A mathematical model of slope development is summarized by the relation
_??_
where
u: elevation,
t: time,
x: horrizontal distances,
a: subdueing coefficient,
b: recessional coefficient,
c: denudational coefficient and
f (
x,
t): arbitrary function of
x and
t, respectively. Effects of the coefficients are shown in figs. 1-(A), (B) and 2-(A).
In order to explain the structural reliefs, the spatial distribution of the rock-strength against erosion owing to geologic structure and lithology is introduced into the equation by putting each coefficient equal a function, in the broadest sence, of
x,
t and
u. Two simple examples of this case are shown in fig. 5.
The effects of tectonic movements, for instance of faulting, are also introduced by the function
f (
x,
t), which is, for many cases, considered to be separable into
X (
x) and
T (
t), where
X (
x) and
T (
t) are functions of
x only and
t only, respectively. An attempt to classify the types of
T (
t) has been made.
Generally speaking, provided the coefficients
a,
b and
c are independent of
u, the equation is linear and canbe solved easily. With suitable evaluation of the coefficients (as shown, for example, in fig. 4-(A)), this linear model can be used to supply a series of illustrations of humid cycle of erosion, especially of the cycle started from faulting.
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