Recent Progress in the Calculation of Potential Flows 



where ' "•' • 



r^ = (x-a)2 + (y-b)2 + (z-c)^ , (30a) 



rj2 = (x-a)2 + (y + b)2 + (z- c)^ , ^3q^^ 



R2 = (x-a)2 + (z-c)2 , - ■ :.:^AiIv ■■-;■>. • ^3Q^^ 



^ o^ (30d) 



and where H^^ > = Jo + iY^ is the Hankel function of the first kind and g is the 

 acceleration of gravity. The term u is the vertical distance measured from the 

 image point. It is assumed in this formula that the free surface is the plane 

 y = and that the region of interest is the half space y < 0. The y-direction 

 may thus be considered vertical and the x- and z-directions horizontal. In 

 particular, R is the horizontal distance between the oscillating point source and 

 the field point. It is evident that the first two terms of Eq. (29) are the poten- 

 tials of two 1' r-type point sources - one at the location of the oscillating point 

 source and one at the image of this point in the free surface. The integration of 

 these two terms over a quadrilateral can be accomplished by the basic method 

 described early in this paper and described in detail in Ref. 1. It is the other 

 two terms that concern us. 



The third term of Eq. 29 is the potential of a 1 r-type line-sink of expo- 

 nentially decaying strength which starts at the image point (a. -b, c) and runs 

 vertically downward through the free surface to minus infinity. This term is 

 denoted the line- source term. The fourth of Eq. (29), which involves a Hankel 

 function, is called the Bessel function term. For large values of iR, it is known 

 that this term oscillates with increasing horizontal distance R at a circular fre- 

 quency of V. Thus V denotes the spatial circular frequency, and its relation to 

 the temporal circular frequency j is given in Eq. (30d), Rapid evaluation of Eq, 

 (29) over a quadrilateral element is the heart of the problem, for otherwise 

 there is no change in the formation of the basic integral equation. 



The basic method chosen for evaluating the line- source term is the Laguerre- 

 Gauss quadrature. However, accuracy becomes poor when the horizontal dis- 

 tance between the oscillating point source and the field point is small. Here, an 

 expansion valid for this condition is developed. A large amount of computing 

 has been done to determine the number of terms required in the Laguerre- 

 Gauss quadrature to meet the specified accuracy. Systematic studies have 

 been made to determine the range of validity of the special expansion. Details 

 of the formulas, as well as tables presenting the accuracy studies are contained 

 in Ref. 12, which is in the nature of a progress report to the Naval Ship Re- 

 search and Development Center on this work. 



The field due to the oscillating source of constant strength distributed over 

 a plane quadrilateral element is found by the multipole expansion method. This 

 method is applied to the last two terms of Eq. (29). Evaluation of the Hankel 

 function term in Eq. (29) is no particular problem, because standard Bessel 

 function subroutines for the computer are available. Several tables in Ref. 12 

 present the results of error studies in evaluating the field of a square element. 



357 



