It is seen that the presence of the slope term, making the depth of active movement
increase, acts to slow the rate of growth with increasing distance. Applying the same
condition on the Q' as in the previous example, Eq. 8 becomes
s
$
-Q
x - xc
dx
=
b
g
(9)
dt W xc - x0 Do - S x
The solution of this equation is
F x - x I+
b
g
S
t
Gx - x J
x - xC = -
C
(10)
ln
H K
Do - S xC
τ′
o
C
where τ′s the characteristic time scale as modified by the channel slope, as
i
F
I
S
G
J
τ′ τ 1+
=
(11)
xC
H
K
Do
If S = 0, Eq. 10 reduces to Eq. 5. With S nonzero, Eq. 10 must be solved by
iteration. A computer program was written for this purpose, and the result with the
same values as in the previous example and for S = 0.01 is plotted in Fig. 4 with a
dashed line. Because more material is required to elongate the spit as it approaches
the (deeper) channel, a longer duration is necessary to extend the same length as for
the case of constant depth of active movement.
Example 4: Spit Growth in Presence of Lateral Force