We, in Midwest America, in Missouri, in Saint Louis are in the midst of a heat wave. This week in July has had a sequence of very hot days — the mornings are hot and humid, the afternoons are like ovens, and the temperature doesn’t let up at night. I got an exposure to this kind of weather when I lived in Nevada. You can  perhaps begin to see my preoccupation with the Sun. And my mission to find some shade, a shadow for some relief. And also, they put those amps and big speakers where I used to play my guitar. So, I have been playing my guitar in the shadow of the Gateway Arch.

Some people developed an elementary mathematical equation for the shape of the Gateway Arch in St Louis. I found a formula at this website.

I don’t know if this formula was used in the design and construction of the Arch, or was it developed later.

In Maxima, one can type in,

f(x):=211.49 – 20.96*cosh(0.03291765*x);

The units for x are meters, and x ranges from -91.2 to +91.2.

I can calculate a plot of the shadow, if I consider an X-Y-Z coordinate system, with Z up, Y positive pointing East, and X positive pointing South. At the Park, East is towards Illinois across the Mississippi river, and South is downstream of the Mississippi.

(%i5) h(x):=f(x)/tan((%pi/180)*el)

(%i6) sx(x):=x-h(x)*sin((%pi/180)*az)

(%i7) sy(x):=h(x)*cos((%pi/180)*az)

The units for the Sun’s azimuth az and elevation el are in degrees.

Here I used az = -10, and el = 45

How fast does the shadow move across the sidewalk? From when I start to play at about 10:30 to when I finish playing at about 11:30, the shadow on the south leg has moved about one tree. I stand at the sidewalk border next to the trees.

Another plot I made with az = -30 and el = 65.

This chart might be more accurate as I am using the Sun position charts I calculated

Saint Louis Sun Chart

Azimuth az = 0 would correspond I think perhaps to az = 90 in the Saint Louis Sun chart. Here’s a plan map of the grassy knoll and the old Court House and the Memorial Ave freeway overpass. I stand at number 13 when I play my guitar.