Can a wave propagate in any substance? Aren't there any prerequisites?

We see waves propagate in air, water, through the cristal of a metal and along a rope. Isn't a wave a wonder of Nature, or is it just a simple phenomenon? Are homogeneity and isotropy necessary properties for the correct propagation of waves?

Update

are a rope, water and space/EM field elastic in the same way?

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I don't believe homogeneity is a necessity, two ropes attached to each other still propagate waves. – Kyle Kanos Aug 13 '14 at 14:48
As a minor addendum to all the correct answers: waves can even propagate in discrete systems. Physical examples would include wave machines (chains of discrete masses and springs), spin waves and ladders of discrete capacitors and inductors, which are used as delay lines in cases where coaxial cables with enough delay or the correct impedance would be impractical. – CuriousOne Aug 13 '14 at 19:30

A wave can propagate in any medium that is:

a) elastic

b) less than critically damped

Neither homogeneity nor isotropy are necessary.

Any elastic system will return to it's original state when deformed, the question is just whether the deformation can propagate, and this is down to how quickly the energy of the deformation is dissipated. If the damping is high enough, this is critical damping, the material will return to its original state with a $e^{-\alpha t}$ dependance on time and no wave will propagate.

For example, in water common experience tells a gravity wave (i.e. a wave) propagates just fine, and a longitudinal wave (i.e. sound) propagates just fine. However shear waves will not propagate because they are too rapidly damped.

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Your answer seems to me entirely correct. Although I have some difficulty to relate the propagation of an electromagnetic wave. Maybe because the view of your response is a bit mechanical. You think? – Martin Petrei Aug 13 '14 at 15:13
@tinchito: the EM field is elastic in the sense that it costs energy to change its value (i.e. deform it) and it will relax back to its equilibrium value afterwards. This is basically no different to a block of rubber. – John Rennie Aug 13 '14 at 16:23

A wave is generated by a disturbance in a medium. For a wave to propagate, do not necessarily need a medium.

For example, an electromagnetic wave can propagate in vacuum, while a sound wave requires an elastic medium to travel.

The requirements for the propagation of a wave, are dependent on the nature of the wave.

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