Lagrangian Solutions of Unsteady Free Surface Flows 



In the examples to follow (see Figs. 4(a) to (d)) satisfactory 

 numerical solutions could be obtained by ignoring all but one of the 

 singularities. The exception was the shoreline, point A, Fig. 4(c). 

 If P is the angle between the tangent to the free surface at A and 

 the horizontal then correlating the two boundary conditions yields: 



(Zj^)a = - e'°/(cot P cos a + sin a) (28) 



Thus the sign of ^ determines the direction of the acceleration up 

 or down the beach. If the fluid starts from rest at t = , |3 = , 

 then (Ztt)t:0 = a-J^d successive differentiation of the basic equation 

 (8) and the boundary conditions yields (for ir rotational motion): 



At A, t = (c^^t|) 'Z'^^f^Z'u^ = 



^tttt '^att'^btt»^ttttt = or oo, unless ^=^ (29) 



Zftittt »Zamt '^^btm = or oo, unless a = ^ or ^ 



These relations suggest a behavior which is logarithmically singular 

 in time at t = unless a = ir/Zn, n integer. Roseau [ 1958] found 

 similar logarithmic singularities in periodic solutions for the general 

 case which excluded a = tt /2.n and another set of particular angles 

 (see also Lewy [ 1946] ). But a systematic analysis of the singular 

 behavior (especially for t =^ 0) has not as yet been completed. Rather, 

 since the relations (29) no longer necessarily hold if the condition 

 of irrotationality near that point is relaxed, the problem was circum- 

 vented numerically by replacing the circulation condition on the single 

 cell in that corner by the condition (28) at the point A and the time 

 t = was avoided by applying (28) at t = t/4 just as was done with 

 the general free surface condition (Section 4C). 



Note that strong singularities could be introduced by unsuitable 

 mapping to the (a,b) plane. 



VI. SOME RESULTS INCLUDING COMPARISONS WITH LINEAR 

 SOLUTIONS 



A, Lagrangian Linearized Solutions 



Linearized solutions to the Lagrangian equations are obtained 

 by substituting X = a+^,Y = B+Ti into the equations of continuity 

 and motion and nfeglecting all multiples of derivatives of i and r|. 

 For incompressible and irrotational planar flow the Cauchy-Riemann 

 conditions 4a = - "Hb* tb = 'Ha result so that i + It) , and therefore 

 Z - c (where c = a + ib) is an analytic function of c. In the absence 



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