The above image is from YouTuber CameraBryan here .  It shows the beautifully built rotary spark gap replica transmitter completed in 2010 by Hal Kennedy, N4GG .

In January '25, our local radio club featured a program by Bill Ruck, The Story of KPH.   It is a fascinating history of the maritime-radio shore station in northern California, beginning with  the 1905 station PH in San Francisco, started by the American DeForest Wireless Company.  See Bill's presentation on YouTube as presented to the California Historical Radio Society here  .

During the Q&A session at our meeting, someone asked about the frequency spectrum of a spark-gap transmitter. 

A few days later, I resumed reading where I had paused in a book, The Science of Radio, by Paul J. Nahin.  I had stopped in Chapter 11, Radio Spectrum of a Spark-Gap Transmitter.  The chapter presents a derivation for the Fourier Series of the spark transmitter's damped oscillation waveform.  Making some spectral plots was just too irresistible.

Encapsulating this project took much longer than I anticipated.  I finally broke it into three parts:

Part-1 gives a background to the spark transmitter's damped oscillation waveform.  It is taken from the excellent 1918 material from the National Bureau of Standards (NIST, today).

In Part-2,  QucsStudio is used as a whiteboard to evaluate Paul J. Nahin's equations for both the time -domain and the frequency spectrum of a damped oscillation waveform. 

In Part-3,  a series RLC circuit is used to simulate a damped oscillation waveform, and the corresponding frequency spectrum.