How does mass of a sound barrier affect the amplitude of sound waves penetrating the room? I am trying to learn what role mass of sound insulation plays in reducing amplitude of sound waves entering a room from the outside.
In a thought experiment, a person has persuaded an elephant to lean against the party wall with a noisy neighbour. Is it fair to reason that any sound wave entering the room through the party wall will make an attempt to 'move molecules of the elephant' during sound transmission, thus reducing the amplitude of incoming sound?
I suspect that sound waves will attempt to travel down a route that is 'easiest to vibrate', avoiding the elephant.
Would the elephant play any role at all in reducing the apmplitude of incoming sound waves?
EDIT: there are no air gaps in our imaginary room.
 A: I don't have the maths to argue this [I'm a sound engineer not a physicist], so let me try to cover it in broad strokes.
Sound isn't all one frequency. To humans it's frequencies from approximately 20Hz to 20kHz [less as you get older]. Higher frequencies tend not to bother humans as we just can't hear them at all. Lower frequencies turn from 'sound' into perceptible 'vibration' - we can feel it even if we can't actually hear it.
Let's consider this like playing back music on a good hifi.
At the top we have cymbals & hi-hats; the bright, fizzy noises.
Below that, the majority of the sounds; guitars, strings, brass, vocals etc.
Then right at the bottom, bass, bass drum [& other lower sounding drums].
The next thing to consider is that higher frequencies take far less energy to generate, but are considerably easier to block. At frequencies from maybe 4kHz & up, you can block them with your fingers in your ears or earplugs, or a wall, or a bit of foam or rockwool.
If you're ever in the market for 'sound absorbing' materials to block sound transmission, always be wary of claims. "Blocks >60dB"  ..at what frequencies? 20Hz? Not a chance, mate.
OK, so lower frequencies take much more energy to generate, but as they're carrying more energy, they also take a lot more effort to block.
Consider the thunder that accompanies lightning. If you are quite close when it strikes [or like they make it sound in the movies], it is a massive 'crack' of pretty much 'all frequencies at once'. By the time you're a mile away [about 5 seconds between flash & sound] then already it's just a deep rumble - the high frequencies didn't even make it that far in clear air. The bass did, though.
A domestic interior wall made of wood frame & drywall [plasterboard] will stop higher frequencies pretty easily, but it will barely stop a human voice - right in the middle of our frequency range, as we discussed earlier, along with the guitars etc.
You'll know this if you have argumentative neighbours/spouse/children etc, or even just an over-enthusiastic dog - slamming the doors & stomping off to another room merely dulls the sound, it doesn't block it entirely.
A light piece of drywall will itself have a resonant frequency, one at which it will vibrate in sympathy. If allowed the freedom to vibrate, it will then transmit that frequency out the other side, almost unimpeded. If we damp it, with foam or rockwool, that will make the energy dissipate, it will waste itself trying to make the rockwool vibrate. If we then add a wooden frame & another piece of drywall of a different thickness & density, this will want to resonate at a different frequency, yet will also be damped by the filler. Things like rockwool don't really have a resonant frequency, but what they will do is just allow very low frequencies to pass as though they weren't there. What they are good at is damping the vibrations of the larger resonant surfaces - in effect making them waste their energy.
This is the basic element of sound transmission prevention. You have layers of different densities, each dissipating energy at a different frequency. The very lowest frequencies need the greatest mass to damp. To do this for a domestic wall would actually take too many layers to be really sensible. Before long, the wood frame would actually be transmitting most of the noise. You'd have to go to additional lengths to prevent transmission through the basic frame the building is constructed on.
Recording studios use this principle, though, by 'floating' a room within a room - literally trying to isolate one structure from the rest of the building, sitting it on sound absorbing [ie non-resonant] rubber.
Concrete in a single 'lump' or 'sheet' also has a resonant frequency - but it also has a lot of mass, making it harder [though not impossible] to vibrate. You could still feel trucks going by, or the very lowest frequencies from a loud hifi. Concrete's 'problem' in this respect is that it's really all one piece, so it does have actual resonant frequencies, like a really thick piece of drywall.
Better than concrete is a wall made of lots of separate stones - non identical, not built in any particular mathematical or artistic pattern. This gives a lot of variation in frequency, each stone & cement joint all being excited by one frequency but damped by all the others around it, which want to vibrate at a different frequency.
Right… so, eventually down to your elephant. Sorry it's taken a while.
Your elephant is, in structure [apart from being alive & probably not too happy at being part of this experiment] most similar to the rough stone wall. It's a bag of liquid, with a lot of different density membranes and an overall skeleton of loosely jointed harder materials. Water will actually transmit sound similarly to air, but as each bit of water in our elephant is in its own tiny cell, the disparity of sizes & interspersed membranes breaks this up like our stone wall.
It has a lot of mass, too.
If you were to persuade your elephant to lean on a resonant wall, it would damp the wall to some extent, depending on how hard it squished itself against it. It would be acting like the world's biggest lump of rockwool or neoprene rubber. It wouldn't really be fabulously efficient used in this manner.
If, however, we were to replace the wall with entirely elephant, you would very probably have a really, really good sound insulator. 8ft of varying densities & substrates, a lot of mass & not a lot of cohesion between each component & its resonant frequency. If we were to get a bit 'icky' then its lungs & rib cage would probably be the most resonant part - so let's wedge him into the wall facing the noise, & we can sit quietly in the room his butt protrudes into.
Sonic bliss, if not olfactory ;)
Note: I've not even touched on reflectivity or diffusion - that would turn this into a full novel ;))
After comments
It is remarkably difficult to achieve total sonic separation in a domestic environment. I once built a 'room within a room' as a home recording studio, in a house basement, of course surrounded by earth - lots of mass. I did manage to achieve sufficient reduction that you could no longer hear anyone shouting or singing as loud as they possibly could, from the room above. My bass (guitar) amp, however, still went through it like a knife through butter. It took the extra attenuation of the building [double skin of old Victorian brickwork] & earth itself to fully damp it to below audibility. You couldn't hear it from outdoors unless it was particularly quiet in the street.
A: In order for sound to go "through" a wall it will first have to enter the wall (i.e. the air molecules bouncing against the wall need to actually move the all) and then the moving wall has to radiate sound into air on the other side.
The wall is a lot heavier than the air, so this is a big impedance mismatch and most of the incoming sound energy is reflected, not transmitted. The mass of the wall plays indeed a role here. But it's not just the mass but also the way it's mounted which usually forms a spring. Most walls do have a resonance frequency and they will transmit sound best at that frequency. Adding mass will lower the resonance frequency(s) and in general the heavier the wall, the less sound will be transmitted.
In practice this type of sound transmission is rarely a factor. Most residential walls do a really good job at blocking sound. The most common sound paths is NOT from air to wall to air, but airgaps (doors, windows) and structure borne sound (walking on the floor, loudspeaker vibrating the floor mechanically, etc).
So your elephant would in most cases reduce transmission through the wall it is leaning against, but it's unlikely that it would make an audible difference since most of the sound you are hearing is not coming this way.
