324 



THEORY OF SEAKEEPING 



Fig. 5 Streamlines of water flow in standing wa\es (from Lamb, I-D) 



5 Standing Waves 



The \va\'e,s discussed in the foregoing propagate in one 

 direction \yith celerity c and are known as "progressive 

 waves," as distinguished from "standing waves"; in 

 these latter, the water particles at crests and troughs 

 move up and down in a vertical straight line, while the 

 particles at nodal points move to and fro along horizontal 

 straight lines. This system can be represented mathe- 

 matically by superposing two progressive wave trains 

 moving in opposite directions, each with the amplitude 

 a/2. It will be convenient first to modify eciuations 

 (13) and (14) by replacing cos k(.v — ct) by sin k{x — ct) 

 and vice versa. This brings the notation to the form 

 used by Lamb (1-D, Art. 228, page 364). Substituting 

 (a/2) for (a) in these ecjuations, forming two pairs of 

 equations with -\-c and —c, and using trigonometi'ic re- 

 lationships for the sum and difference of two angles, the 

 following expressions for the velocity potential and 

 stream fimction are obtained : 



aq cosh k(y + h) , 



<p = }^, '- cos kx cos cot 



CO cosh kh 



^P = + 



ag sinh k{y -\- h) 

 u> cosh kh 



sin kx cos ui 



(70) 



(71) 



The corresponding streamlines are shown in Fig. 5. 

 The relationship between the circular frequency oj, wave 

 length X (through k = 27r/X) and water depth /( is given 

 by (Lamb 1-D, page 364): 



gk tanh kh 



(72) 



The water ^•elocities are foiuid by partial differentiation 

 of efjuation (70) with respect to x and y, 



u = 



d(t> _ agk cosh k{y-\-h) 

 d.c CO cosh A7( 



agk sinh k{y -\- h) 



dy CO 



cosh kh 



sin kx cos coi (73) 



cos kx cos coi (74) 



and displacements of water particles ^ and r) from the 

 still-water position (.r, y) are obtained by integration of 

 the foregoing equations with respect to t, and coml)ining 

 with etjuation (72) for co-, 



cosh k(y -\- h) . , . , ,__, 



5 = -a f^V — - «'" ''■•i' sni o3t (75) 



smh kh 



snih k(y + /() , 



V = a . cos Am: sui coi 



smh kh 



(76) 



These eciuations show that the motion of each particle 

 is rectilinear and simple harmonic, the direction of each 

 motion \-arying from \-ertical beneath the crests and hol- 

 lows {kx = 0, -K, 2ir, etc.) to horizontal beneath the nodes 

 {kx = 7r/2, 37r/2, etc.). 



The shape of the free surface is obtained from (76) by 

 letting y = 0, 



a cos kx sin wt 



(77) 



Two cases of standing waves are of greatest interest to 

 naval architects; namelj', that formed by interaction of 

 the wave reflected by a beach or a vertical cliff with the 

 oncoming sea wave, and that formed by water oscillating 

 in a rectangular tank. The first case needs only to be 

 pointed out, since all relationships given in the foregoing 

 applj' for the case of a complete reflection. Li the case 

 of partial reflection, the result is a weaker standing- wave 

 system superposed on the progressive wave system. 

 This is often found in towing tanks when the wave- 

 absorbing beach at the end of a tank is not sufficiently 

 good. Li the actual sea this phenomenon is often mani- 

 fested in a sea condition which is much more unfavorable 

 for ships near the shore than in the open sea; the Bay of 

 Biscay is particularly known in this respect. It has been 

 deri\'ed theoretically and verified experimentally (Penny 

 and Prince, 1-1952; Taylor, 1-1953) that the minimum in- 

 cluded angle at the wave crest of a standing wave is 90 

 deg, as against 120 cleg for the progressive wave system; 

 i.e., much steeper waves can be encountered in a stand- 

 ing-wave system than in a progressi\'e one. 



5.1 Waves in Rectangular Tanks. The sloshing of 

 water in rectangular 1anks is an instance of a standing- 

 wave ."system which is of interest to naval architects. The 

 basic principles of the problem are discussed by Coulson 

 (1-A, Art. 49, page 75) and Lamb (1-D, Art. 190, pages 

 283 and Art. 257, pages 440, 441), and in application to 

 swimming pools on hoard ships by Comstock (1-1947). 

 All the boundary conditions discussed earlier in connec- 

 tion with waves in water of limited depth h are applicable 

 in this case, and in addition the normal velocity at the 

 tank walls must vanish. E(|uation (72), connecting the 

 wave freciuency, wave length, and tank depth applies in 

 this case, but the possible choice of A--values is now 

 limited to certain ratios of tank length or width to wave 

 length, 1/2, 1,2,..., or specifically 



