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

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).

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. 🙂

Room modes in non-rectangular rooms

I recently programmed a method for easily doing basic 2D finite element analysis of acoustics. I did this for a different kind of project, but thought it would be cool to try it out as a method of analyzing how sound behaves in L-shaped rooms. I used the following dimensions in the software (the room consists of the L-shape in the picture):

Room dimensions

Obviously we're simplifying things a lot, as we're leaving the height of the room out of the calculations. Still, the calculations will give us a lot of information of how sound behaves from the perspective of the most interesting dimensions of the room.

Let's try feeding in a plane wave from the top of the room, just to see what happens. Try moving the slider around a bit to see how the sound field forms in the room. Please note that it can take a while for the content to load.

Some central things to note:

• When the plane wave reaches the convex corner, the corner will radiate sounds in all directions (also to the right).
• A sound field quickly forms in both the top-down and the left-right direction

The red dot represents a microphone in the room, in case you're wondering. Let's see what the microphone gives us:

The signal arriving at the microphone in the room

We can clearly see when the first diffracted sound arrives at the microphone. Two reflections arrive shortly afterwards, in close succession. After that, the sound field quickly becomes complicated.

Let's check out the frequency response as measured by the microphone:

Frequency response as measured by the microphone

We can clearly see at least a few room modes. Let's try examining the modes more thoroughly.

The plane wave consists of a gaussian pulse. We can't feed too sharp of a pulse into the room, as that would lead to errors in the calculations. By increasing the width of the pulse, we can get a pulse which the calculations will be able to handle.

Additional: How is the response of the room calculated

I cheated a bit. The calculation model I used doesn't account for damping, which in practice means that the room would continue reverberating indefinitely. I calculated the response for 0.5 seconds and approximated damping simply by multiplying the non-fading response with a decaying curve. Which isn't something that really should be done.

The gaussian pulse has the following frequency content (magnitude):

Using this, I calculated the response up to 200 Hz by deconvolving the pulse from the response. Deconvolution, in this case, means that I took into account the varying frequency content of the excitaiton. This can be done by dividing the frequency content of the response by the frequency content of the excitation.

Room modes

I've written quite a bit about room modes recently, but not from the perspective of irregular rooms. Let's see what they look like in this case! Below are the 6 lowest room modes. Many of them are far from obvious, as can be seen. They bear very little resemblance to the lowest room modes in a rectangular room.

Room mode at 32 Hz

Room mode at 49 Hz

Room mode at 76 Hz

Room mode at 80 Hz

Room mode at 86 Hz

Room mode at 91 Hz

Conclusions

If the geometry of the room differs even slightly from a rectangular layout, and one wishes to calculate an approximation of the room modes of the space, there really doesn't seem to be that many options available. Numerical modeling using the finite element method, as used in this post, is a method which works nicely.