How can one, in a smooth way, produce a semicircle wave, as opposed to a sinusoidal wave ?

Imagining the machinery and linkages behind the movement. I mean the trace. For example a point fixed in wheel of a car may engender a cycloid. Others curves may be a Cyclogon, or a curve produced by a point fixed inside a sharkwheel, when rolling(not gyration).

The motivation behind the question, it is because in the usenet, there is a user called Archimedes Plutonium that has semicircle waves instead of sinusoidal. I have curiosity if it will be possible to produce such kind movement in at least one way. The mechanism that produce the curve does not need to be contained inside a plane.

Other way of thinking on it may be in the sense of producing such movement in a non-trivial way, similar to producing a straight line using the Peaucellier–Lipkin linkage.

Question re-phrased by @Transistor:

enter image description here

Figure 1. A cycloid can be generated by a point on the circumference of a circle. Image source: Britannica.

Can a similar waveform consisting of a series of semi-circles be generated by a mechanical linkage?

  • $\begingroup$ I have edited your question. Please check to see if I have interpreted correctly what you are asking. $\endgroup$
    – Transistor
    May 15 at 9:44
  • $\begingroup$ Thanks @Transistor, I have researched a bit more, even there a couple of topics in MO and SE, but, I am not satisfied with the anwers provided in those topic. In the sense of not just producing one semicircle alone, rather, a bit more than one, in the "semicircle waveform" ;) $\endgroup$ May 29 at 0:14

This can be done very easily in air, as follows:

Make an audio sound file using a semicircular waveform instead of a sinusoid. You will probably have to do this by hand and approximate the waveform with lots of thin "slices", and then spline them together and smooth them using audio sound synthesis software.

Play the resulting audio file through an audio amplifier and into a loudspeaker. The sound waves in air will now be semicircular instead of sinusoidal.

  • $\begingroup$ make me remember the phonopaper, browsing the pixilang web. Rather a unconventional approach... $\endgroup$ May 29 at 0:18
  • $\begingroup$ The sound waves aren't actually shaped like a waveform drawn on paper however. The y axis represents pressure, not vertical displacement I think. $\endgroup$
    – Drew
    Oct 19 at 17:08

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