what are the basic properties must a medium posses to have the propogation of a mechanical wave?

A wave can be propogated ina medium if it is continuous, it should have inertia and elasticity.To account for  the continuity of the medium Think of a rope which is tied at one end and the other end is free then how can you think of a wave pattern to be setup in tthe rope because wave motion causes a certain type of movement in the medium particles and how can you think of this movement if the medium is not same throughout.

While the elasticity and inertia accounts for the vibrational motionof the medium particles when the wave propogates.