Good sound absorption is the only important quality of any sound-insulating material. You can get a good idea of how the absorption principle works by listening to your voice for a while, then going into a living room or a bedroom and comparing how this changes. This difference depends on what happens to the sounds you make when they come in contact with a wall, floor or any other surface.
The best soundproofing materials and products are the ones that absorb the most sound. As a rule, the most absorbent materials are porous, which means that the structure of the material actually traps the moving air waves. Fabrics, for example, tend to create effective sound-absorbing surfaces because the fibers easily trap the waves instead of letting them bounce back. The same goes for foam - the structure of this material is exceptionally good at catching waves when they hit it, making it a great choice for soundproofing.
You can imagine sound waves as a series of small spheres that can bounce back from solid surfaces such as concrete, but would be trapped in, for example, a net or a surface with a series of holes in it. When fewer 'sound spheres' bounce back from the latter surface, you will hear less sound and thus have a higher sound absorption level.
This is how sound insulation works for airborne sound or sound that moves through the air and hits a surface. But sound is also transmitted through surfaces. You can hear transmitted sound when someone walks on a floor above and you hear them through the ceiling. To reduce transmission, it is necessary to break the 'route' that a sound takes. Imagine the difference between these balls rolling down a sloping slope (high sound transmission) and through thick grass (low transmission). Again, the energy of the sound must be absorbed.
When a sound wave is trapped in a material with good sound absorption properties, the energy within the absorbent panels wave simply becomes heat. Not much heat, of course - the energy absorbed from a high-playing album would be barely a hundredth of the heat from a match. A material with good heat absorption properties will also not necessarily provide good sound absorption. Polystyrene insulation, for example, captures heat efficiently but does not absorb much sound.
It is also worth noting that different sound frequencies require different choices of sound insulation. The biggest factor is thickness, and the greater the thickness of the material, the deeper the frequency of sound that is absorbed.
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