We use the same development as above to obtain the dimensionless profile celerity for a

fixed value of φ on the diffusion wave profile, which in simplified form is

-*G*t /*G*x

=

(23)

~

.

1/ 2

π*t*

*x *+ *c*kt

(*x *+ *c*k

erfc

exp

*D*

(4*Dt*)1/ 2

4*Dt*

The forms of the linear diffusion wave and dynamic wave solutions have little apparent

resemblance to each other. The kinematic wave celerity appears in each term of the diffu-

sion wave solution (eq 20), but not in the dynamic wave solution (eq 14). The dynamic

wave celerities in the downstream *c*+ and upstream *c* directions both appear in *z *of the

dynamic wave solution, and *c*+ provides an upper bound on the speed of disturbances

moving downstream. There is no similar restriction on disturbances traveling downstream

in the diffusion wave solution. The diffusion coefficient *D *is an important parameter of the

diffusion wave solution, while η*c*0 and η are corresponding parameters of the dynamic

wave solution. The Froude number *F*0 appears often in the dynamic wave solution, but is

absent from the diffusion wave solution unless the inertial diffusion coefficient is used.

Nonlinear monoclinal wave solutions that are analogous to the linear dynamic wave

and diffusion wave solutions will now be developed and compared. The term "monoclinal

wave" refers to the classical solution, and "monoclinaldiffusion wave" is the solution de-

veloped after neglecting the inertia terms of momentum equation (eq 2). A monoclinal

wave profile does not exist for the kinematic wave equation because diffusion is not

wave development of Whitham (1974), emphasizing the contribution of inertia to the solu-

tion.

We seek a solution that depends on a single variable *X *= *x * *Ut*, where *U *is the constant

profile celerity, and rewrite eq 1 and 2 as

-*Uy*′ + *vy*′ + *yv*′ = [y(*U *- *v*)]′ = 0

(24)

(*v *- *U *)*v*′ + *gy*′ = *gS*0 - 2

(25)

*

respectively, where Chezy conveyance coefficients, dimensionless *C *and dimensional *C*,

*

are related as

.

*

The first term on the left side of eq 25 follows from the inertia terms of eq 2. Integration of

eq 24 yields

(26)

where the constant of integration *B *represents a wave overrun unit discharge.

The profile celerity *U *is obtained from eq 26 using the flow states on either side of the

wave as

1/ 2

(

)

3

3

2

= *g C** S0

= *v*f + *v *+ *v*

2

(27)

* y *-*y*

0

0

f

f

7