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*

=

b

g

(9)

The solution of this equation is

F x - *x *I+

b

g

Gx - *x *J

(10)

ln

H K

where τ′s the characteristic time scale as modified by the channel slope, as

i

F

I

G

J

=

(11)

H

K

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.

Wave-induced longshore currents and the direct impact of waves on the exposed

end of a spit protruding into an inlet will tend to curve the distal portion of the spit

away from the ocean. The flood current combined with wave-induced longshore

currents and sediment transport around the distal end of a spit act together in curving

the spit bayward. Focus of wave energy by refraction over an ebb-tidal shoal

increases the trend of curving a spit.

Curving of a spit can be represented phenomenologically in a simple analytic-

solution context by tracking a cross-shore component of movement of the tip of the

spit. As an example, suppose the cross-shore (directed bayward) migration speed of

the tip is a linear function of distance to the channel from some location far up drift

of the inlet, as in previous examples. Then the cross-shore migration speed *v*S of the

tip of the spit can be represented as

(12)

where *v*C is the cross-shore migration speed of the spit at the center of the channel.

The cross-shore coordinate of the spit tip *y*S is given as

(13)

Kraus

8

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