Similar, yet different: Quadratic vs. Itself?

For this installment of “Similar, yet different,” we will take a classic welled-quadratic sound diffuser,The Model Q, and compare its performance to itself – only installed backwards!

Taking “similar” to the extreme in this case, we are testing the difference in performance of a 1-dimensional, welled-quadratic diffuser installed in the standard welled configuration, and then installed reversed – with the sound impeding on the back side of the wells. For a bit of history, the Classic Quadratic Diffuser (or Schroeder diffuser) was designed with a grid separating the reflectors – creating wells of different depths proportional to the remainders of n2(mod N). This design has some interesting facets.

  • They are inherently symmetric if left in the original sequence.
  • They are periodic (i.e. they repeat.)
  • The discrete Fourier transform of the exponentiated sequence has constant magnitude.

The design principal is simple if you tear apart the math, and it’s simply wells that have a different effect on different frequencies, depending on the geometry of the wells. TheModel Qis an advanced 1D-Quadratic with angled well-bottoms, which assist in smoothing out the performance and widening the 1D polar radiation. So if this design is relying on the wells to be effective, why would we reverse it?

An acoustically diffuse environment develops due to many factors, and while the frequency focus of the wells is useful, there are other scenarios where different methods may be preferred. If the geometry of the elements were flipped around, you would get the same (albeit reversed) ratio of distance, but you lose the containment and channeling that the wells provide. This imparts a diffraction on the unrestrained elements. This also allows for a different interaction between the elements, as the face of the unit is no longer planar.

Let’s look at the effect this has on the performance of the device at some different frequencies – starting low and moving up…

First, we will look at the 1150Hz performance of the devices… standard welled-install on the left, reversed on the right.

1150Hz Performance of the Model Q - standard on left, reversed on right.

At 1150Hz, there is a little variation in the performance. Both are front focused, with a strong 1D horizontal polar response, but they are not identical. The welled-design (left) shows a broad frontal response, while the reversed design has a smoother vertical response, sharper front lobes, and stronger side performance. Overall, this difference is relatively small at this frequency.

Now, we will look at 2300Hz.

Again, we have two similar looking balloons, but there seems to be a bit more variation. The welled-design (left) shows a smoother 1D pattern in the front as the wells release sound within the same plane – at the front face of the wells. On the right you will notice sharper and more discreet lobes, but you will also notice that it has wider horizontal performance again, as it isn’t as front focused due to its free standing elements. The vertical performance is also a bit different – the welled design is broad and smoother vertically, while the reversed installation shows sharp lobes again.

Step up to 2800Hz, and we see some more drastic differences.

The performance of the standard welled-install (left) stays smooth and front-focused, while the lobes of the reversed install (right) have become even more distinct. Interestingly, the side lobes are even larger, showing an even wider polar pattern than before. These two instances show a marked difference between the smooth front-focused wells and the wide sharp scattering of the unrestrained elements. These two configurations are both very different, but are still both very effective at helping to disperse the incoming energy. Remember that the room develops diffusion through sound travelling in many different directions – these are not simple reflectors sending the specular energy in a single direction.

Now at 3650Hz we see a shift toward the reverse installation.

At around 4K the welled-installation (left) begins to move back front and center. It’s primary method of diffusion uses the wells to channel the energy, and at higher frequencies sound becomes much more directional. This directionality is used to create a temporal shift in the sound, as the reflections will occur out of phase from the source, and controlling that reflection is paramount to tuning this method of diffusion. However, as stated before, there are other mechanisms contribute to diffusion. The unrestrained elements on the right balloon, have hit their stride and still maintain a wide 1D polar pattern. The lobes are still sharp, showing the interaction of the elements with sound. This installation is showing the strength of its spatial dispersion, which will send acoustic energy in more directions and use the travel through the space to create a diffuse environment. It loses some of the frequency tuning of the wells, but makes up for it in the wide polar pattern.

Now for the super high frequencies – we jump straight to 10Khz.

最后一组显示了两个扩散器推到limits. The welled-installation (left) is a very narrow focused beam now. You will note that it has some variance due to the interactions with the walls of the wells but all of its work is done through phase shifting at this point. In contrast, the exposed elements (right) are still allowing for a bit of diffraction to occur, and the angled faces are still allowing for a bit of spatial redirection. Also note that these polar patterns were generated with a sound source directly in front of the device at 0° incidence, and the exposed elements would offer more exposure to its surface area than a welled design at wider angles of incidence.


Diffusion develops using many different variables, including the untreated walls of the space. While both of these installations are functioning in nearly identical frequency ranges due to their geometry, the mechanisms which they work are slightly different and have different strengths. The welled-design (in classic temporal Schroeder configuration) uses the wells to channel sound and address the frequencies in a tuned and controlled fashion. By simply flipping the device around, however, you change its performance from a controlled time shift, to an unrestrained spatial redirector, which imparts time shift through dispersion, diffraction, and distance travelled – further reducing intensity by having a wide 1D diffusion polar pattern. Both have scenarios which one configuration would be preferable over the other, makingthe Model Q diffusera very versatile device.

Both configurations are literally two sides of the same coin… they work in different ways, over the same frequencies, providing results – no matter how you flip them.