Understanding Baffle Step and Diffraction
Baffle step and diffraction are two very important concepts to understand in speaker design. Most people who are in to performance audio have probably heard the terms before, but if you don’t really understand what they are or how they affect a speaker, this article is for you.
We’re going to start with a look back at something you probably learned in an early physics class, and that is how two waves interact with each other in different waves. If you look below at figure 1, we are displaying a graphical representation of a sine wave with the x-axis represented as time and the y-axis represented as amplitude (or output level). When a wave moves, it moves in both positive and negative cycles across the x-axis with a peak and a trough defined by the initial output amplitude and a period, or cycle time, defined by the frequency being generated. Cycle time is the length of time to go from the zero position on the y-axis, through both the positive and negative peaks, and back to zero. In this instance, let’s assume this is a 1 kHz sine wave. This would mean we have a cycle time (T=1/f or in this case T=1/1000) of 1 millisecond. This means it takes one millisecond for a 1 kHz wave to complete a full period or cycle.
Now let’s add a second 1 kHz wave of exactly the same amplitude to the first and play them together at the same time. In figure 2, this situation is displayed. The red wave shown is laying directly over the top of the blue wave of figure 1 and together, they create the new wave shown in magenta. In this instance, the two waves are equal in both period and amplitude which creates a second wave of the same period but with double the amplitude (magenta line). This is called constructive interference. Constructive interference is when two waves add to each other’s amplitude to create a higher amplitude than the original.
But what if we started the second 1 kHz wave later? Let’s start it 0.5 millisecond later so that the second wave is perfectly out of phase (180 degrees) from the first. This means that the second wave’s positive peak is occurring at the first wave’s negative and vice versa. This produces the flat magenta line in figure 3. The two sine waves have cancelled each other out through destructive interference and now have no measurable amplitude.
Now let’s look at what happens if we start the second wave only slightly out of phase. Let’s start the second wave only 0.25 milliseconds later (90 degrees out of phase). In this instance, we get a mix of both constructive and destructive interference which creates a new wave that has slightly more output ins some areas and slightly less in others. You can see from the graph that the peak of the wave is higher than either the blue or the red because both waves have positive values at this point and add to a higher amplitude. But if you look where the magenta line crosses the x-axis, you can see that one of the waves has some positive value and the other has some negative value of an opposite amount, which is why the magenta line goes to zero.
If you'd like to play with this yourself, you can check out this handy tool.
Getting to the Point
You might be wondering what this has to do with diffraction and baffle step. Both diffraction and baffle step are technically part of the same phenomenon, but when people refer to baffle step, they are generally referring to the region when a wavelength gets long enough that it is no longer interacting with the front of the box. When you put a woofer in a box and measure it, you will see a gradual roll of in the low frequencies, and this is baffle step loss.
We have to cover a little bit more theory to really explain what is going on, so just hang in there. Let’s look at the image below, which represents a standard tweeter on a standard square-edged baffle. When a tweeter emits sound, it emanates as a complex spherical waveform that tries to radiate in all directions at once.
When a wave encounters a sudden transition, like the edge of a baffle, it can cause the wave to scatter in all directions at once. That occurs when the physical length of the wave is approximately the same distance as the distance to the edge of the baffle. This causes some of the wave to be reflected back into the forward direction and interact with the waves that were emitting from the tweeter itself. This causes additions and subtractions to the original wave like described above.
An object must be sufficiently acoustically large before full diffraction can occur. So what is sufficiently acoustically large? An object has to be greater than ½ of the wavelength at whatever frequency you are looking at to be considered sufficiently acoustically large. Wavelengths above this value see full diffraction effects. Below half the wavelength, the diffraction effects get progressively smaller until about one tenth the wavelength. At that point the object essentially becomes acoustically invisible and has no measurable impact on the diffraction signature of the wave.
As the wavelengths get longer in nature, or in other words get lower in frequency, the cabinet becomes too small to act on those waves, and you start to have a roll-off in the SPL at those frequencies. Depending on cabinet size and shape, this usually starts between 1000-2000 Hz and continues down to about 200 Hz.
That together is diffraction and baffle step. And that is what takes a nice looking, flat factory response in shown in gray below to the black line on the graph. In the black line, we now have a small peak at 1500 Hz and a roll-off below so that we are a full 6 dB down at 200 Hz.
2Pi versus 4Pi
Now the question might be, why don’t we see that from graphs from a manufacturer? When you see a measurement posted by a manufacturer, it is typically the infinite baffle, or 2Pi response of the driver, also known as half space response. When we measure a driver in a cabinet, it is generally the 4Pi response. The terms 2Pi and 4Pi are referring to a coordinate system using radians on an x/y axis. This can also be done in degrees. If you were to lay a circle centered on both the x-axis and y-axis and trace around it’s perimeter, you would have gone 360 degrees or 4Pi when you get back to your starting point. When we refer to 2Pi, we are referring to half that space, or 180 degrees of change.
We’ll look first at 4Pi. If you assume that you have a point source (or a source emitting sound that is infinitely small) in free space, it has nothing to act on it which could cause any form of diffraction because the wave is free to radiate in all directions at once. But if we place that point source on an infinitely large wall, you get something a bit different, and that is the 2Pi interaction. The large wall reflects acoustic energy, which wants to radiate in all directions. This contains the wave to one side of the wall (or 180 degrees). This effectively doubles the output in the forward direction and adds an additional 6 dB to the perceived response of the point source.
Finally… the point
When you change the wall to something the size of a typical baffle, you end up with something in between. At higher frequencies that see the baffle as sufficiently acoustically large, you end up with a 6 dB increase in output. At lower frequencies as you go through the transition region, you see a gradual reduction of output until the baffle becomes acoustically transparent and you are at a full 6 dB of reduction. And in that transition region at the edge of the baffle, you get that scattering effect described earlier which causes a series of peaks and dips. The severity of that is dictated by the size of the driver and the shape of the edge.
Smaller drivers radiate higher frequencies better off axis, so more of the energy reaches the edges and cause diffraction artifacts in the form of peaks and dips in the on-axis response. Larger drivers see less of this because the high frequencies don’t radiate as strongly off axis, so there is not as much interference on-axis.
I also want to mention is that there are several online calculators available to calculate a baffle step circuit for a speaker. In reality, good designers never use a separate baffle step circuit and the circuit used generally doesn’t even look similar to those online calculators. Good designers incorporate the compensation into the shaped response of the driver at the crossover with more typical crossover circuits, and you can see this in the crossovers we design if you look at the circuit as well.
Finally, if you want to simulate baffle step and diffraction of a speaker, you don't even need to know all the math behind how it works. There are hand tools for free online like Jeff Bagby's Response Modeler, which allow you to simulate it by putting in your cabinet dimensions and loading in a driver's .frd file.