Antenna wires in the breeze

When I was 7 or 8 years old, I was given a crystal diode radio.  It made a big impression on me.  The wire antenna that was hung outside my bedroom window would sway in the breeze, and it was fun to gaze at it.  Overhead wires continue to capture my attention, particularly if they sag quite a lot and sway in the breeze.

The Hyperbolic Cosine and the Catenary Equation

In Calculus class, one of the few facts I tucked away was the professor's comment that the hyperbolic cosine described the shape of a hanging chain.    45 years later, I decided it was high time I learned what he was talking about.  There is much interesting reading on the web about the history and use of the catenary equation which governs hanging cables and the inverted curve, the catenary arch.   See for example, Wikipedia articles Catenary  here and Catenary Arch here. 

For the purposes here, the equation which describes the shape of a suspended wire or cable is

y  =  C cosh ( x / C )

where C, called the catenary coefficient, equals H/W, the ratio of Horizontal tension to Weight (per unit length) of the cable.

John F. Nash

I searched for how to calculate the sag of a suspended wire.  The first surprising result was for the 1945 paper written by John Nash, Sr and his son, the famous  mathematician John Nash who received the Nobel Prize and is the subject of the book A Beautiful Mind by Sylvia Nasar and the movie of the same name.  The paper was written when the younger Nash was 19, attending Carnegie Institute of Technology, and his father worked for Appalachian Electric Power in Bluefield, West Virginia.  The life of Nash is both fascinating and tragic, and his mind, indeed, beautiful.  I read Nasar's book before the Best Picture movie came out, but only on rechecking did I realize that she mentions the 1945 paper near the end of chapter one.

The Nash paper is "Sag and Tension Calculations for Cable and Wire Spans Using Catenary Formulas," AIEE Transactions, October 1945, pp. 685-692, (submitted Feb 23, 1945).   Also see  Discussions, pp. 984- 987.

The paper presents a table, worksheet and procedure for finding sag and tension, so I spent considerable time duplicating the table and sample problems in an Excel spreadsheet.  This was fine, but it underscores the distance we have come from days of using tables.  With tables, it is not much trouble to work forward or backwards, i.e., to find either y, given x, or x, given y, in an equation like y = cosh( x ) .   For either direction, one finds the nearest values to those called for in a problem and interpolates between them.   A similar procedure applies when using a table for the catenary equation in a suspended wire problem. 

In the present world of calculators and spreadsheets, we have built-in functions for cosh and inverse cosh, but if we want to solve the catenary equation in both directions, we are forced to either use an approximation or numerical iteration.  In my younger days, I would have quickly decided in favor of iteration, but for a spreadsheet-like approach I considered approximation.

I might mention here, that I found the Nash paper treatment of transmission line cables over inclined terrain a bit hard to follow.  Instead, I used the equivalent formulas given by Marsh F. Beal in a paper, "How to Determine Low Point of Sag without Use of a Template," in March 17, 1945 Electrical World, p. 114, available here from the archive.org  web site.

I bumped into John Nash a short time later, when translating The Enciphered Letter, and found his letters to the NSA available online at the National Cryptologic Museum.   Fascinating to see his hand-written letters here  and read about them here .

Approximations

I spent considerable time inventing approximations, but the results were humbling.  Then I tried using the approximation found in an internet search.  It is essentially treating the catenary curve as a parabola.  This is quite practical for wires with very little sag, but becomes intolerable for droopy antennas or power lines that are over steeply inclined terrain.  

The popular approximation is very clearly presented in the 1911 AIEE paper by Harold Pender and H.F. Thomson, "The Mechanical and Electric Characteristics of Transmission Lines."   A copy may be here at the Zenodo open-access repository.   The approximation is based on the Taylor Series for cosh( x ), expanded about the point x = 0.   It occurred to me the thing to do is a similar expansion about the point x = 1.  Then, it would be a simple thing to divide the range, choosing whichever approximation has less error in a given problem, and in so doing, cover a wider range. 

The JavaScript Calculator

I learned just enough JavaScript to make the calculator on this site.  In the process, I decided that for my purposes, I don't need to invert the catenary equation after all, but instead just provide a slider to vary C, which in essence varies the tension, and observe the resulting sag.  The slider was abandoned in favor of buttons that adjust the value in 10% increments.  So, in the end, I didn't need an approximation at all, but can use the catenary with its beautiful exactness.  

A Range Extending Approximation

Although I did not need to use it, perhaps someone will find a use for the approximation given in the slides below.