115
_pg d \dB B, dB*^j r* / cosh 2k[h0 + r¡) cosh 2k(h + z)
4 dx¡ [dxf dx( J J-h \2fcsinh2A:(h0 + f¡
PI
4
4 dx
P9 d
__n .
, t , J ... x r¡) 2k sinh 2k{h0 + fj))
dB B* I dB* b\ ( C08b2M^ + ^-) /L -1 Binh2k{h0 + fj) \
dx( dx( dx( J \ 21; sinh 2k(h0 + fj)' 0 ^ 4fc2 sinh 2k{h0 + fj) J
r 1 '2khcoeh2kh sinh2fch
dB dB* J /
t [dx( + dx( B\ (
4fc2 sinh 2fch 4A:2 B\nh2kh)
dB
dB*
4dx¡&B+d^B
Expanding the derivative term in E!q. (B.16)
dx(
dB dB*
B* + -B
dx
(
dx.
d2B d2B* dB dB*
rB* + -5-5-B + 2-
dx^
dx^
dx, dx
f
Recognizing that f is a free index so that = V*
d2B
dx
5S7
= VB = -Jfc2B
d2B*
dx?
= Vj[B* = k2B*
(B.16)
(B.17)
which results from use of the Helmholtz equation which governs diffraction in water of
constant depth and
2^§!r=2|vB|! (B18)
so that the expanded derivative term (B.17) becomes
d2B. d2B* dBdB*
dx2 B + dx2 B + 2dxf dxf
= 2|V*B|2 2Jfc2|B|
(B.19)
and the result is
/-l ("/," ^~Z) = T (|VS|1 2F 1] (B.20)
The author wishes to express some reservations concerning the validity of Eq. (B.20).
The shallow water asymptote for this expression is zero since for small k(h+fj), coth 2Jt(h +
fj) 2k(h+fi) i 2fc(h+^)coth2fc(/+^)-l 0. In deepwater, however, coth2A:(h+^) 1
so that 2k(h + fj) coth2(/i + fj) 1 2 2kh. If the derivative terms are non-zero, the Szz