X and the City: Modeling Aspects of Urban Life (7 page)

For
T
_c
= 1 s,
a
≈ 0.22; for
T
_c
= 0.5 s,
a
≈ 0.39, and
T
_c
= 0.1 s,
a
≈ 0.92. That’s more like it! A graph illustrating the dependence of
a
on
T
_c
is shown in
Figure 3.3
. Again, note the rapid fall-off when
T
_c
< 1 s.

In reality, the physics of concert halls must be very much more complicated than this, but you didn’t really expect that level of sophistication in this book, did you?

Figures 3.4
and
3.5
show, respectively, rebuilding at Ground Zero in New York City, and an office tower at Delft University of Technology in The Netherlands. In both cases we see tall buildings, very much taller than most kinds of trees. But trees sway in the wind, right? So why shouldn’t the same be true for tall buildings? In fact, it
is
true. The amplitude of “sway” near the top of the John Hancock building in Chicago can be about two feet. At the time of writing, the world’s tallest building has opened in Dubai. The Burj Khalifa stands 828 meters (2,716 ft), with more than 160 stories. It may not be long before even this building is eclipsed by yet taller ones. How much might such a building sway in the wind? Structural engineers have a rough and ready rule: divide the height by 500; this means the “sway amplitude” for the Burj Khalifa is about 5.5 feet! The reason for such swaying is intimately associated with the wind, of course; wind flows around buildings and bridges in a similar fashion to the way water flows around obstacles in a stream. Careful observation of this dynamic process reveals that small vortices or eddies swirl near the obstacle, be it a rock, twig, or half-submerged calculus book. The atmosphere is a fluid, like the stream (though unlike water, it is compressible) and buildings are the obstacles. If wind vortices break off the building in an organized, rhythmic fashion, the building will sway back and forth. Or, to put it another way, a skyscraper (or a smokestack) can behave like a giant tuning fork!

Figure 3.4. Rebuilding at the site of Ground Zero, New York City. Photo by Skip Moen.

Figure 3.5. Tower at the Delft University of Technology, Delft, The Netherlands. Photo by the author.

The oscillations arise from the alternate shedding of vortices from opposite sides of the tall structure. Their frequency depends on the wind speed and the size of the building. Not surprisingly, such “flow-induced” oscillations can be very dangerous, and engineers seek to design structures to minimize them [
9
].
Figure 3.6
illustrates in schematic form how such vortices can develop around a cylindrical body and alternatively “peel off” downstream.

There are two dimensionless numbers that are very important in a study of flow patterns around obstacles. One is called the
Reynolds number
, denoted by
R
, and is defined by

Figure 3.6. Schematic view of vortex formation around a cylindrical obstacle.

where
L
and
U
are characteristic size and (here) wind speed, respectively, and
ν
is a constant called the kinematic viscosity. For small values of
R
(
R
< 1) there is no separation: the cylinder just causes a symmetrical “bump” in flow. For
R
> 1 things get more complicated; in the range 1 <
R
< 30 small vortices develop behind the obstacle, but symmetrically about the axis of symmetry, so there is nothing yet to drive oscillations. For
R
> 40 this symmetric flow becomes unstable, and vortices are shed alternately from side to side (as viewed from above). This pattern can exist for Reynolds number up to several thousand (even up to 10
⁵
if the obstacle is very “smooth”).

The other important number is the
Strouhal
number,
S
. It is named after
Vincenc Strouhal
, a Czech physicist who in 1878 investigated aeolian tones—the “singing” of wires set into oscillation by the wind. Again, in terms of the wind speed
U
and the diameter
d
of the wire,
S
is defined by

f
_s
being the frequency of the vortex shedding. Naturally it is to be expected that the Strouhal number is related to the Reynolds number, but for a wide range of the latter,
S
is almost constant, varying at most between 0.15 and 0.2. When
f
_s
is close to the natural frequency of vibration of the obstacle, the latter can “capture” the former, and the resulting resonance can be very dangerous, for obvious reasons. This phenomenon is called “lock-in.”

And speaking of swaying buildings, a few minutes before I wrote this, my office building started to sway. I’m on the second floor, and didn’t feel
anything, but I heard the bookcases behind me move back and forth. They have a lot of books in them, and are very heavy. Subsequently I heard tales from across campus of swinging lights and moving floors—yes, Virginia, we just had an earthquake! Initial reports indicated it was 5.8 on the Richter scale, with an epicenter midway between Charlottesville and and Richmond, Virginia (and about 80 miles from Washington, D.C.). Tremors were felt as far as New York, Massachusetts, Ohio, Tennessee. and the Carolinas, and the U.S. Capitol and the Pentagon were evacuated. And as I write, Hurricane Irene, currently a Category 3 storm, is making her way steadily toward the Eastern seaboard! Thankfully, by the time it made landfall the storm had weakened to Category 1, but it still caused one death and extensive property damage in the Outer Banks, North Carolina, New Jersey, New York, and Vermont.

Note that in the UK, mall = shopping center = shopping centre!

More and more frequently, malls are being located out of towns and cities, perhaps as a single “megastore” or as a mall-like complex; however, many are still built in cities with pedestrian walkways. What follows is a very simplistic model for the competitiveness of two such malls, carried out by considering how one “stacks up” against the other, as measured by the number of trips (
N
) made per unit interval to a given location.

An important (but rather subjective) question for urban planners is “How attractive is the mall likely to be?” Factors such as the variety of stores within it, ease of access, and parking facilities all contribute to the answer. Essentially, the attractiveness of the mall determines whether one will prefer to travel farther to get there, as opposed to shopping at a nearer but less attractive one. Another question, fundamental for developing a mathematical model for the competition between malls (or specialty stores and shops, for that matter) is “How does
N
depend on the distance
d
from the mall?” [
10
]. Many factors, including those mentioned above, must be contributory, and it is therefore unlikely that
N
will be a simple function of
d
. Nevertheless, that is beyond the scope of this book, and we shall content ourselves with a simpler approach to illustrate some basic principles involved.

Obviously we expect
N
to decrease with distance
d
, but how rapidly? Another question concerns what we mean by distance here, that is, should we use
the standard Euclidean “metric” in the plane, or a modified version, weighted in some manner to account for geographical or social factors? In all likelihood the latter, but again for simplicity we will stick with circular symmetry, consistent with models that appear later in the book. (However, see
Appendix 3
for a brief introduction to the so-called
taxicab
or
Manhattan
metric.) To that end, we define a constant “attraction factor”
a
_i
> 0 for each mall (here
i
= 1, 2) and write