Non-Linear Shtp Wave Theory 
(y = 7) (2) 
Im Wdz = 0 (3) 
u-l (x — - o) (4) 
Im Wdz = 0 (jxl<1,.yrtet(x)) (5) 
where t(x) =t' / T' is the dimensionless thickness distribution and 
w=utiv. 
For a given body shape w depends on z,e,h and F. 
We consider now a ''thin body expansion", i.e. an expansion of w 
for e= o(1) and F, h = 0(1). This is the basic linearization pro- 
cedure used in ship wave resitance theory. 
Hence, with 
2 
Wize €, EF, bh) = 2 + ew, (z, F, h)+ e w, (Z, Fwd). 4 
Z; 
fi(zecojeiech pare + ef, (z, F, h) + « f, (z, BY OF? fee 
2 
ix, ek, Ooh) =renrte, Be bh) date 15 
; (S50 Ree ta) + 2 (6) 
the free surface (2), (3), radiation (4) and body boundary condition (5) 
become at first and second order 
df 
Im GF === £.) =p. (x)=0 (7) 
dz 1 1 (y Sys 
a ee (8) 
a (x — — ©) (9) 
hei tee Ox|<t, y= =k. 4 O) (10) 
df 2 
Im (iF? = - f,) = P(x) thd Bm) +,) fay (11) 
, 3 (y=0) 
i aeriety - Le (12) 
