Theory of Short and Long Waves 



This gives the two equations: 



duldr+8wlrd& = and iw\dr-du\rdd = 0. 



The wave profile in the vicinity of A is given by >] = a— rcosa. 

 Bernoulli's theorem has the form of 



P 1 



- H — q 2 -\-g(a — rcosa) = constant , 



e 2 



if the total velocity along BA is q. 



This requires that, with p constant (free surface) 



25 



Fig. 13. Transformation from rectangular into polar co-ordinates. 



In the vicinity of A, therefore, q must be proportional to j/r or «= C y rf(&) and w = 

 = C|/r(#). With the previous equations, we then obtain %f(&)+g'(&) = and ig(&)— /'(#) = 0, 

 which leads to the differential equation 4g"(&) + g(d) = 0, with the solution & = Acos^#+ Bsin|#. 

 With & = 0, w = and we obtain u — — Cj/r cos|# and h> = + Cj/r sin £#. In each point (r,&) 

 the current vector forms with the -x-axis the angle J#. This also applies to the surface, where & = a, 

 for which a+\a = 90°, or a = 60°, 



A simpler derivation of this condition can be obtained starting with the stream function accord- 

 ing to Lamb (1932, § 63 (3), p. 69). If the stream function in polar co-ordinates {r,&) has the form 

 ip = Cr"cosn& with the condition that y> = when & = ±a, this will lead to na = £tt. From this 

 we obtain q = nCr n ~ 1 , where q is the resultant fluid-velocity. But since the velocity vanishes at 

 the crest, its value at a neighboring point of the free surface will be given by q- = 2gr cos a. 

 Comparing q and q 2 , we see than n = | and therefore a = \ji = 60°. 



The condition of uniform pressure at the free surface along the entire 

 wave profile (y> — 0) here again leads to a relation between wave velocity 

 and wave length. The total velocity is obtained from u and w from (11.15), 

 and the square will be c 2 + j4V z — 2Ace" z cosxx. Therefore Bernoulli's theorem 

 gives for the free surface 



— + ^- + ^ A 2 e 2 *i — cAe*' 1 cos xx + g>] = const. 



q 2 2 



