476 HYDROLOGY
also plotted in Figure 7. Note the general agreement in shape
of the C 0 and curves, etc. Exact agreement is lost because
of the diminishing accuracy of the d O /d t terms, etc. as ∆ t
increases. Note also how rapidly the C values change when
∆ t is approximately equal to K.
These remarks are not necessarily meant to dissuade
hydrologists from using the Muskingum method. The inten-
tion is to illustrate some of the pitfalls so that it may be pos-
sible to evaluate the probable validity or constancy of the
coeffi cients (Laurenson^22 ). The worst situation appears to be
when ∆ t ; K, because in real rivers K decreases with rising
stage and the C values are very sensitive to whether ∆ t is just
less than or greater than K.
A very real problem in the application of the Muskingum
method, and in fact of any channel routing procedure, is the
problem of lateral infl ow to the channel reach. Given the
infl ows and outfl ows for the reach as functions of time, it
is necessary to separate out the lateral fl ows before best fi t
values of K and x can be determined. The lateral fl ows will,
in general, bear no relationship to the pattern of main stream-
fl ows. Sometimes it is possible to use fl ow measurements
on a local tributary stream as an index of total lateral fl ow.
The cumulative volume of lateral fl ow can be determined
by subtracting summed infl ows from summed outfl ows. The
measured tributary fl ow can then be scaled up to equal thetotal lateral fl ow and these fl ows can then be subtracted from
the reach outfl ows. This residual outfl ow can be used in the
determination of the Muskingum coeffi cients.
Sometimes it is possible to fi nd periods of record where
lateral fl ows are small or perhaps have a more predictable
pattern, such as during recession periods. Also, the routing
coeffi cients can be refi ned by an iterative procedure and by
using various sets of data, although not infrequently it is
found necessary to defi ne Muskingum coeffi cients for dif-
ferent ranges of fl ow, because real stream fl ow is not linear
as is assumed in the model.KINEMATIC WAVE THEORYIt has been known for many years that fl ood wave movement
is much slower that would be expected from the shallow
water wave theory result of
gy.
Seddon^25 showed as early
as 1900 that fl ood waves on the Mississippi moved at about
1.5 of river velocity. The theoretical reasons for this slowness
of fl ood wave travel have only been realized during the last
twenty years. The theory describing these waves is known
either as monoclinic wave theory or kinematic wave theory.
Kinematic wave theory has suffered from neglect
because of its usual presentation in the literature. The
common approach is to start with the continuity equation
and a simplifi ed Manning equation. These equations are
combined to yield Seddon’s Law that fl ood waves travel at
about 1.5 times the mean fl ow velocity. Such an approach
restricts the method to an unalterable fl ood wave translating
through the channel system. Although such a simple model
has the benefi t of approximating the real situation, much
more useful and general results are available by setting up
the two equations in fi nite difference form and carrying
out simultaneous solution on a computer. In particular the
method handles lateral infl ow with great ease and also cal-
culates discharge against time, eliminating the guess work
from the time calculation.
Kinematic wave theory can be seen to be a simplifi cation
of the more general unsteady fl ow theory in open channels,
for which the equation of motion and the continuity equation
can be written (Henderson and Wooding 4,26 )Sy
xv
gv
xgv
tv(^0) cR
2
2
1
∂
∂
∂
∂
∂
∂
(36)
Bed-Slope Depth Slope Convective Acceln. Local Acceln.
Friction term
∂
∂
∂
∂
( )
Q
x
B
y
t
dilateral inflow.
(37)
In different situations various terms in equation (36) become
negligible. In tidal fl ow the friction and bed-slope terms
are sometimes omitted, although this simplifi cation is not
strictly necessary. In this way the positive and negative
–.7
–.6
–.5
–.4
–.3
–.2
–.1
0
.1
.2
.3
.4
.5
.6
.7
.9
.8
1.0
VALUES OF C's
K
∆T
LEGEND
C-Generated Values
C'-Estimated Values
C 0
C' 0
C' 2
C 1
C' 1
C 2
0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6
FIGURE 7 Muskinghum coefficients as a function of storage
and routing period.
C008_003_r03.indd 476C008_003_r03.indd 476 11/18/2005 10:29:28 AM11/18/2005 10:29:28 AM