This is possibly a bit of a stretch (in relation to point 2); in that it concerns a modelling assumption which led to a variety of mathematical models which attempted to explain Marconi's transmission of radio waves across the Atlantic in 1901. The assumption was that the radio waves could propagate large distances beyond the horizon primarily via diffraction from the surface of the earth/ocean. The most well-known supporters of this idea (to mathoverflow users ... I assume), in the early twentieth century, were Poincar\'e and Sommerfeld, who amongst others, studied a variety of boundary value problems with this assumption playing a central role.

The paper that effectively closed this line of mathematical inquiry (almost 20 years old), was written by Watson (at the request of van der Pol) https://doi.org/10.1098/rspa.1918.0050. It is probably fair to say that experimental evidence also contributed to the demise of this idea (see Austin-Cohen reference in the paper ... and amateur experimental evidence) and an earlier paper by Nicholson which is referred to in Watson's paper.

The wider effects of this topic are discussed in detail Yeang's book (see review in the following link).

http://www.reeve.com/Documents/Book%20Reviews/Reeve_BookReview_ProbingSkyRadioWaves_Yeang.pdf