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.

While making the slides, I did not find a schematic for a spark transmitter that I found satisfying.  Afterward, though, I came across the schematic below that seems just right.  It is from the April 1961 issue of Wireless World, a special issue celebrating the 50th anniversary of the publication.  The associated  history article is a good read.  Find it at World Radio History here .   The so-called "Jigger" in the schematic is the name used by Marconi for a capacitively tuned RF transformer, used in both transmitters and receivers, and described in British Patent No. 7,777 of 1900 and US patent 763,772 of 1904.   See  the the Appendix in the 1933 book,  Over Thirty Years, by  R.N. Vyvyan, available from Internet Archive here .