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Some materials like fibreglass are really good at absorbing sound, aren't they ? (not blocking sound, as mentioned here.)

Sound propagates when particles oscillate. But what happens to these vibrations when they reach a sound absorbing material ?

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Simple answer:

Because the wall itself takes over a portion of oscillatory motion which is then either absorbed (turned in heat) or transmitted to another location (say a neighboring room). Of course, for the purposes of noise reduction, the first option is favorable.

A bit more elaborated answer:

The simplest approach is using appropriate boundary conditions. At the wall of an ideal reverberant room all the impinging sound is being reflected and therefore the net acoustic energy in the room is being conserved. Formally for acoustic velocity and pressure:

$$ \mathbf{v}\cdot\mathbf{n} = 0 \Leftrightarrow \nabla_\mathbf{n} p = 0 $$

at the hard wall boundary.

In reality there is always nonzero absorption. Acoustic velocity and pressure at the wall are then governed with so-called Robin boundary condition which then implies that the amount of acoustic energy in a source-free room must be diminishing.

For easily computable and measurable approach google Sabin-Franklin-Jaeger theory dealing with such phenomena in kind of statistical point of view.

Inside the sound absorbing material the waves are attenuated due to the friction (kinetic energy to heat transfer instead of kinetic energy to potential energy). Practical techniques to do so are various and beyond this question.

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