height the true 

 surface profile, which is practically trochoidal. deviates 

 increasingly from e<iuation (18). However, the profile of 

 such a wave can be represented by a Fourier series of 

 simple harmonic waves, and in that ca.se equation (18) 

 represents the first term (the first harmonic) of such a 

 series. Ec[uations (15), (16) and (18) represent the 

 simplest form of gra\-ity wa\'e. As such, they are in- 

 dispensable as the basic material from which any type 

 wave can be constructed by a suitable superposition of a 

 number of simple forms. 



2.3 Pressure and Pressure Gradient. A wave charac- 

 teristic which is often ncetleil in aiiah'ses of engineering 

 structures (inclutling ships) is the pressure acting at any 

 point in water. Within a first approximation, equation 

 (9) for example, the pressure is given by 



where p is the mass density of water 2 lb per cu ft for sea 

 water. For water of limited depth h, on the basis of 

 equation (12) 



