Is there any kind of material medium through which sound does not travel at all? In my book it is written,

the conditions necessary for the propagation of sound are:

*

*A vibrating source for the production of sound.

*A material medium for the sound to travel through.

*The material medium must be continuous and elastic.

*A receiver to receive the sound.


My question is: If the material must be elastic, is there any kind of non-elastic material through which sound does not travel at all ? (I know sound cannot travel through vacuum, but it is not a material medium.)
 A: This is why I often add the qualifiers passive, continuum, stable or similar when making very broad generalizations about materials.
I don't believe one can go to a materials supplier and walk away with a sample of bulk material that doesn't conduct any sound and will never conduct any sound. This may be sufficient to address your question.
At its core, one could think of acoustic wave propagation as a tradeoff between a material's stiffness and its density. The stiffness drives a rapid return to the equilibrium atomic/molecular spacing and arrangement. The density resists this return. In this way, we obtain a sound speed that scales up with increasing stiffness and decreasing density.
Do all materials have a stiffness and a density? I had a professor who used to hedge his bets to similar material-property questions by saying something like "You're not going to walk up and show me a chunk of material with zero bulk modulus (say, or negative density, or negative shear stiffness, or so on)." I like this and use it instead of saying, "All materials have a positive bulk modulus.", for example.
Because then someone will bring up an active material, externally powered, that expands a little when you pressurize it. Or an explosive that detonates. Or some reentrant geometry on the inside that does something unusual. And so on. (Sometimes these details lead to insight, as with auxetic materials, for instance, whose study illuminated certain thermodynamic requirements for material stability.)
So for the "Gotcha!" police, yes, you could design a material that would effectively damp a certain wavelength. Or you could provide a material that's unstable to the slightest overpressure that—technically—won't conduct sound because it disintegrates (although the remnants would presumably conduct sound).
Anyway, this gives you a sense of the exceptions that come to mind. All stable, passive, continuum materials are elastic to some degree, and therefore all conduct sound.
A: One can imagine a soundproof substance based on known physics, as follows.
You want something that is dense, to reflect the low frequencies. So we will choose a barium compound, like barium carbonate. A sound wave will mostly bounce off a slab of barium carbonate, and go away.
Then you want something with lots of internal dissipation so the waves that do enter the slab rub themselves into frictional oblivion before they make it all the way through the slab. So we crush the barium carbonate into dust and mix it with heavy oil.
Now we have a big mess. So we throw all that stuff away and use beach sand instead. Fairly dense, lots of internal dissipation, cheap- and no goop!
