Free X and the City: Modeling Aspects of Urban Life by John A. Adam

quantity Δ
T
above, assumed to be small enough to neglect its square and higher powers. The parameter
γ
depends on several constants including gravity and air flow speed, and is related to the Reynolds number discussed in Chapter 3 . The non-dimensional equation for
θ
(
x
,
z
) is

     
    The basic method is to seek elementary solutions of the form

     
    where “Re” means that the real part of the complex function is taken, and
k
is a real quantity On substituting this into equation (6.8) the following complex biquadratic polynomial is obtained:

     
    There are four solutions to this equation, namely,

     
    but for physical reasons we require only those solutions that tend to zero as
z
→ ∞. The two satisfying this condition are those for which Re
σ
< 0,

     
    Using these roots, the temperature solution (6.9) can be expressed in terms of (complex) constants
c
1 and
c
2 as

     
    Note from equation (6.12) thatUsing specified boundary conditions, both
c
1 and
c
2 can be expressed entirely in terms of
σ
1 and
σ
2 , though we shall do not do so here. The authors note typical magnitudes for the parameters describing the heat island of a large city (based on data for New York City). The diameter of the heat island is about 20 km, with a surface value for Δ
T
≈ 2
C
, and a mean wind speed of 3 m/s.
Exercise: Verify the results (6.10)–(6.12).
     

Chapter 7
NOT
DRIVING IN THE CITY!

     
     
     
    As we have remarked already, cities come in many shapes and sizes. In many large cities such as London and New York, the public transportation system is so good that one can get easily from almost anywhere to anywhere else in the city without using a car. Indeed, under these circumstances a car can be something of an encumbrance, especially if one lives in a restricted parking zone. So for this chapter we’ll travel by bus, subway, train, or quite possibly,rickshaw. Whichever we use, the discussion will be kept quite general. But first we examine a situation that can be more frustrating than amusing if you are the one waiting for the bus.
X
=
T
: BUNCHING IN THE CITY
     
    In a delightful book entitled
Why Do Buses Come in Threes?
[ 8 ] the authors suggest that in fact, despite the popular saying, buses are more likely to come in twos. Here’s why: even if buses leave the terminal at regular intervals, passengers waiting at the bus stops tend to have arrived randomly in time. Therefore an arriving bus may have (i) very few passengers to pick up, and little time is lost, and it’s on its way to the next stop, or (ii) quite a lot of passengers to board. In the latter case, time may be lost, and the next bus to leave the terminal may have caught up somewhat on this one. Furthermore, by the time it reaches the next stop there may be fewer passengers in view of the group that boarded the previous bus, so it loses little time and moves on. For the next two buses, the cycle may well repeat; this increases the likelihood that buses will tend to bunch in twos, not threes.
    The authors note further that
if
a group of three buses occurs at all (and surely it sometimes must), it is most likely to do so near the end of a long bus route, or if the buses start their journeys close together. So let’s suppose that they do . . . that they leave the terminal every
T
minutes, and that once they “get their buses in a bunch” buses
A
and
B
and
B
and
C
are separated by
t
minutes (where
t
<
T
of course). A fourth bus leaves
T
minutes after
C
, and so on. The four buses have a total of 3
T
minutes between them, initially at least. When the first three are bunched up, the fourth bus is 3
T
− 2
t
minutes behind
C
(other things such as speed and traffic conditions being equal). If you just missed the first or the second bus, you have a wait of
t
minutes for the third one; if you just missed that then you have a wait of 3
T
− 2
t
minutes. Thus your average wait time under these circumstances is just
T
, the original gap between successive