= b q,

(9)

Equations 8 and 9 are simultaneous non-linear equations for channel depth *z *and width *x *as

a function of the input rate (which can be time dependent) and time. Equation 8 indicates

that *z *will increase more rapidly as the width decreases, and Eq. 9 indicates that the width

numerically for a general situation with time-dependent variables. An analytic solution

approach for rapid desk study is given next. The analytic solution reveals physical

Ongoing maintenance of channels will not allow the depth to become less than project

depth or allow the width of the channel to be greatly reduced. These conditions are

equivalent to stating mathematically that practical applications concern a relatively short

time interval after dredging as compared to the total time required to fill a channel

completely. For this situation, the equations can be linearized under the reasonable

assumptions *z*/*z*0 << 1 and *x*/*W*0 << 1. By expansion of denominators, Eqs. 8 and 9 become,

=

(10)

and

=

(11)

which are now simultaneous linear equations for *z *and *x*.

Differentiating Eq. 10 with respect to time and substituting Eq. 11 into the resultant

equation to replace *dx*/*dt *gives,

+ 2*b * - *cz *= *d *,

(12)

where the quantities *b*, *c*, and *d *are defined as:

(13)

2

2 *W*0 z0

A second initial condition for *z *was introduced through the first derivative as determined

from Eq. 8 evaluated with the initial conditions on *x *and *z*. The solution of Eq. 12 is found

to be,

(14)

Kraus and Larson

6