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In physics, the de Broglie equations tell us that things can matter behave both like a particle and like a wave and that its wavelength can be calculated by the de Broglie formula. So my question is then, do electrons "orbiting" a nucleus travel as a wave "around/orbit" this nucleus in a wave as well?

If so (since electrons are not a form of electromagnetic wave) through what medium are the electrons propagating as they orbit the nucleus as a wave? Also an added question: do electrons in the same orbital shell ever constructively or destructively interfere with other electrons with each other?

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So my question is then, do electrons "orbiting" a nucleus travel as a wave "around/orbit" this nucleus in a wave as well?

Both are nucleus and electrons waves, but you will see that the mass of the proton alone is 1,836.15267 times the mass of the electron. The electron is a bound wave, represented by some wavefunction.

If so (since electrons are not a form of EM wave) through what medium are the electrons propagating as they orbit the nucleus as a wave?

There is no medium except the electrons themselves, that is the change in ideology from the ether theories.

Also an added question: Do electrons in the same orbital shell ever constructively or destructively interfere with other electrons with each other?

In a sense, but that is a bit of an oversimplification. Using that language, the electrons actually interfere with themselves.