Ogilvie 



I have taken the trouble of writing out the inner expansion of the 

 outer expansion in three ways just to point out how, in this problem, 

 there is an additional term in the lowest-order expression each time 

 we add another term of higher order in the outer expansion. Each 

 of the three terms included in (2-44) contributes to the e term in 

 (2-45c). This phenomenon occurs frequently, and its occurrence is 

 the reason that one must proceed step-by- step in the matching. In 

 the present problem, one would be in some difficulty if he tried to 

 write down an arbitrary number of terms in each expansion and 

 inmmediately start nnatching. 



Next we formulate the near- field problem. Instead of making 

 the formal changes of variable, x = cX and y = eY, we sheill simply 

 understand now that, in the near field, 



x = 0(c) and y = 0(e); also 8/ax=0(e'') and 8/8y=0(€''). 



Of course, differentiation with respect to z does not ciffect orders 

 of magnitude. 



The Laplace equation can be written in the form: 



<l>xx + <^yy = - <l»zz. (2-46) 



where the right-hand side is c^ higher order than the left-hand side. 

 The boundary condition on the body is: 



= ^,(g^ ± h,) - 4»y + <)),(g, ± h,) on y = g ± h. (2-47) 



The last condition is equivalent to requiring that B^/bn = on the 

 body, where 9/8n denotes differentiation in the direction normal to 

 the body surface. An alternative statement is the following: 



BJf (^g, + h,)«<>, T<()v ■ (hz±gz)(^z ^^ y^ g±h, (2-47') 



™ V[ 1 + (gx ± Kf] V[ 1 + (gx =t h/] 



where 8^/9N is the rate of change in a plane perpendicular to the 

 z axis, measured in the direction normal to the body contour in 

 that cross section plane. Note that the left-hand side is 0(<j>/c), 

 since differentiation in the N direction has the same order-of- 

 magnitude effect as differentiation with respect to x or y. The 

 right-hand side, on the other hand, is 0(<j)c) , since g and h are 

 both 0(€). 



Now let there be an inner expansion: 



702 



