# Airborne sound insulation

I thought it would be cool to demonstrate some of the vary basics behind airborne sound insulation using a finite element simulation. This is also something I’m planning on utilising in real projects (i.e. it’s not just for playing around with).

## Some background

Let’s assume that we have two rooms with nothing but a small piece of wall between them. The rooms are perfectly separated from each other, they’re connected by nothing but this one piece of wall. This means that the situation is analogous to laboratory measurements (i.e. when measuring the sound reduction index R, or the single-number quantity Rw). Situations on the field are different from the situation described here, as flanking transmissions are not taken into account in laboratory measurements.

Let’s also assume that this simple, homogenous piece of wall is perfectly sealed. It’s modeled as simply supported (we’re allowing for rotation on the boundaries) in 2D.

I’m not going to go any deeper into the material parameters I used here, but here’s some background on the theory:

• The simulation is done using a Timoshenko model for the piece of wall (taking shear locking into account using reduced integration).
• I used linear shape functions for both the fluid and beam domains.
• The bottom boundary is completely absorbing.
• There are two completely separate fluid domains, which are both coupled to the piece of wall in the middle.
• I did the simulation in python, and exported the results to a binary file which is read by the javascript viewer below.

## Simulation 1

http://kaistale.com/blog/131125coupling/viewer.html

The simulation is a bit heavier than in the simulations I’ve usually posted, as I had to use more elements to get a nice looking result. Here’s a very short summary of what’s happening:

1. A sound wave travels in the lower room.
2. The sound wave arrives at the wall.
3. The sound wave consists of positive pressure (as compared to the surroundings and other side of the wall), and will as such exert forces on the wall.
4. The wall will deform as a consequence of the force. Note that the deformations of the wall are exaggerated in the visualization!
5. As the wall deforms, it moves the air above it, creating new sound waves.

## Simulation 2

I also made the following simulation, which is more complex and more difficult to understand, but looks way cooler. I especially like it that you can see the sound moving faster in the wall than in the air when the first wave hits the wall (compare the sound waves below the wall to the sound waves above the wall).

Note:  I switched the colors, here red represents a positive sound pressure. Also, the wall is clamped (rotation isn’t allowed at the ends of the wall).

http://kaistale.com/blog/131126coupling/viewer.html

## Epilogue

The most important thing to note is that when you’re hearing sound through the wall, as in the simulated situations, what you’re hearing are the deformations of the wall. It’s the wall that radiates sound into the room. If the wall wouldn’t move at all as a consequence of the pressure waves hitting it, you wouldn’t hear anything. This is why heavy structures, such as concrete, isolate sounds so well.

The situation becomes a whole lot more complicated with separated structures (i.e. light structures or drywall), and when flanking is taken into account. Measuring the sound reduction index with a diffuse sound field is another interesting task I’ll definitely have to do. I’ll most likely return to these topics in later posts. 🙂