This equation is also obtained directly from Eq. (3.37) for the stated assumptions of no
currents, no dissipation, and constant depth. Substitution of (3.40) into the Helmholtz
equation gives
Axx + 2ikAx + AVV = 0 (3.42)
and employing the above mentioned assumption concerning derivatives of Ax yields the
linear parabolic equation
2 ikAx + AVV = 0 (3.43)
Another method of obtaining a parabolic equation is to use a splitting method in which
it is assumed that 4> is composed of forward moving and backward moving components that
are uncoupled.
4> = + 4>- (3.44)
Uncoupled equations for + and are developed and since the equations are uncoupled
only the equation for + is retained. In the following the work of Kirby (1983) and Booij
(1981) is closely followed.
The second order elliptic equation
d_ /1££\
dx Vqf dx )
+ -yS = 0
(3.45)
can be split exactly into equations governing forward and backward disturbances $+ an $
which are given by
d$+
dx
d$~
dx
+7$+
7*~
Booij chose the relation
(3.46)
(3.47)
(3.48)
*