Acoustics of small open plan offices

In this post, I'm going to examine a hypothetical small open plan office, and the optimal way to treat the space acoustically. Check out the publication related to some of the theory I'm going to base this on here. I'm going to make the example geometrically simple, so the result will be clear and somewhat intuitive.

The setup

Small open office setup

Small open office setup

A hypothetical simplified small open plan office is shown in the picture to the left. The spheres and cubes represent the possible positions for the office workers.

I'll assume that sturdy office screens are placed air-tightly against the wall and floor, so that sound doesn't leak through the edges of the screens. I'll also ignore any sound diffracted over the screens.


First degree reflections

First degree reflections

The sturdy office screens isolate sounds very well; this means that sound doesn't travel directly from one position to another, but instead through diffraction (which I assume to be negligible) and reflections.

First and second degree reflections

First and second degree reflections

First degree reflections are relatively easy to predict. Second degree reflections are already significantly harder to predict. Third degree reflections are very hard to predict without computer simulations. Third order reflections (and above) are often already far from intuitive.

The goal

I wish to hear as little as possible of my coworkers. The office screens already attenuate direct sound. But this is not enough. If I don't consider the other routes the sounds travel from one position to another, the screens will function as little more than visual barriers.

So what do I want to do? It turns out that early reflections are almost always the most important reflections to consider when one wishes to affect speech intelligibility. Another important factor is the background noise level, but I'll assume that the ventilation provides a decent amount of masking noise. Keep in mind that by early reflections I mean reflections arriving early on in time, without taking any notice of how complicated the path the reflection has traveled is.

I'll make the following goal: I want to get rid of the early reflections as effectively as possible, using a relatively small amount of absorbing material, such as acoustic panels. Let's assume that I can't place anything on the floor, as it would make cleaning (and walking around the room) too difficult. What is the optimal way to place the absorption?

The result

The result

The result

The figure to the left shows the places where absorbing material should be placed, with dark blue representing the most important positions. There are two places where the placement of absorbing material is very central in this example; the ceiling above the office workers and the wall on the opposite side of them. In this simple case, the answer is fairly intuitive. For more complex situations, this is not always the case.

Binaural Sound with the Web Audio API

The simulation

Use headphones and click on any point around the person below to choose a direction for the incoming sound. The blue dots are in perpendicular directions relative to the listener. Try different head-related impulse responses (HRIR). Some of them will work better than others, depending on the individual. Note that the simulation has only been tested on Firefox and Chrome! Also, some people get errors with their web audio context having a different sample rate than the HRIR:s*.

  You need headphones for the following simulation!

The theory

Head-related transfer functions describe the cues we receive that enable us to determine the direction a sound arrives from. We only have two ears. To be able to determine the direction the sound arrives from in 3D, our brain has to use all the information it can.

For example, the sound will often arrive at the other ear with a small delay. Also, there will often be a difference in the sound level at one ear, as compared to the other (especially at high frequencies). But, additionally, there is a ton of information available for our brain to use. Our shoulders reflect sound. Sound reflects and diffracts around our external ears (pinna). As our features, such as the shape of our pinna, are individual, so is the way our brain perceives sound in 3D.

Still, our heads are often similar enough, which enables us to approximate 3D sound by ready made head-related transfer functions. Once we have a description of how sound arrives at our ears from different angles, we can take any sound and play it back from some direction in 3D.

The simulation in this post uses head-related transfer functions from the CIPIC HRTF database. This paper by Henrik Møller provides some nice additional information about head-related transfer functions.

The source code

The source code is here:

*) If your web audio context has a different sample rate as compared to the HRIR:s sample rate (44.1 kHz), the audio won't work. Apparently the sample rate of the context isn't definable (please correct me if I'm wrong!), so the HRIR should be resampled for it to work.

Predicting the absorption coefficient of micro-perforated plates

Aluminium Honeycomb perforated panel

Aluminium Honeycomb perforated panel

I was playing around with the topic of this blog post on my laptop in the train a while ago (while reading the paper "Predicting the absorption of open weave textiles and micro-perforated membranes backed by an air space" by Kang et al), and thought it might make a good topic for a blog post.

Micro-perforated plates

What are micro-perforated plates/membranes? I'll just quote Wikipedia's concise explanation: a thin [...] plate, made from one of several different materials, with small holes [...] in it. An MPP offers an alternative to traditional sound absorbers made from porous materials.

The basic idea is this: because the holes are so tiny, sound gets absorbed while passing through the holes due to friction. A nice example of micro-perforated plates are transparent membranes, which you can see through while they still offer sound absorbing qualities.

Designing micro-perforated panels

I wrote up some equations from the paper in JavaScript. The equations give the absorption coefficients for normal incident sound and a diffuse sound field. Note that the following equations are only meant for designing flat, micro-perforated plates with a relatively small closed air space behind them. Also, I take no responsibility for any errors in the calculations, although I noticed that the calculations did seem to agree with the plots in the paper.

This is currently for non-metallic substances only (as given by the paper), let me know if someone out there wants me to implement metallic membranes as well (or equations given by some other paper)!

If you're at a loss as to what to try, you can try the following values: hole width: 0.3 mm, hole spacing: 3.3 mm, cavity depth: 0.1 m, mass: 0.19 kg/m2, thickness: 0.17 mm.

Synthesizing thunder using JavaScript

Have you ever considered what actually causes the sound of thunder? The past summer brought with it a few thunder storms, which got me thinking about a topic I find very fascinating and cool: simulating thunder!

The simulation

Note that the simulation can be quite heavy for more complex lightning strikes (some are even unrealistically complex) and for longer distances from the strike. Calculating the result can take quite a bit of time, depending on your computer. A sample frequency of 22 kHz is used and the simulation is monaural. The lightning consists of a single discharge between the cloud and the ground.

If the simulation appears to jam up, please reload the page. Once again, I recommend Chrome for the simulation. Take a look at the source code here - sorry, I was somewhat lazy with the commenting of the code!

What is lightning?

Let's use the definition of lightning given by google:

The occurrence of a natural electrical discharge of very short duration and high voltage between a cloud and the ground or within a cloud, accompanied by a bright flash and typically also thunder.

I believe the geometry of the lightning channel becomes clearer when one considers the part of the following video starting at 1 min 15 s:

Note that the person in the video talks about the lightning consisting of "roughly 50 yard segments". These segments, referred to as the tortuosity of the lightning channel, are usually between 5 and 70 meters long [Rakov et al. 2003].

In our case, let's simplify the lightning as consisting of pretty much straight lines, with a random length of 5 to 70 meters. The lines zig-zag constantly, with a random variation of about 16 degrees between each line. Also, we'll need to add a small statistical deviation in the vertical direction.

Here is an example of what this looks like (calculated in 3D, and projected to 2D):

Lightning channels generated using a simple algorithm

Lightning channels generated using a simple algorithm loosely based on the statistics given by Rakov et al.

What causes thunder?

Ok, so what causes the sound of the lightning? Let's consider what happens when the discharge happens. We very quickly warm up a channel of air (the orange area in the cut plane image below). As the temperature in the channel rapidly rises to ~24000K [Orville, 1968], the pressure of the air in the channel rises enormously (to about 10^6 Pa). This pushes the the air outwards at speeds exceeding the speed of sound, causing a shock wave expanding at roughly 3000 m/s [Few, 1986].

The shockwave caused by lightning

The shockwave caused by lightning

After this, the air in the channel quickly cools down. The pressure behind the rapidly expanding shock wave will momentarily drop below atmospheric pressure due to the inertia of the outwards traveling air mass. The shock wave will travel some distance (the "relaxation radius"), after which it will dissipate, leaving behind what is called the weak shock wave. This weak shock wave can now be plotted as a function of pressure:

Weak shock wave [Brode, 1955]

Weak shock wave [Brode, 1955]

Never mind the scales for now (y-axis represents atmospheric pressure), but note this: the pressure wave will propagate towards you from the lightning so that the "sharp" part of it will reach you first.

What is thunder?

Ok, so now we now that the really hot lightning channel causes a traveling pressure wave. We also know that changes in pressure equals sound. So what we're hearing is the pressure waves caused by the rapidly heating air in the lightning channel.

But what causes the rumbling sound? Why does the thunder keep on rumbling for many seconds? What makes a close lightning strike sound (sort of) like a clap, while a distant strike can only be heard as rumble?

Here are some of the reasons:

1. The size of the lightning is huge

Sounds from different parts of the lightning will arrive at different times

Sounds from different parts of the lightning will arrive at different times

Consider a lightning strike some distance from you, as in the image above. If you measure the distance to each part of the lightning, you will see that the distance can vary by miles/kilometres! Considering that sound only travels at about 340 m/s (1,125 ft/s), there will be multiple seconds between when the sound from the nearest part of the lightning strike arrives at your position, as compared to the sounds form the more distant parts.

The situation can also be thought of according to Huygens' principle, which states that any source can be thought of as a series of spherical sources (kind of like in the image above). This is how the simulation, presented at the beginning of this post, works. The lightning is divided into multiple small segments, each modeled as a separate spherical sound source.

2. Sound attenuates by distance

It is, perhaps, obvious that more distant sound sources are quieter. But, additionally, it should be noted that higher frequencies attenuate much faster! Thus, when sound travels a distance, it gets "muffled" by air. Wolfram alpha is a great resource for this, it calculates this attenuation according to ISO 9613-1:1993. This causes distant sounds to "rumble" while the closest sounds are sharp and discernible.

3. Other stuff

There are loads of things at play in real life (for example atmospheric diffraction, which is the reason for the thunder sometimes being completely inaudible even when the lightning strike is clearly visible). If you're interested in learning more about the topic, I found the following books useful:

  • Vladimir Rakov & Martin Uman - Lightning, physics and effects
  • Geophysics study committee - Earth's electrical environment
  • Hans Volland - Handbook of Atmospheric Electrodynamics, Volume 2

4. Future ideas

It would be really cool to make the simulation in stereo, so that the sounds from the lightning channel are panned to their respective position  (or even using head related transfer functions!). If someone else is up for the task and has some knowledge of acoustics and DSP, feel free to contact me! 🙂