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There are many types of waves - sound waves, water waves, light 'waves' etc.

What are the common properties of the media in which these various types of wave travel? And how these properties enable the wave propagation?

I'm especially interested in a mathematical description of these properties. (If it's reasonable to ask for it.)

Qmechanic
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Jen
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Light does not need a medium.

Mechanical waves need a medium with inertia and a restoring force.

For longer waves on a water surface it is gravity that provides the driving force to a flat surface. The motion of water below the surface is not so easy to describe mathematically.

Sound waves in air or water are pressure waves where the elasticity provides the restoring force. These are longitudinal waves. Transverse waves cannot exist in a fluid because there is no restoring force for a shearing deformation. Solids have an elastic shear modulus so there will also be transverse waves.

Mathematically one often ends up with the wave equation, a differential equation that for mechanical waves is derived from Newton's law $F = ma.$ The solutions are then functions of position and time that can be written as $$f(x,t) = f(x-vt),$$ which propagate with a velocity $v.$

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    Just out of curiosity, one thing I've never understood - how come that "light does not need a medium"? Every wave needs a medium, even if it's just a single particle. There has to be something that wiggles back and forth, because a "wave" isn't a thing in itself - it's a pattern of movement. How does light get away with it? – Vilx- Jan 12 '20 at 10:03
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    @Vilx- In mechanical waves the energy oscillates between kinetic energy and elastic energy. In an electromagnetic wave the energy oscillates between the electric field and the magnetic field. This is derived from Maxwell's equations instead of from $F=ma$. The differential equation happens to have the same form, with the same type of propagating solutions. But there is nothing that is wiggling, no movement of any material. –  Jan 12 '20 at 10:12
  • Ahh, I see... I think. Thanks! – Vilx- Jan 12 '20 at 10:33
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    @Vilx- In other words, the "medium" of light is the electromagnetic field. – JiK Jan 12 '20 at 18:01