*Figure 11. Case IV monoclinal and monoclinaldiffusion profiles at*

*two distance scales for depth ratios ranging between 1.1 and 10.*

similar to case III with distance scales substantially increased. At *y*r = 2 the profiles begin to

separate at a dimensionless depth of about 0.3, and an overrun of the leading edge occurs at

*y*r = 10 up to a dimensionless depth of 0.03. Case V, presented in Figure 12, is qualitatively

similar to cases I and II except that the largest diffusion coefficient produces the longest pro-

files of all the cases.

General results from these comparisons are that monoclinal and monoclinal-diffusion

profiles agree for all values of *y*r with *F*0 ≤ 0.2, and that profile length increases with η or *D*.

At small *F*0 the depth ratio needed to produce a shock is large, and the shock height and

overrun distance of the leading edge are small. These results agree with the Lighthill and

Whitham (1955) contention that dynamic waves are subordinated at "well subcritical"

Froude numbers. As *F*0 increases the monoclinal waves differ over a larger portion of the

profile, shocks occur at smaller depth ratios and their dimensionless amplitudes increase,

and for a given η the profile length decreases. General dimensionless monoclinaldiffusion

profiles for each depth ratio are given in Figure 13 as a function of *x *, and include the profiles

~

of all cases as indicated by eq 35. Similar dimensionless monoclinal profile plots in Figure 14

are almost as well-behaved, but differences near the leading edge occur for each *y*r due to

their *F*0 dependence.

Steady flow rating curves relate river stage or mean depth at a given location to a unique

discharge. The governing equation for linear waves holds with either *v *or *y *as the depen-

dent variable, indicating a fixed steady flow rating. In unsteady flow the discharge relating

to a given stage generally varies from that for steady flow, depending on the rate of rise or

fall of the hydrograph. We will develop and compare dimensionless monoclinal wave and

steady flow ratings. Using eq 26, an equation for the monoclinal wave unit discharge can be

written in terms of depth and depth ratio as

18