How can describe electromagnetic waves with mechanical waves?

Mechanical waves are mostly longitudinal waves. It means the constituents of the naterial through which the wave is moving are oscillating in the same direction as the wave itself.

In an electromagnetic wave the motion of the electric and magnetic fields is transverse to the direction of propogation of the wave. So the E, B and k vectors are perpendicular to each other. Also unlike a mechanical wave, an electromagnetic wave does not require a medium to travel. It can travel in completely empty space.

The matter wave or de Broglie wave comes from the wave particle duality of a quantum mechanical object such as an electron or an atom etc. This wave can also travel in empty space. Due to the strange nature of quantum mechanics, a quantum object manifests itself as a particle or a wave depending solely on the type of measurement or observation. So for example if one lets a stream of single electrons to fall upon 2 closely spaced slits then each electron would behave as a wave and lead to interference fringes on the screen behind the slits.

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