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Specifically why do they have $V = \frac{c}{β}$ as stated in this paper written by De Broglie: http://aflb.ensmp.fr/LDB-oeuvres/De_Broglie_Kracklauer.pdf ?

I follow the proof of “phase harmony” between the reference frames, but i’m getting confused between them and i don’t see why $V = \frac{c}{β}$

P.S i know matter waves are kind of an archaic concept, this is purely a historical exercise.

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