How thick must granite walls be to isolate monks?

Question

Part of a monastic order's lifestyle involves a ring of twelve cells — one per month of the 372-day year, each with a certain spiritual function. A single tunnel outfitted as an acoustic anechoic chamber (henceforth an "airlock") is bored through each wall between the cells; the doors on each end of the tunnel are coated in soundproofed padding.

As a month goes by, each room's occupant maintains their cell in a good condition for the next monk in addition to their monastic duties. Once a month is up, the monks simultaneously enter the "airlocks", simultaneously shut the doors they entered through, then simultaneously open the doors in front of them, entering a new cell to ring in the new month.

The monks are never to directly speak with one another — only to leave messages the next monk to occupy the cell reads, and, in emergencies, communicate via rabbit tubes bored through the walls. Therefore, audible sound cannot be permitted to pass between rooms. The doors and "airlock" between them are specifically designed to block sound, but the walls between the cells may still transmit it if they're thin enough.

These rooms have been made in an outcropping of the most durable stone the order could find; for the sake of this question let's assume it's granite. How thick must the granite walls between the cells be to ensure the monks cannot speak, tap, or otherwise directly communicate through them?

Answer 1

Sound waves travel through substances via vibration energy transmitted through atoms in a material. Sound is changed in a material either through diffusion/absorption (ie, molecules within the material dissipating and diffusing the sound over time altering its frequency) or separation (so molecules cannot influence the other surface, reducing noise levels).

This means, contrary often to popular belief, that more dense materials transmit sound easier than less dense materials. The properties of the material determines the frequencies at which the sound is altered.

The reduction of sound from one side of a surface to another can be measured, and often translated in architectural terms to two measures, Rw and Ctr. Rw means the reduction of overall from one side of a material by one dB (Decibel). Ctr is a factor to adjust for less lower frequency and is a negative number to indicate dB reduction of lower frequency noises. These values are added together to provide a 'percieved' weighted reduction value.

In your case rock is similar to concrete, it is a very dense material and therefore transmits sound very easily - the Rw Ctr measure for a 100mm wall of this nature could be around Rw 20 Ctr -2, meaning a weighted reduction only of 18dB. In comparison the same thickness in wood with insulation is around Rw 41 Ctr -6, a Rw+Ctr of 35dB. This is likely because air pockets help reduce vibration transfer, and molecular bonds also dissipate high frequency noise.

A 250mm thick concrete wall has Rw+Ctr of 42dB.

In terms of perception, the following table provides data on dB levels that could be heard:

So in terms of sound reduction, a 250mm thick heavy mass wall will only reduce levels of a nail tool from 95dB down to 53dB - easily perceptible as general conversation.

In terms of perception - the human ear can perceive noise at quite low dB levels, (at 15dB you can hear your own heartbeat - often you can still perceive noises lower than this to around zero - dB levels are a logarithmic scale). Your wall would need to be at the very least 600mm thick, and even then some monks (especially if they place their ear on the wall) could likely perceive it as sound reduction can never reach zero unless there is complete separation.

Answer 2

A compact and dense material alone is probably the worst choice when it comes to insulating against sound, because sounds travels very well through it.

Look at modern houses: to reduce the amount of sound going from one house to the neighbors through the common walls (i.e. when going up and down the stairs) they don't build thicker walls using thicker bricks, but rather put an air gap between the step and the wall.

In short, to prevent sound from propagating well through something, you want to build a steep gradient in acoustic impedance in that something. You can achieve that embedding air filled cavities in the wall.

Doing that the thickness of your walls is going to be dictated more by the airlocks than by the soundproofing.

Answer 3

As an addendum on the other answers, I'd like to add an answer more from a construction perspective. This is minimal, there is a lot to be found on the internet on how to sound proof a room in your house, this is not different:

• Mass is good, but not the holy grail. You want to isolate the room physically from eachother. If they share a wall you can knock on it. Sure, a thick granite wall would be a hard knock, but just take a spoon and tap.
• Layers your walls. Have a wall, and air gap and a wall. Now your rooms are physically seperated. If you make it 20cm/3cm/20cm the spoon doesnt make a chance. Instead of air you could also add something fluffy like wool, which would help more. You're looking for a material which is "airy", but doesnt allow for air to really move (and pass sound through). Now the sound vibrations cant travel to the other side of the wall.
• Seperate the wall from the floor. Actually the spoon still has a chance: Through the floor. Tap the wall close to the bottom and the tap might travel wall->floor->other wall. The first part is an easy fix: Place something 'soft' (it has to hold granite!) at the bottom. Now there are losses from wall to floor and back again to the other wall.
• Seperate the monk from the solid floor. The monk can stomp on the floor which is the same solid thing as the neighbour. This can be done fairly easy and depends on your setting. You can make a floating floor (eg full floor, not touching the walls, on something like rubber pads or other dampening material). Or just sand.
• Ambiance sounds. Absolute silence is very difficult to achieve. If you add just a tiny amount of natural sound (like flowing water, or wind, or ...), you would have an very difficult time seperating sounds from each other.