Why a wave cannot propagate in a non-elastic medium
We know that wave is a distrubance and carries energy. In this sense let imagine fall of dominoes, which carries disturbance and energy. Here fall of dominoes is non-elastic and we can see that wave propagates. Can I call it as a wave?
Waves can propagate in an inelastic medium, and the fall of dominoes is a wave. So when people say that a wave can't travel in a non-elastic medium can't, this statement applies to particular kind of waves.
When talking about waves one often means perturbations that are periodic in space and periodic in time at every point. This is not the case in an inelastic medium, where due to damping, such a wave would decay.
Yet, for the case of dominoes such terms as solitary wave, shock wave or soliton could be quite appropriate - the kind of single "waves" associated with explosions, tsunami, propagation of cracks, etc.
I think it goes like this. A mechanical wave is indeed a periodic perturbation the particles of the medium through which it travels, and if the medium is non-elastic, as most media are, then energy will be lost at each 'passing on' of the energy from one particle to the next. This is the cause of the attenuation of mechanical waves over distance. The more elastic the medium, the farther the wave will travel. A wave can travel in a non-elastic medium, but not very far. Although it is not quite the same thing, consider the speed of sound in air (331 m.s-1), water (1,403 m.s-1), ice (3,838 m.s-1), iron (5,120 m.s-1) & diamond (~12,000 m.s-1). As we progress through that list, the medium becomes more elastic(/brittle), and sound will travel farther.
