He realised that spherical In and Out-Waves removed the need for a separate particle, as the Wave-Center of the Spherical Waves created the particle effect.
He then discovered that when one spherical standing wave was moving relative to another the Doppler shifts gave rise to BOTH the de Broglie Wavelength AND the Mass increase of Albert Einstein's Relativity . (. Wolff demonstrated that when two charged particles (Wave-Centers of two SSWs) are moving relative to one another they gives rise to beats of interference (caused by the Doppler shifting of the In and Out Waves due to relative Motion) which were identified in experiments as the de Broglie wavelength y=h/mv, and also gave rise to the frequency increases and thus energy/mass increases (as E=hf =mc 2 ) of Special Relativity.
In the second part of his 1924 thesis, de Broglie used the equivalence of the mechanical principle of least action with Fermat's optical principle : "Fermat's principle applied to phase waves is identical to Maupertuis' principle applied to the moving body; the possible dynamic trajectories of the moving body are identical to the possible rays of the wave." This equivalence had been pointed out by Hamilton a century earlier, and published by him around 1830, in an era where no experience gave proof of the fundamental principles of physics being involved in the description of atomic phenomena.