Dagan 
We begin with the body correction. To satisfy (15) we have 
to cancel arround the source j, on |z - Z I= €€); , the velocities 
induced at z) by the first order terms of (16), excepting the first 
term representing the source itself. In other words we require 
we =(df, jk /dz) -[e; In (z - Z| )/2n]> 0 for | z - z | see". To 
satisfy this requirement at second order we have to superimpose a 
source of strength e%e: uv (x, -h) to the original source of 
strengthee; and also a vertical doublet of strength - ee) vj (x), - h). 
Since we consider only second order terms, we disregard the vertical 
doublets which contribute only at third order. Adding the appropriate 
sources at Z) and z, and carrying out the computations (for 
details see Dagan, 1972a) we obtain finally, for the second order 
streamfunction far downstream, the following expression 
Zz 2 Me 
b =ix?/ FE 2 is /E 2it,\/ Fa 
5 Oe J € 
¥2 jk (x, O)S= Tm € {(eve ees )iB + 
2 a 
ix, /F ise e~! = 
. J = k 
ve ei Lle e ee Baie ae 
Zz 
i b 
pertk/F toe lt (x — es) (19) 
jk 
where 
2 
Spgs /F 
Zz 
EP Liemeor [ n° Op( bo = oe9] 
“lag a aa ee moe a 
jk ek (Fe 2 ie, gE ), jk F 
2 
i sae 2h / F cost (20) 
jk jk 
In deriving (19) and (20) we have assumed that k>j, 
x, > x and jk= (x, apy aga ee 
We consider now, the second order free surface correction 
which results from the linearized pressure P, (x) in (11), acting on 
the free surface. In the case of a pair of sources P, can be written 
as follows 
ig ee €. € 
P a eee (21) 
2,jk Mawes Sip te aye Buk 4m P ix 
