January 13, 2004
14:37
WSPC/101CEJ
00097
520
N. C. Kraus
which, for a small initial perturbation, yields xy ∼ 2at, a special case of Eq. (8).
Equations (9) and (10) make apparent the significant role the dimensions of the
initial perturbation take on the course of breach development. Although Eqs. (6)
(10) govern incipient breaching, they have limited value in being restricted to very
short elapsed time. More informative solutions are obtained by solving Eqs. (4) and
(5), which incorporate modification and control of breach growth by the inclusion
of equilibrium.
Closedform solution of Eqs. (4) and (5) is not found feasible for arbitrary values of
the factors a and b, but, for the situation of a = b (signifying QS = QB = Q), the
following solution emerges:
x = xe(1  f (x)et/τ )
(11)
and
z = ze(1  g(z)et/τ )
(12)
in which,
Ve
τ=
(13)
Q
where Ve = xezeL is the volume of the breach at equilibrium, and
1/α
1
1
x0
 αx/xe
f (x) =
(14)

xe
1  αx0/xe
x0/xe  z0/ze
α=
(15)
(1  z0/ze)x0/xe
1/β
1
1
z0
 βz/ze
g(z) =
(16)

ze
1  βz0/ze
z0/ze  x0/xe
(17)
β=
(1  x0/xe)z0/ze
Iteration is required to evaluate Eqs. (11) and (12) because of the appearance of the
dependent values in Eqs. (14) and (16). Equations (4) and (5) can also be solved
numerically, as through a RungeKutta procedure, and this was done to confirm the
validity of the closedform solutions above, as well as to perform calculations for
a = b.
In addition to making basic dependencies of breach growth explicit, Eqs. (11) and
(12) provide closedform solutions with which to check numerical solutions of the
simultaneous, nonlinear governing Eqs. (4) and (5). This simple morphologic model