Can you give us a simple equation? The corresponding full wavelenght will be 16'' which corresponds to about 850hz, this part I can't workout. Essentially your rule helps figure out if the midrange will be effected by baffle difraction?
I mean for arguments sake we can't use a waveguide on the midrange. And we have accounted for the general lobing using 1.4 times wavelenght at XO and the XO will be around 2700hz or 2800 maybe even 2900hz. What I am trying to achieve a narrow baffle?
Also, I am curious about how baffle edge diffraction is mitigated in a speaker that is layed vertically. If you look in my profile photo top right corner I have a 3 way, it is a trusted design by a trusted deisgner but am curious if by looking at it the side which has the 8inch woofer has a baffle edge far enough to mitigate the issue(as mentioned in your other post)? And I am presuming that the other side(to the side of the tweter and midrange to the baffle) is also narrow enough?
You ask good questions, and I wish I could give equally good answers to all, but I'm not sure. You want a simple formula of some sort, but I'm not sure this is reasonable because of the gradual nature of the effect. What I can do is have a try at a better explanation, although this is clearly not what you want. Sitting here at the keyboard I started typing and ended up with something much longer than I had intended to write. (This always happens to me.) Should I delete it, or should I go ahead and post it? It might be helpful to you, and it isn't going to do any harm.
A good source of information on this topic is found in Linkwitz' general discussion of speaker design. Last I checked it is still found in the website that is still being maintained. One section of that broad discussion of speaker design is devoted to baffle edge diffraction. He did a bunch of experiments with tweeters mounted on simple baffles of various size, set on a tall pole. His experiments echoed earlier experiments done by Harry F. Olson. Speaking of Olson, his two major books on acoustics are now available for free on line, and therein you will find his experiments and conclusions on baffle edge diffraction and the effect of the shape of the enclosure. Somewhere on the ASR site there is a list of recommended reference material, and somewhere toward the end of it, you should find links to online locations where you can grab copies of Olson's books. If you want.
Before I go any further I also need to apologize to Rick and ctrl for dumping this in their Directiva thread. They've both been very generous to people who've taken the thread off on divergent paths. If there is ever a cleanup of the Directiva thread, maybe this post could be moved somewhere else.
Baffle edge diffraction is intimately related to the baffle step effect. Usually these two phenomena are thought of as separate phenomena, but they cannot be cleanly separated. As concerns the baffle step, we know that at adequately high frequency, where the wavelength is much shorter than the baffle dimensions, the radiation is into hemispherical space. This implies +6 dB of gain (for the area directly in front of the speaker) vs. low frequency. For a perfectly spherical speaker we would expect a very smooth, gradual step increase from low to high frequency, amounting to +6 dB overall, with the bulk of it occurring within a spread of just a few octaves. At wavelength much greater than the biggest dimension of the speaker and baffle, the wave envelopes the speaker and pays no heed to whether the speaker enclosure is spherical or rectangular or whatever, because the irregularities of the speaker enclosure are dimensionally small in comparison to the wavelength. As frequency increases and the wavelength shortens, eventually the wavelength is short enough such that the irregularities of the speaker enclosure are no longer dimensionally small in comparison to the wavelength.
When the wave emitted by the driver reaches the edge of the baffle there is an abrupt decrease in acoustic impedance. A secondary wave propagates from the baffle edge (all along the baffle edge, all the way around). When an abrupt change in acoustic impedance is a decrease, the wave propagating from the point of diffraction is shifted by 180 degrees relative to the incident wave. This is one of two sources of phase offset, for the diffracted wave relative to the wave reaching the listener directly from the driver. The other source of phase offset is the difference in the two distances: the distance from the listener directly to the driver center, vs. the longer, indirect distance that takes a detour. For a listener on-axis the distance to the edge where diffraction occurs is approximately the same as the distance from the listener to the center of the driver. Thus, the difference in the two distances is approximately the distance from the center of the driver to the edge of the baffle. The amount of phase shift that this distance implies depends on how great this distance is in relation to the wavelength. And to this amount of phase shift, you add the other 180 degrees, to obtain the full phase offset for the diffracted wave.
At very low frequency where the wavelength is very long compared to the baffle dimensions, the amount of phase shift associated with the distance from the center of the driver to the baffle edge will be very close to 0, which means that the total phase shift will be the 180 degrees associated with the soft reflection. 180 degrees of phase shift implies mutual (partial) annihilation; the -6 dB baffle step loss can be explained by way of the phase cancellation.
As frequency increases and wavelength shortens, the distance from the center of the driver to the baffle edge becomes an increasingly greater fraction of the wavelength. The amount of associated phase shift increases from 0 degrees to something bigger than 0, eventually to 180, such that the total phase shift gradually migrates from 180 degrees to 360 degrees (or to 0 degrees, as you prefer to think of it). Since the phase cancellation effect decreases in strength, the response rises (i.e., the step). Eventually, the distance from the center of the driver to the baffle edge (the greatest of the several baffle edge distances) will be half of the wavelength. At this point each of the two sources of phase shift contributes 180 degrees, which that the total phase shift is 180 + 180 = 360. This of course means reinforcement rather than cancellation. At this wavelength, there is a prominent hump superimposed on the erstwhile smooth baffle step. This hump is located at wavelength equal to twice the distance from the center of the driver to the baffle edge. (More specifically, the baffle edge furthest from the driver.)
With many speakers, tall/skinny speakers in particular, the distance along that furthest edge of the baffle will be significantly shorter than the baffle edges that are nearer the driver. It is nevertheless useful to examine the special case where the baffle is perfectly square and the driver is located at the center of the baffle. In this case, the full wavelength, being equal to twice the half-wavelength, is equal to the width of the baffle. Thus, all we need to do, to obtain the corresponding frequency, is to take wavelength equal to the baffle width and translate this to frequency. For anyone who has any difficultly remembering how to do this, it can be helpful to think about the units. Since frequency is expressed in reciprocal seconds, it is apparent that you divide the speed of sound (which is expressed in units of distance divided by time, i.e., 1125 ft/s), by wavelength, which is expressed in units of distance. The distance units cancel and you are left with reciprocal seconds. For example, if the baffle width is 1 ft, you divide 1125 ft/s by 1 ft and get 1125 reciprocal seconds, i.e., 1125 Hz. This is the frequency at which the prominent first hump in the diffraction ripple is found, for a 12" x 12" baffle with driver mounted at the center.
As the frequency increases further and the wavelength shortens further, eventually the distance from the center of the driver to the baffle edge will equal a full wavelength. At this point in frequency the total phase difference will be 180 degrees, which means that instead of a peak, there will be a dip. For the square 12"x12" baffle this occurs at 2250 Hz. Then another peak at 3 x 1125 Hz, and another dip at 4 x 1125 Hz, etc.
Mr. Linkwitz subtly suggested making the baffle 2x or 4x or even 8x greater in one dimension vs. the other, and locating the driver 1/3 of the way from one edge, for both the vertical and horizontal dimensions. This leads to mutual cancellation among the dips and peaks in the different ripples associated with the different edges, sort of. This isn't a complete solution, for two reasons. One reason is that this mutual cancellation effect only works for listening positions directly on axis. For off-axis listening positions it doesn't work, because the phase relationships get out of whack. The other reason is that there isn't any dip anywhere that would cancel with the prominent first peak. If the driver is located 1/3 of the way from one of the side edges, the first peak associated with the shorter distance will coincide in frequency with the first dip associated with the longer distance. Still, there will be some suppression of that strong first peak, owing to the fact that only one edge is generating that prominent first peak, rather than both acting in unison. For a tall skinny speaker with rectangular baffle in the 4:1 or 8:1 ratio, the prominent first peak will most likely be suppressed adequately by a combination of other effects, including the effect whereby sound intensity weakens with increasing distance from the source. Still, it should be remembered that the mutual cancellation effect works only for listeners directly on axis. As you move off-axis in any direction, the phase relationships change in a way that is theoretically predictable yet chaotic for intents and purposes.
The question of how you know which drivers are affected is not an easy question. In order for a driver to not be affected, the upper limit of its frequency band needs to be an octave or so lower than the calculated frequency of the first peak in the ripple, using the greatest driver-center-to-baffle-edge distance. In general it is not necessary to completely avoid this effect. It is desirable to avoid this effect to the extent that is practical, and it is up to the individual speaker designer to decide what "practical" means. This effect is one among a slew of reasons why real speakers cannot achieve perfection. One of the best mitigating strategies is the use of strong beveling, where the bevel surfaces are nearly as prominent as the baffle itself, such that the enclosure is a crude approximation of a sphere, a tall cylinder, or cone, or some hybrid of these shapes.
