Environmental Engineering Reference
In-Depth Information
which can be easily verified. It should be noted here that the equation can be of
a different type at different points in the domain, but for our purpose we shall assume
that the equation under consideration is of a single type in a given domain.
We suppose first that
a
11
,
a
12
and
a
22
are non-zero. Let
be the new
variables such that the coefficients
A
11
and
A
22
in Eq. (1.4) vanish. Thus
ξ
and
η
2
x
2
y
2
x
2
y
A
11
=
a
11
ξ
+
2
a
12
ξ
x
ξ
y
+
a
22
ξ
=
0
,
A
22
=
a
11
η
+
2
a
12
η
x
η
y
+
a
22
η
=
0
.
These two equations are of the same type and hence we may write them in the form
a
11
z
x
+
a
22
z
y
=
2
a
12
z
x
z
y
+
0
,
(1.5)
. Dividing through by
z
y
,Eq.(1.5)
in which
z
stands for either of the functions
ξ
or
η
becomes
a
11
z
x
z
y
2
2
a
12
z
x
z
y
+
+
a
22
=
0
.
(1.6)
Along the curve
z
=
constant, we have
d
z
=
z
x
d
x
+
z
y
d
y
=
0
.
Thus
d
y
z
x
z
y
, and therefore, Eq. (1.6) may be written in the form
a
11
d
y
d
x
d
x
=
−
2
2
a
12
d
y
d
x
−
+
a
22
=
0
,
(1.7)
the roots of which are
a
12
−
a
12
+
a
11
a
22
d
y
d
x
=
,
(1.8)
a
11
a
12
−
a
12
−
a
11
a
22
d
y
d
x
=
.
(1.9)
a
11
These equations, which are called the
characteristic equations
, are the ordinary dif-
ferential equations for families of curves in the
xy
-plane along which
are
constant. The relation between
x
and
y
specified by Eq. (1.7) can be represented by
a curve on the
xy
-plane, called a
characteristic curve
. Since the characteristic equa-
tions are first-order ordinary differential equations, their solutions may be written
as
ξ
and
η
ξ
(
x
,
y
)=
c
1
,
c
1
=
constant
,
η
(
x
,
y
)=
c
2
,
c
2
=
constant
.
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