Lagrangian Solutions of Unsteady Fvee Surface Flows 



where Aa is the a difference across a cell and the second and 

 third terms constitute truncation errors. Inspect this in the light of 

 a linearized standing wave solution (see Section 6A) , i.e. , 



X = a - M cos Vk t sin ka e 



Y = b + M cos Vk t cos kae*^"^ 



where the variables are non-dimensionalized as in Section 4C auid 

 k is the non-dimensional wave number in the a,b plane. Then, 

 the second and third terms of Eq. (26) will be insignificant provided 



kW.«l and ^«1 

 6 24 



respectively. Or, in terms of a wavelength, X = 2ir/k: 



-^ » 2 and t « 4 -^ AX (27) 



since Aa » AX, the X difference between points on the free surface. 

 The first condition states the inevitable; namely, that the solution 

 will be hopelessly inaccurate for (a,b) plane wavelengths comparable 

 with the mesh-length Aa, Given that the first condition holds then 

 the second says that t « SAX, For a travelling wave system the 

 same condition states that t should be less than the time taJcen for 

 a wave to travel one mesh length. This constitutes a restriction on 

 T which is usually more stringent than that of Section 5A, If, for 

 example, the depth of the fluid is divided into N intervals and the X 

 difference across each cell is of the same order as the Y difference 

 then T should be less than 8/N. 



A more difficult problem arises when the first condition is 

 considered alongside the fact, ascertained in the previous section, 

 that the field equation provides little or no resistance to disturbances 

 whose wavelength is equal to Aa. The only resort would seem to be 

 to some artificial dannping technique which would eliminate or sup- 

 press these small wavelengths. The technique used in the examples 

 to follow was to relax the W values on the free surface such that 

 W = ^w''^^ + (1 - P)W* where W''^^ was the value Indicated by the 

 free surface condition, W the value which would make the numeri- 

 cal equivalent of "^aaaa ^^ zero at that point and P was slightly less 

 than one half. 



131 



