No, the different accelerations of charged particles are not the cause of the various frequencies of electromagnetic waves such as blue light, X-rays, gamma rays, microwaves, and radio waves. The frequencies of electromagnetic waves are determined by the specific properties of their sources, not by the acceleration of charged particles.
The frequency of an electromagnetic wave corresponds to the rate at which its electric and magnetic fields oscillate. For example, blue light has a higher frequency than red light, X-rays and gamma rays have even higher frequencies, while microwaves and radio waves have lower frequencies.
The frequency of an electromagnetic wave is not directly linked to the acceleration of charged particles. Accelerated charged particles can indeed emit electromagnetic waves, but the frequency of the emitted waves depends on the properties of the source and the nature of the acceleration process. The acceleration of charged particles can give rise to a broad range of electromagnetic waves, spanning from radio waves to gamma rays, depending on the specific conditions and energies involved.
As for your second question, the proof that light has a frequency comes from extensive experimental evidence and the development of the electromagnetic theory of light. Numerous experiments and observations have shown that light exhibits wave-like behavior, including interference, diffraction, and polarization, which are characteristic properties of waves with well-defined frequencies. The understanding of light as an electromagnetic wave was significantly advanced through the works of scientists like James Clerk Maxwell and the subsequent experimental verifications of his electromagnetic wave equations.
Today, the wave nature of light is well-established and supported by a vast body of experimental evidence, confirming that light does have a frequency. Additionally, the wave-particle duality of light was further elucidated by the development of quantum mechanics, which describes light as both waves and particles known as photons.