Instantaneous Point-source Solution - IfH

Section 5: Point-source solution with advection 19
5. Point-source solution with advection
The final step is to substitute the coordinate transformation for the
moving reference frame into the point source solution (35). In the
moving coordinate system, this solution is
( )
M
C(ξ, τ) =
A √ 4πDτ exp − ξ2
. (37)
4Dτ
Our coordinate transformation was ξ = x − (x 0 + ut) and τ = t.
Substituting into (37) gives:
(
M
C(x, t) =
A √ 4πDt exp − (x − (x 0 + ut) 2 )
. (38)
4Dt
Figure 5 shows the schematic behavior of this solution for three different
times, t 1 , t 2 , and t 3 . A second animation shows this solution
in action.
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