3.1. Superposition of Fundamental Solutions
53
FIGURE 3.2. A representation of
the line source problem.
which is located at z' with mass mdz', is
Y
,P(x,Y,Z) /'
I
Y 1
/z
/'
_ _ _ ....1/
x
d _ (m/8)dz'
[ x 2 + y2 + (z - Z' )2J
C - 8(nDt)3/2 exp
4Dt
.
x
Integrating along the z axis, the solution of the line source problem is
obtained, i.e.,
m/8 Joo [x 2 + y2 + (z - Z' )2J '
C(x, y, z) = 8( )3 / 2
exp -
dz
nDt
-00
4Dt
m/8
(x 2 + y2) (z - z') 1
00
= - 8nDt exp - 4Dt erf 2JDt -00
= m/8 ex (_ x
2
+ y2).
4nDt p
4Dt
(3.1.15)
Using a similar method, the solution of the plane source problem can also be
obtained. Assume that the solute mass, injected instantaneously into the unit
area along the yz plane, is 11,. The plane source may be divided into many
small strips which are parallel to axis z and have differential width dy'.
Each of them may be approximated by a line source (see Figure 3.3). From
Eq. (3.1.15), the differential concentration at point (x, y, z) caused by a line
source located in y' is
_ {j1j8)dy'
[x 2 + (y - y' )2J
dC - 4
exp -
.
nDt
4Dt
(3.1.16)
The solution of the plane source problem is obtained by integrating along the
y axis, i.e.,
_ /1/8 Joo [x 2 + (y - y' )2J '
C(x, t) - 4-exp -
4
dy
nDt -00
Dt
(3.1.17)