January 13, 2004
14:37
WSPC/101CEJ
00097
519
Analytical Model of Incipient Breaching of Coastal Barriers
means of assuring robust results though imposition of morphologic constraints  in
Insertion of Eq. (3) into Eqs. (1) and (2) and taking the limit yields the following
coupled, firstorder nonlinear governing equations for breaching:
,
1
QS
x
a
dx
a=
=
, x(0) = x0 > 0
(4)

dt
z
xe
L
and
,
1
dz
b
QB
z
=
,
z(0) = z0 > 0
(5)
b=

dt
x
ze
L
Equations (4) and (5) cannot be solved without specifying a nonzero perturbation
of the barrier (representing an indentation or "pilot channel" through the barrier),
as given by a finite initial width x0 and initial depth z0. Equation (4) describes a
onesided breach such as constrained by a jetty in nature or a wall in a flume. If both
sides of the breach can move, then the value of QS in Eq. (4) should be doubled.
3.1. Solution for xe and ze → ∞
This special case corresponds to very short elapsed time of incipient breaching, for
which the governing Eqs. (4) and (5) reduce to dx/dt = a/z and dz/dt = b/x. The
solution of this simplified set of coupled nonlinear equations is,
b/a
x
a
a+b
x
a+b
x = 0 a + (a + b) 0 t
(6)
z0
and
a/b
z
b
a+b
z0
a+b
z=
t
(7)
+ (a + b)
b
0
x0
If the initial perturbation or pilot channel is small, the crosssectional area of
the breach is found to grow as
QS + QB
t
(8)
xy ∼ (a + b)t =
L
indicating a linear increase in area immediately after the breach occurs. For the
special case of a = b, Eqs. (6) and (7) simplify to,
1/2
x
x0
2
(9)
t
x=
0 + 2a
z0
and
1/2
z
z0
2
z=
+ 2a t
(10)
0
x0