Figure 6. Dimensionless monoclinal profile celerity and overrun discharge as a function of depth ratio

Figure 6. Dimensionless monoclinal profile celerity and overrun discharge

as a function of depth ratio. The stable profile range spans a depth ratio range

from 1 to an upper bound, indicated by the dots, that decreases as the Froude

number increases. Panel b is an expanded view of the shaded area of panel a.

For monoclinaldiffusion waves eq 41 has a value of 1, and the deviation from 1 indicates

the relative importance of inertia. The same cases analyzed for linear waves are used to

depict monoclinal waves, with time deleted from the parameters considered and depth

ratio across the wave representing amplitude added. Evaluations of eq 41 for each case are

presented in Figure 7 as a function of yr for selected values of y , all with limit 1 - F02/4 at yr

= 1. At low Froude numbers the part of the profile affected by inertia is very close to leading

edge (small y ), and then only when depth ratios are large. Differences between monoclinal

and monoclinaldiffusion profiles near the leading edge increase with yr and F0. Negative

values of eq 41 indicate that the profile point y is located on a shock. Cases I and II have a

calculated stability limit of yr = 51, where the dimensionless shock amplitude is smaller

than 0.001. In case III with high F0 much more of the profile is affected by inertia, and larger

shocks occur at relatively small depth ratios. Case IV is intermediate between these condi-

tions, and case V is similar to cases I and II.

Monoclinal and monoclinaldiffusion dimensionless depth profiles for case I, presented

in Figure 8, are in exact agreement except for the leading edge at yr = 50, where the diffusion

solution leads. The front half of these profiles shorten and steepen as yr increases to 10. At

yr = 50 the wave front lengthens, and the steepest portion continues forward to the leading

edge. The profile comparisons and trends for case II in Figure 9 are identical to those of case

I, except that profile lengths are significantly increased as a result of much higher diffusion.

Case III, depicted in Figure 10, has monoclinal and monoclinaldiffusion profiles that pro-

gressively separate below a dimensionless depth of about 0.4. The leading edge of the

steeper monoclinal profile lags behind that of the diffusion profile. At yr = 5, outside the

stable profile range, overrun of the leading edge of the monoclinal profile indicates shock

formation up to a dimensionless depth of about 0.1. Case IV in Figure 11 is qualitatively