THEORY OF SIMPLE WAVES 



317 



+ y, T,v 



Table 1 Properties of Harmonic Deep-Water Waves 



Surface profile r] = <t cos kix — el) 



Velocit}- potential </) = —ace''" sin k {x — ct) 



Stream function xp = —aci'''" cos k(x — ci) 



\ II = —d(i>/dx = —dip/dy 



I = kace'''' cos k{x — ci) 



\ c = -d<t>/d!i = diA/dx 



I = kace''" sin k{x - ct) 



Horizontal water velocity. 

 Vertical water velocity . . . 



Wave celerity c=— =- = (' — ) 



T CO V-t/ 



= ^ = ■2:2C,\/x* 

 '2ir 



2Trc- 27r(/ gT 



— = 5.127'=* 

 27r 



Wave length X = 



!/ 



27r w' g 4-7r^ 



\ ~ g ~ e- ~ gT'' 



Pressure** p = apgc''" cos k(x — el) 



Horiz. pressure gradient** dp/dx = —apgke'" sin k{x — ct) 



Vert, pressure gradient**, dp/di/ = apgkc'''" cos k{x — ct) 



2Tra 



X 



Wave number k 



Ma.ximiun wave slope 



ka 



* X in feet, c in feet per sec. 



Exclusive of hydrostatic. 



sumptions are made in evuluatinj;; boundary renditions 

 (ii) and (iii). In the first ea.se it is assumed that condi- 

 tion (ii) is fulfilled at the undisturlied water level ;/ = 

 instead of at y = -q, i.e., condition (ii) is written as 



(ii-simplified) br)/bt = ( — d(^, d(/)y=o (8) 



In the second case the squares of the small perturba- 

 tion velocities u and v are neglected in comparison with 

 i><i>/'dt, and (iii) reduces to 



1 d(/> 



(iii-simplified) -q 



g N 



(9) 



The simplified conditions (ii) and (iii) can be combined 

 to form 



(iv) 





- 0, 



t .i/ = Oi 



(10) 



which is usually referi-ed to as the free-surface condition- 

 In the present work the notation confoi-ms to the 

 sketch at the top of Table 1. The origin of co-ordinates 

 (x = 0, // = at < = 0) is taken at the still-water level 

 under the wave crest. The .r-co-ordinates, the wave 

 celerity c, and the horizontal component u of the water 

 velocity are positive to the right. The ordinates jj and 

 the vertical components v of the water velocity are posi- 

 tive in the upward direction; i.e. below the still-water 



level, y has negative values. The watei' (lei)lh li is taken 

 as a positive quantity. 



The Laplace equation (3) and the boundary conditions 

 (i) and (iv) are satisfied by the expression for the com- 

 plex potential (JXIilne-Thomson, 1-F, p. 357, modified 

 for change in the origin of the co-ordinates from nodal 

 point to crest and from bottom to still-water level); 



^ , ., sin(fo -I- ik-k - kct) 



w = 4> + iiP = -ac . , ,, (11) 



smh kh 



where z = .r + iy, k is written for 27r/X and a is the 

 amplitude, or more exactly half the wa\'e height meas- 

 uretl from trough to crest. Separation of real and 

 imaginary parts gives the velocity potential; 







cosh k(ii + h) . -, , 

 ac . y.. sni k(x - ct) 



^inh /,/( 

 and the stream f miction 



^ = -ac '^'"h kjy + fe) 

 sinh kh 



cos k(x 



ct) 



(!-') 



(13) 



By the u.se of expression (19) for c", and noting that At 

 = w (where w is the circular fre((uency in radians per sec), 

 ecjuation (12) can be rewritten in an alternate form; 



<t> = 



ga cosh kdj + h) 

 0) cosh kh 



sin kU 



ct) 



(14) 



For very deep water (i.e., large h: in practice, h > X/2), 

 these expressions become 



(j> — —ac f'" sin /,■(.)■ — ct) 

 ip = —ac c*'" cos /i(.r — cl) 



(15) 

 (1(5) 



The components of the water velocity at any point 

 p{x, y) in the water are obtained by partial differentia- 

 tion, as shown by equations (2). The resulting expres- 

 sions are listed in the summarj^ tal)les of wave proper- 

 ties, (1) and (2). 



The elevations of lines of equal pressure are obtained 

 by using expressions (14) and (15) for 4> in equations (9), 

 as follows; 



For water depth h, 



cosh k(y + h) _ 



ri = a ^y-; cos k(x "') 



and for deep water. 



cosh kh 



ac'"' cos /,'(.r 



(17) 



ct) 



(18) 



In particular, the surface wave profile is obtained by 

 letting y = 0. In this ca.se the factors containing hyper- 

 bolic functions and the exponential become unity. 



The velocity of wave propagation c is found by in.sert- 

 ing the expressions for (^ in the surface condition (iv) ; 

 i.e., equation (10), .so that: 



For water of tlei)th h, 



'i- tanh 



irh 

 X~ 



(19) 



