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CONFIDENTTAtT™
Assume, then, that there is a uniform outward flow of population governed by the
diffusion equation
(1/k) (Of/0t) = (0°f/Ax?) + (Af/dy’)
(A20)
and subject to the initial condition that at t= 0 the population distribution is given by
Eq. A9. Fourier’s solution is simply
f= c/a) fr" [> exp (-—£& — uv’): (1/2m¢.7.) exp {- [L(x + 2heVt)?/20?
(A21)
~—[(y + 2kuvt)?/277]} dé du,
which evaluates at once as
[1/2av/ (a? + 2kt)(7?2 + Qkt)] exp {— [2?/2(02 + 2kt)] — [y?/2(7?+ Qkt)]}}.
(A22)
This leads to Result 5.
Result 6. Under a uniform outward flux of population the distribution remains bi-
normal but the variances becomelinear functions of time. That is, the relative position
and orientation of the city do not change; the distribution surface simply becomes flatter
and flatter.
.
The constant & is found from the network of roads and streets the city possesses.
Actually there are more traffic lanes near the center of the city than in the outskirts and
therefore the center should drain out relatively quickly. Rather than a flattening hummock,
the population distribution would probably resemble an expanding doughnut.
If concentric circles are taken about the center of population and people are periodically displaced outward through the available traffic lanes, a series of ‘‘snapshots”’ of the
diffusing population can be obtained. These can be plotted radially and fitted by
(r/o?) exp (— 12/202).
(A23)
A simple application of Fisher’s Method of Maximum Likelihood indicates that the best
value to take for o? is half the second moment. If these variances are plotted against
time it may be seen that Result 5 is an adequate approximation to the truth, unless the
road network is very spotty.
80
ORO-R-17 (App B)