316 



THEORY OF SEAKEEPING 



more mathematical presentation will be found in Lamb 

 (1-D pages 421-423), :\lilne-Thomson (1-F pages 381- 

 385), and, in a more complete form, in Kochin (1-C pages 

 441-448). The extension of the theor^y to shallow water 

 is given liy O'Brien and Mason (1-G pages 28, 29). 



Historically, introduction of the trochoidal theory into 

 Naval Architecture was rather unfortunate. As men- 

 tioned before, it cannot be used in most hydrodynamic 

 problems. It fails in defining the maximum possible 

 wave height. The vertical position of the troclioidal 

 curve on the water surface is not included in the basic 

 definition, and has to be evaluated independently. 

 Finall\', it does not indicate the "mass transport," i.e., 

 the greater amount of water surge in the direction of 

 wave propagation at wave crests than in the return back- 

 ward movement at wave troughs, which is clearly ob- 

 servable in nature. All of these characteri.stics form an 

 integral part of the potential theory of gravit.y wa\-es of 

 finite amplitude. Moreover, the potential theory of 

 waves of finite amplitude gives a wave profile which, like 

 the trochoidal one, is in agreement with observations 

 for wa\'es of moderate height. In other words, the 

 trochoidal form and the form of waves of finite height 

 are identical within the obtainable accuracy of experi- 

 mental observations although mathematically they 

 always differ in the terms of higher order. 



2 Potential Theory of Surface Waves 



2.1 Outline of Theory, Velocity Potential, Wave 

 Elevation and Celerity. The theory of two-dimensional 

 gravity waves on the surface of water is usually known 

 as "potential theory of surface waves." The wave crests 

 are of infinite length, uniformly spaced and parallel to 

 each other, and are advancing in the direction normal to 

 them at a certain celerity c. The word "celerity" is used 

 for this velocity of propagation of irave form, to dis- 

 tinguish it from the water-particle velocity U, which is 

 usually very much smaller than c. With infinitely long 

 and uniform wave crests, the various functional rela- 

 tionships involved in the wave description remain un- 

 changed for any change of position along the direction 

 parallel to the wave crests. Only the distances x meas- 

 ured in the direction of wave propagation, and the verti- 

 cal distances y have any effect on these functions. Such 

 waves are therefore often referred to as two-dimensional. 

 Since the wave form advances with celerity c, the wave 

 elevation (as well as all other functions involved, such as 

 pressure for instance) depends on time t, as well as on the 

 distances x and y; that is, all relationships involved 

 have the general form F{x, y,t). 



Water is assumed to be inviscid and incompressible; 

 its motion can therefore start from rest only by pres- 

 sures or impulses acting normally to any water surface 

 or boundary. Such a motion is called "irrotational" 

 or "potential" and is characterized by the existence of 

 the velocity potential and of the stream function \j/ 

 such that 



Horizontal comi^onent of water velocity 



u = — d</) d.r = — Si/- di/ 



■ (2) 

 Vertical component of water velocity 



V = —d<f>dy = bip/dx, 



where the origin of co-ordinates is taken at the water 

 surface, positive x is in the direction of the wave propaga- 

 tion, and y is positive in the upward direction; i.e., it is 

 negative throughout the region occupied by the water. 

 In any form of potential flow, the functions <t> and <p 

 satisfy Laplace's eciuations: 



and 



V-</> = (dV/d.r=) + (d-</)'dy=) = 



W = {d-4'/dx-) + (d^f d//-) = 



(3) 



(4) 



Solution of a hydrodynamic problem (in this case the 

 wave motion) consists of finding relationships between 

 the function 4) or xp and the co-ordinates .r and y such that 

 the Laplace equations are satisfied. In addition, such a 

 solution must satisfy the geometric and dynamic condi- 

 tions existing at various boundaries of the fluid. In the 

 case of two-dimensional wa\'es in water of uniform depth 

 h, three boinidary conditions are to be satisfied: 



(i) At the bottom { — y = h), the vertical velocity is 

 nil; i.e.. 



d0 



= 



(5) 



(ii) At the free surface, the vertical velocity of the 

 water particles, v = —d(t>/dy, must be the same as the 

 vertical velocity of the surface itself, d-q/dt, where r? de- 

 notes the wave elevation; i.e., 



dr, 



dt 





(6) 



(iii) The atmospheric pressure acting on the free 

 surface is uniform. 



In connection with condition (iii), it should be noted 

 that the theory of surface waves treats the propagation 

 of free wa-\-es solely under the action of gravitational and 

 inertial forces. Neither the method of generating waves 

 nor the exciting or damping effect of air motions is in- 

 cluded here. These are treated separately in Chapter 1. 

 Condition (iii) is put into mathematical form by use of 

 Bernoulli's equation for time-dependent flows with 

 gra^^ity force gri: 



gv + '/2(ii' + r'-) - d4>/dt = ^(0 



where F{i) is an arbitrary fiuiction of time resulting from 

 integration of Euler's equations of motion. 

 Including Fit) in b<t>/dt: 



iu' + V"-) 



(7) 



= 1 ^ _ 1 



In the first-order theory, usually referred to as the 

 theory of "waves of small amplitudes," simplifying as- 



