Falttnsen 
The lowest-order equations become 
Z 2, 2 
(2 . ) eye yh Bae: (28) 
ay’ az" 4 
0 = 
(ere a) ies => 0 on s4—e-810 ; (29) 
ae yi) = - Cn ke” on the body. (30) 
In addition, Y, must match with the far-field solution. 
A one-term far-field solution is assumed to be the potential 
associated with a line distribution of sources of density 
spread along the line y=z=0, -L/2 < x < L/2. That solution has 
been obtained in a previous chapter, and a one-term inner expansion 
of a one-term far-field solution can be found from (19). For any fixed 
x greater than -L/2, it is obvious that a one-term inner expansion 
is 
and that the second-order term in the inner expansion is of order 
V2 compared with the first-order term, 
(31) should match with a one-term outer expansion of y 
i(wt- vx . 1 
e , as determined from (28), (29) and (30). 
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