320 



THEORY OF SEAKEEPING 



X or y, giving: In the horizontal direction, for shallow 

 water, 



{dp/d.v) = -apgk '"''' ^l^t ''^ sin k(x - ct), (29) 

 cosh kh 



anil for deep water 



(dp/dx) = -ap^fre'-'sin k(x - ct), (30) 



in the \'ertical direction, for shallow water 



sinh k{y + h) 



(dp /by) = apgk 

 id for deep water 



cosh kh 



cos A:(.r - ct), (31) 



{dp/dy) = apgke"" cos k{x — ct) 



(32) 



In applying the expressions for thf pressure gradients, 

 the distinction between the effects of the hydrostatic and 

 dynamic gradients should be remembered. In a hydro- 

 static case, the force acting on a body is given by 

 — {dp/dx) times the volume of the body. It has been 

 shown by G. I. Taylor (1-1928) that when the pressure 

 gradient is caused by velocities and accelerations in the 

 fluid, this relationship is modified and becomes: 



Force = — (dp/d.r) (body volume) (1 -|- A',) 



(33) 



where A'x is the coefficient of accession to inertia in the 

 .r-direction in the case of the horizontal force. A similar 

 expression is obtained for the vertical force by using the 

 derivative dp/dy, and the A'„-value for the vertical direc- 

 tion. For a cylindrical body A-j, = 1 , and the force due 

 to a dynamic pressure gradient is double that due to a 

 hydrostatic one. 



Now the complete forms sin A-(.r — ct) and cos k{x — 

 ct), which had to be kept throughout the development, 

 since various derivatives had to be taken, can be simpli- 

 fied for two typical cases. 



If a ship, relatively long with respect to the wave 

 length, is analyzed at a specified instantaneous location 

 on the wave, the time t at this particular instant can be 

 considered zero, so that the expressions become sin(27ri:/ 

 X) and cos(27r.r/X). In another case an analysis may be 

 recjuired of time-dependent pressures and forces acting 

 on a small body at a fixed location, which can be taken 

 as x = 0. The expressions become then functions of 

 {2Tct/\), and are expressed more conveniently in terms 

 of the circular frequency co as sin wt and cos uit. 



3 Waves of Finite Height 



3.1 Stokes' Waves. In the previous section the ap- 

 proximations made in writing the boundary condition 

 equations (8), (9) and (10) made the analysis exact only 

 for vanishingly small wave height. It has been men- 

 tioned already that in the case of a finite wave height this 

 analysis gives correct expressions for the first harmonic 

 of the wave form expressed by a Fourier series. A brief 

 discussion of the theory of waves of finite height will now 

 be given in order to support this statement, to demon- 



strate the significance of the approximation made, and 

 to compare the results of the potential theory with the 

 trochoidal one. The theory of waves of finite amplitude 

 was formulated first by Stokes (1-1847), and was further 

 developed and confirmed by Levi-Civita (1-1925), and 

 Struick (1-192(1). A brief exposition of it is given by 

 Lamb (1-C' par. 250, pages 417-420), and the results are 

 stated in a simple and convenient form by O'Brien and 

 Mason (1-G, pages 14-24). The subject is completely 

 missing in other basic texts on hydrodynamics. 



Stokes (1-1847) gives the following expres.sions for the 

 wave in deep water: 



To the second order, 



(r,/\) = (o/X) cos A-.r + 7r(a/X)2 cos 2kx (34) 



and 



c = {gX/'lirY^^ (as in the linear case) 



to the third order, 



{r}/\) = (a/X) cosA'.r -|- tt (a/X)- cos2kx + 



37r-(a/X)3 cos 3Au- (35) 



and 



(gA/27r) 



1 + 



-0' 



For a wave of length-to-height ratio X/2a = 20, the 

 correction to c in the latter case is only 1.25 per cent. 



According to Stokes, equation (35) for a potential 

 wave of finite amplitude to the third order of approxima- 

 tion coincides with the sum of the first three harmonics of 

 the trochoidal form. This is above the accuracy attaina- 

 ble in practice in wave generation and measurement. For 

 the equation of tj to the fourth order, and for correspond- 

 ing expressions for water of depth h the reader is referred 

 to O'Brien and Mason (1-G pp. 15-22). 



From the work of Rayleigh (1-1876) (see Lamb, 1-D, p. 

 417, equation 1) it is seen that expression (15) for the 

 velocity potential </> in deep water is valid within a .second 

 approximation ; correction is needed only for third-order 

 and higher approximations. Correction of the fu-st-order 

 theory to a second approximation, therefore, involves only 

 the relationship between wave height and potential, 

 which was mutilated in making the approximations 

 shown by equations (8) and (9). Part of the correction 

 is to reinstate the terms in velocities u and v of equation 

 (7). Denoting this correction to 17 by Air/ using expres- 

 sions for u and /' listed in Table 1 (for y =0) and substitut- 

 ing c- = g\/'2Tr, 



AiV = - 



[sin^ k{x - ct) + cos^ A-(.i- - ct)] (36) 



The second necessary correction A^?; to the 17 given by 

 equation (8) can be accomplished by the method of R. 

 Guilloton (see Korvin-Kroukovsky and Jacobs, 1-1954, 

 pages 26, 27). To the first order of approximation equa- 

 tion (8) represents a function F(.v, 0) and it is desired to 

 find F{x, 7i). Taking the first two terms of the expansion 

 in a Taylor's Series 



