322 



THEORY OF SEAKEEPING 



A 



Fig. 4 Sketch used in connection with Section 4.1 



1.3(1 fps; i.e., ;i shallow floating object will drift 19.3 

 nautical miles in a day. Because of the factor of 2 in the 

 exponential, the drift velocity diminishes very rapidly 

 with depth. 



4 Wave Energy and Group Velocity 



4.1 Energy per Unit of Sea Surface. The energy 

 connected with water motion can be considered from two 

 points of view: (a) Energy contained per unit area of sea 

 surface, and (6) energy transported by waves through a 

 vertical plane normal to the direction of wa\-e propaga- 

 tion. 



In considering the first of these aspects, distinction is 

 nnule between the potential and kinetic energy. The 

 potential energy is due to the weight of water pg, and its 

 ele\ation or depression with respect to the still-water 

 le\-el. For a unit width of the sea surface area, and for 

 one wave length, it is defined as 



V2 pg SI v'd.i 



(50) 



Using the artifice of making tiie wave motion steady by 

 superposing the water velocity -c, which is e(iui\-alent to 

 taking r; at t = 



a cos Am- 



(51) 



and theref 



:>reiore 



Ef, = ^/ta-pgX (,52) 



The kinetic energy contained in a body of fluid is de- 

 fined in terms of the conditions existing at its boundaries 



E^ = "2 f^<f>(d<f>'c>n)(h (53) 



where ds is an increment of length taken along the 

 boundary. Consider a mass of water contained between 

 the free surface, the bottom and two vertical control 

 planes spaced one wave length X apart, as shown in Fig. 4. 

 As the integral is taken following the.se boundaries in the 

 clockwise direction, the contribution due to the control 

 planes \anishes, as it is equal and opposite in sign on the 

 two planes. At the bfittom (d4>'dn) = 0, so that the 

 only non-vanishing contrilnition comes from the free 

 water surface. With the low wave height assumed in the 

 linear theory, the normal surface velocity -^0/d?i can 

 be replaced by the vertical one —c)(t>/dij. Substituting 



the values of 



f.r — ac sin kx for y = (54) 



dcfi/dij = —ack sin kx (55) 



and omitting the constant term resulting from ex, the 

 kinetic energv due to wa\'e motion is found to be 



E, 



(56) 



' iCi'pgX 



i.e., the potential and kinetic energies in wave motion are 

 ecjual and the total energy is 



E = E„ + El, = ', ••a-pgX per wave length (57) 



or 



'/■>a-pg per unit area of sea surface 



(58) 



4.2 Energy Transfer. The rate at which the wave 

 energy is transferred in the direction of wave propagation 

 can be evaluated by considering the rate at which the 

 work is being done on the water to the right of a vertical 

 control plane (Lamb, 1-D, page 383). The control plane 

 can be assumed to be located at .r = 0. This rate of work 

 is evidently a product of the pressure p and the horizon- 

 tal water-particle ^'elocity u. Using the previously de- 

 rived values of these, the instantaneous rate of work is 



dE dl = /i\, pii fhj = pga-kc cos- k{x - ct) /»„ c'"" 



= ^/ipga-c cos- k{x —ct) (59) 



The mean value of cos- /i'(.r — ct) over a cycle is 1 '2, so that 

 finally 



Mean dE/dl = '/.pga-(c/2) 



(60) 



i.e., the total energy of wave motion in deej) water is 

 transferi'cd at half the wave celerity. 



4.3 Group Energy. The following is e.s.sentially 

 (luoted from Lamb (1-D Art. 236, pp. 380, 381), with 

 minor changes and some abbreviation : 



It has often been noticed that when an isolated group 

 of waves, of sensibly the same length, is ailvancing over 

 relatively deep water, the velocity of the group as a whole 

 is less than that of the individual waves composing it. 

 If attention be fixed on a particular wave, it is seen to 

 advance through the group, gradually dying out as it 



