Speaker driver beaming frequency formula

[Tom Danley]

Well, using the 1/4 wave rule (dividing by 4 instead of 2) really is the most conservative but the problem is that it's pretty much impossible to build a modern speaker with less than a 1/4 wavelength between two drive units, even if you modify flanges to make it happen. So, the 1/2 wave "rule" is "close enough".

Well, it isn't impossible to get that spacing, it may require re-thinking where crossover points are and what drivers you use however.

While nearly all the products I have designed more recently are based on that patent and are horn loaded loudspeakers, fig 8 here shows how that can be done with direct radiators to also produce coherent addition and a single point of radiation.

https://patentimages.storage.googleapis.com/f7/b8/42/601accac9016b2/US8284976.pdf

Fig 7 shows how this can also be accomplished by modifying a conventional coax driver and adding an external horn.

I see your in north Alabama, not too far from North Georgia. Do you ever go to "Iron City" in Birmingham or see the Tide play?;
Best,
Tom Danley
Danley Sound Labs

 

[617]

This is an interesting topic. A while back I made a graph showing the db of drop off vs frequency and diameter:

I was trying to find the ideal crossover points and diameters for a 3 way speaker. What I found was that direct radiators are always a compromise in terms of off axis response. If you want to cross a 26mm tweeter over to a midrange, even a 70mm midrange will produce 3db difference at 40 degrees off axis. That's a small midrange!

Revel's bigger speakers show the ideal approach, with a small waveguide on the tweeter crossing over to an upper mid, then a lower mid, and finally woofers. If you really want a smooth DI curve, big waveguides or some other acoustic intervention is needed.

The other technology is big or geometrically complex arrays with shading and phase/frequency manipulation, but this is outside the expertise of most DIYers at least at the moment.

 

[hardisj]

Well, it isn't impossible to get that spacing, it may require re-thinking where crossover points are and what drivers you use however.

I agree. I was thinking about your synergy horn when I said made the note about it being practically impossible with conventional speakers (mid and tweeter on a baffle; conventional).

I see your in north Alabama, not too far from North Georgia. Do you ever go to "Iron City" in Birmingham or see the Tide play?;
Best,
Tom Danley
Danley Sound Labs

No, I sure don't, Tom. But I'm only about an hour north of Birmingham and have been there countless times for one thing or another (been shot at a couple times there, and gotten in a fight there, too! ). Lots of good times back in my BMX days and some crazy stories to tell from then, too.


I just double checked your location... I didn't realize you're in Gainesville! For some reason, I thought you were in North Carolina. My brother lives about 30 minutes from Gainesville and teaches school at East Hall County. Maybe, if you're game, I could come by for a visit sometime and maybe even possibly get a loaner of your product to test and review?

 

[MrPeabody]

There are a couple simple points on the graph that is easy to remember.
When a simple piston is about 1 wavelength in diameter, the radiation pattern is about 90 degrees.
The point it departs from being an omni source (and reaches the knee in the radiation resistance curve) is when it's diameter is K=1 about 1/3 wl or 1wl in circumference. Complex radiation begins (and produces a narrower lobe with sub lobes and nulls) when the piston is still larger

Examine the polar plots here;

http://www.acousticfrontiers.com/20...kers-open-up-your-acoustic-treatment-options/

The little debate taking place here, over the “correct” formula, may be attributed to the fact that formulas deal with a threshold that isn’t sharply defined. As such, it seems to me that the formulas aren’t as complex as they need to be. They need to include a parameter that represents the quality of the dispersion.

I like the polar plots in the link you provided (which is why I made this post a reply to your post). However they can be little difficult to interpret, one reason being that the ka number is defined in terms of piston circumference rather than diameter. Since this is all only a crude approximation anyway, it seems reasonable to replace circumference with 3 x diameter.

There is a huge difference between the ka = 2 plot, where the piston diameter is 2/3 the wavelength, vs. the ka = 5 plot, where the piston diameter is 5/3 the wavelength. In the interest of dumbing this all down:

- Directivity will be excellent (wide) for wavelengths that are at least twice greater than the piston diameter.
- Directivity will be fairly good for wavelengths that are at least half-again greater than the piston diameter.
- Directivity won’t be horrible for wavelength equal to the piston diameter, however at wavelengths slightly shorter than the piston diameter, directivity will be narrow with multiple lobes and suck-outs in the forward hemisphere.

If there is to be a formula (there always has to be a formula) to assist in deciding on crossover points, it should be just complex enough for people to not argue pointlessly over whether it is correct. The following formula seems useful:

f < 13500 / (N x D_in_inches), where N = 2 for excellent directivity, N = 1.5 for good directivity, N = 1 for respectable directivity

It goes without saying that D is the actual piston diameter, not the diameter of the frame.

For N = 1 and D = 1", this formula gives 13.5 kHz. In other words, for a 1” tweeter, directivity is respectable at 13.5 kHz. Directivity is fairly good at 9 kHz, and excellent at 6.75 kHz.

For a 6" mid-woofer crossed over to a tweeter, and using N = 1, f is 2.25 kHz. For N = 1.5, f is 1.5 kHz, and for N = 2, f is 1.125 kHz.

The same formula but with D expressed in millimeters:

f < (343 x 10^3) / (N x D_in_millimeter), where N = 2 for excellent directivity, N = 1.5 for good directivity, N = 1 for respectable directivity

One could argue that the constant 13500 (inches/second) should be higher or lower in order for the frequency returned by the formula to better correspond to the subjective terms “excellent”, “fairly good”, and “respectable”. Then the debate ends up over what constitutes “excellent” directivity, what constitutes “fairly good” directivity, and what constitutes “respectable” directivity. Regardless, making the formula any simpler than this isn’t going to help.

 

[Tom Danley]

Hi
It is frustrating to figure what the pattern is for the cases in between the points on a graph and K=1 is handy as a greasy screwdriver handle.

At work, getting the good results in a large room depends on the right speaker pattern and right aiming dimensions because most of them have a very strong polar pattern.
The design tool for those speakers might be of interest here though as one of the source choices is an onmi speaker and you assign how many and where x,y,z they are and so on AND select the frequency to see the polar pattern as an intensity plot.

I don't use this tool often (my job is done at this point haha) , this is more for the guys at work and consultants that design the sound systems but might be fun here in the question "what do two or more sources do when xx x". It is free too.. This video shows a quick start using actual speakers but onmi's are a choice too. The Narrator is Doug Jones a friend of mine who made the LEDR stereo recordings in the 80's.

I took that down, I pasted only a link not a bill board

Tom Danley

 

[fluid]

If you really want a smooth DI curve, big waveguides or some other acoustic intervention is needed.

This does not have to be the case if a low DI is the target. The Magico A5 as seen with measurements here https://www.audiosciencereview.com/...magico-a5-spinorama-cea2034.23866/post-802416 is a good example of how the DI can be very smooth. There is still a waveguide but it's quite minimal. There are tweeters with faceplates that would offer a similar directivity.

 

[fluid]

If there is to be a formula (there always has to be a formula) to assist in deciding on crossover points, it should be just complex enough for people to not argue pointlessly over whether it is correct.

I think using basic formulas (other than for early preliminary thoughts) is the wrong direction to go in if an actual speaker design is the goal. By using tools like Vituix you can see the effect of distance between drivers, matched or unmatched phase, crossover types and almost any DSP or passive component on all angles all at the same time. It is free and not that hard to learn, easier than arguing over formulas anyway

The shape and size of the baffle, any waveguide and the shape of the cones of drivers plus the crossover and other components all play a part in the final directivity of a speaker. BEM/FEM/FEA modelling or making and measuring prototypes is the only practical way of seeing how all of these things will interact. It is very complex and the effects are not always intuitive.

 

[ctrl]

f < (343 x 10^3) / (N x D_in_millimeter), where N = 2 for excellent directivity, N = 1.5 for good directivity, N = 1 for respectable directivity

For a very rough classification of the different chassis sizes, such a loudspeaker cabinet-independent formula is useful, especially if you have no other option.

So let's apply all formulas for a 6.5'' woofer. Specifically, it should be the 6.5'' Purifi driver, with about 0.131m effective diameter.
Besides this formula, other formulas were mentioned. Here is an overview of all formulas:

c = speed of sound
dia = effective driver diameter

f-beam < c/N*dia = 2600Hz / 1700Hz / 1300Hz (N = 1, 1.5, 2)
f-beam = 2*c/pi*dia = 1700Hz
f-beam = 2*c/dia = 1300Hz
f-beam = c/dia = 2600Hz

BEM/FEM/FEA modelling or making and measuring prototypes is the only practical way of seeing how all of these things will interact. It is very complex and the effects are not always intuitive.

Exactly, nowadays, unlike 40 years ago, every developer has tools at his disposal that more or less accurately predict the radiation behavior.

For very accurate predictions the effort is quite high (because the behavior of the whole driver has to be simulated), for good predictions (around +-1 or +-1.5dB) simulations are relatively easy to set up.

In addition to the obvious influences such as cone shape and driver surround, the shape and dimensions of the loudspeaker cabinet also have a major impact on the radiation pattern of the driver (as already mentioned by others) - the depth of the cabinet also plays a role, which is often not taken into account during preliminary considerations.

So let's take our driver from above and put it into different speaker enclosures.
1) a very narrow cabinet with 17cm width
2) a wide cabinet with 30cm width
3) a very narrow cabinet of 17cm width and shallow depth
4) an asymmetrical cabinet with a width of 8cm at the top and 33cm at the bottom



As a measure of beaming, we consider the -6dB point (sound pressure level is lowered by 6dB) relative to the normalized axis frequency response, for each enclosure shape:

After the transition from full-space to half-space, the loudspeaker cabinet significantly determines the radiation pattern.

- Wide cabinets show a significantly different radiation pattern than very narrow ones (green and cyan curves).
- The cabinet depth also has a significant influence on the radiation pattern (cyan and red curves).

Only above 2.2 - 2.6 kHz the driver itself determines the radiation.

For all those who think it would change fundamentally, if one does not normalize to the axial frequency response, here the same analysis normalized to the 30° frequency response:


For all those who can't do anything with the upper diagram, here are the simulations of the frequency responses 0-180deg (normalized on-axis):


How well the above formulas apply, everyone must decide for themselves.

 

[MrPeabody]

For a very rough classification of the different chassis sizes, such a loudspeaker cabinet-independent formula is useful, especially if you have no other option.

So let's apply all formulas for a 6.5'' woofer. Specifically, it should be the 6.5'' Purifi driver, with about 0.131m effective diameter.
Besides this formula, other formulas were mentioned. Here is an overview of all formulas:

c = speed of sound
dia = effective driver diameter

f-beam < c/N*dia = 2600Hz / 1700Hz / 1300Hz (N = 1, 1.5, 2)
f-beam = 2*c/pi*dia = 1700Hz
f-beam = 2*c/dia = 1300Hz
f-beam = c/dia = 2600Hz



Exactly, nowadays, unlike 40 years ago, every developer has tools at his disposal that more or less accurately predict the radiation behavior.

For very accurate predictions the effort is quite high (because the behavior of the whole driver has to be simulated), for good predictions (around +-1 or +-1.5dB) simulations are relatively easy to set up.

In addition to the obvious influences such as cone shape and driver surround, the shape and dimensions of the loudspeaker cabinet also have a major impact on the radiation pattern of the driver (as already mentioned by others) - the depth of the cabinet also plays a role, which is often not taken into account during preliminary considerations.

So let's take our driver from above and put it into different speaker enclosures.
1) a very narrow cabinet with 17cm width
2) a wide cabinet with 30cm width
3) a very narrow cabinet of 17cm width and shallow depth
4) an asymmetrical cabinet with a width of 8cm at the top and 33cm at the bottom

View attachment 133291 View attachment 133292 View attachment 133293 View attachment 133294

As a measure of beaming, we consider the -6dB point (sound pressure level is lowered by 6dB) relative to the normalized axis frequency response, for each enclosure shape:
View attachment 133302

After the transition from full-space to half-space, the loudspeaker cabinet significantly determines the radiation pattern.

- Wide cabinets show a significantly different radiation pattern than very narrow ones (green and cyan curves).
- The cabinet depth also has a significant influence on the radiation pattern (cyan and red curves).

Only above 2.2 - 2.6 kHz the driver itself determines the radiation.

For all those who think it would change fundamentally, if one does not normalize to the axial frequency response, here the same analysis normalized to the 30° frequency response:
View attachment 133303

For all those who can't do anything with the upper diagram, here are the simulations of the frequency responses 0-180deg (normalized on-axis):
View attachment 133307 View attachment 133308 View attachment 133309 View attachment 133310

How well the above formulas apply, everyone must decide for themselves.

All very good and very insightful. In fact, in order for the original question to be a meaningful question, constraining conditions are needed. My assumption, had I thought about it, would have been that the baffle is an approximation of an infinitely large baffle. Otherwise it is kind of meaningless to even be talking about the directivity of the piston per se. I vaguely recall Harry Olson having done some studies of the effect of the piston diameter on directivity, and I would assume that he did these studies using a very large baffle, since otherwise it wouldn't have made a lot of sense. The simple calculators likewise have to be based on such an assumption; ditto for the polar plots that are often presented.

Notwithstanding the obvious limitations of any simplistic formula such as the one I hastily cobbled together, simplistic formulas of this sort are useful to me because they give me a rough sense of the directivity match to expect from a driver of a given size crossed over at a given frequency.

Time and again I have reached the conclusion that except for tiny bookshelf speakers with mid-woofers no bigger than 12 cm, good loudspeakers should have at least three drivers different in size. I'm itching to get back to where I have a workshop so I can build. I am encouraged to begin building again, partly by the fact that I think I will be able to coax ctrl (and some of the others here) into helping me with the crossover design. Like Rick and ctrl, I occasionally spend time thinking about what sort of enclosure design will simultaneously satisfy the need for it to be easy to construct and the need for it to work well acoustically. And thinking about where you get the most difference by spending money, and where you don't get much return for the money spent. Driver selection is without doubt the most important determinant of the cost/quality ratio for a speaker. This is where it gets fun, to prove that "my speaker that uses a dome tweeter is as good as your speaker with that expensive ribbon, and mine costs a lot less to build than yours". And in the same vein, "but if I don't use that faddish woofer it will sound just as good and won't cost nearly as much." I need a workshop.

 

[fluid]

In addition to the obvious influences such as cone shape and driver surround, the shape and dimensions of the loudspeaker cabinet also have a major impact on the radiation pattern of the driver (as already mentioned by others) - the depth of the cabinet also plays a role, which is often not taken into account during preliminary considerations.

I did not specify it directly although in my head it was included I agree that the depth of the cabinet is almost never considered as a factor by many DIY speaker builders, probably in part due to the fact that without BEM or cut and try the diffraction calculators only consider the front face and edges. Your graph shows clearly what happens. Getting the transition to omni at the right point for the driver and baffle to produce a smooth result is not that hard with the right tool, but without it quite easy to make a mess of it.

I like the beamwidth graph you made in VACS, how did you do that?

My assumption, had I thought about it, would have been that the baffle is an approximation of an infinitely large baffle. Otherwise it is kind of meaningless to even be talking about the directivity of the piston per se.

It's not meaningless because at some point the driver size and shape will dictate the directivity, but if you don't consider the baffle and overall enclosure you won't see the full picture.

 

[MrPeabody]

It's not meaningless because at some point the driver size and shape will dictate the directivity, but if you don't consider the baffle and overall enclosure you won't see the full picture.

Not that this matters in the least, but I don't follow your meaning. Certainly "at some point" the driver will dictate the directivity. My point was that implicitly, the baffle is assumed to be large enough such that the directivity is determined entirely (or nearly entirely) by the driver, and that otherwise the question of how piston size affects directivity is a meaningless question. To me, this is self-evident. Maybe you misunderstood what I was saying.

 

[fluid]

Maybe I didn't explain it very well. When the diameter of the radiating area is comparable to the wavelength then that size and shape will dominate the directivity pattern from there upwards. An infinite baffle will free the response of low frequency diffraction but it is that diffraction on a finite sized baffle and box that will determine the low frequency directivity below the point where the driver sets the response.

To me it seems more useful to consider the whole mechanism rather than abstract it back to only the driver but as frequency rises the driver becomes dominant so in that sense it is not meaningless to consider separately if that is all that is possible.

 

[ctrl]

I vaguely recall Harry Olson having done some studies of the effect of the piston diameter on directivity, and I would assume that he did these studies using a very large baffle, since otherwise it wouldn't have made a lot of sense. The simple calculators likewise have to be based on such an assumption; ditto for the polar plots that are often presented.

I don't know the studies you refer to, but your statements make sense to me. In particular, if the polar plots only cover +-90 degrees, that is an indication that simulations were done under half-space conditions (infinite baffle).

Our example would then look like this (ignore the tweeter):

The -6dB frequency response then looks like this:

...and the corresponding polar plots at 1.3, 1.7 and 2.6 kHz:

For comparison, here are the polar plots for the 6.5'' driver in the speaker enclosures from our example in Post#48.

1) a very narrow cabinet with 17cm width

2) a wide cabinet with 30cm width

4) an asymmetrical cabinet with a width of 8cm at the top and 33cm at the bottom

As a very, very rough approximation, one can use the formulas mentioned or to simply explain certain facts, for example why it is not a good idea to pair a 12'' woofer and a tweeter at a crossover frequency of 2kHz.

But I can't imagine that anyone who seriously designs loudspeakers uses them to calculate the "optimal" crossover frequency.

I like the beamwidth graph you made in VACS, how did you do that?

Right click in the spectrogram => New marker => set z-Value to -6

 

[Tom Danley]

I agree. I was thinking about your synergy horn when I said made the note about it being practically impossible with conventional speakers (mid and tweeter on a baffle; conventional).





No, I sure don't, Tom. But I'm only about an hour north of Birmingham and have been there countless times for one thing or another (been shot at a couple times there, and gotten in a fight there, too! ). Lots of good times back in my BMX days and some crazy stories to tell from then, too.


I just double checked your location... I didn't realize you're in Gainesville! For some reason, I thought you were in North Carolina. My brother lives about 30 minutes from Gainesville and teaches school at East Hall County. Maybe, if you're game, I could come by for a visit sometime and maybe even possibly get a loaner of your product to test and review?

Hi Erin
I would be interested in getting your opinions about what you hear, in fact there are two or three different hifi prototype speakers you could hear.
I think this might be valuable to us as this market area is not where the company itself has focused.

We have a big demo out of town next week but I think the week after would be ok for a demo if you want to come to Gainesville.
We ran into an amplifier supply issue on the small amplifiers after the first run which is being taken care of but until we get more, there wouldn't be any small speakers to lend out for measurements.

Hey north Alabama seemed like a nice part of the country at least where i was in Huntsville, I was down there about 40 times in the 80's and 90's but didn't see much outside of MSFC and a steak restaurant there I can't recall the name of haha.
Best Regards
Tom Danley

 

[hardisj]

Hi Erin
I would be interested in getting your opinions about what you hear, in fact there are two or three different hifi prototype speakers you could hear.
I think this might be valuable to us as this market area is not where the company itself has focused.

We have a big demo out of town next week but I think the week after would be ok for a demo if you want to come to Gainesville.
We ran into an amplifier supply issue on the small amplifiers after the first run which is being taken care of but until we get more, there wouldn't be any small speakers to lend out for measurements.

Hey north Alabama seemed like a nice part of the country at least where i was in Huntsville, I was down there about 40 times in the 80's and 90's but didn't see much outside of MSFC and a steak restaurant there I can't recall the name of haha.
Best Regards
Tom Danley

Oh, man, that would be so cool. I'd love to just get to hang out with you and chat with you. Definitely a lot of respect from my end towards yours. I wouldn't consider myself a golden-eared person but I would welcome the opportunity to get to listen to your speakers and provide input if it helps.

I actually worked on MFSC for a few years. Started my career there as a co-op and shifted to full-time after graduation. I'm on the army side now, though. I live in Decatur (if you happen to know where that is).

Should I reach out to you in a couple weeks or do you have an email for me to contact you?

Thanks again for the invite!

- Erin

 

[jtwrace]

 

[yourmando]

Oh, man, that would be so cool. I'd love to just get to hang out with you and chat with you. Definitely a lot of respect from my end towards yours. I wouldn't consider myself a golden-eared person but I would welcome the opportunity to get to listen to your speakers and provide input if it helps.

I actually worked on MFSC for a few years. Started my career there as a co-op and shifted to full-time after graduation. I'm on the army side now, though. I live in Decatur (if you happen to know where that is).

Should I reach out to you in a couple weeks or do you have an email for me to contact you?

Thanks again for the invite!

- Erin

That’s so cool! Dreams can come true!

 

[puppet]

Starting to archive thoughts from contemporary masters of the industry ... just fuckin' awesome.

 

[MrPeabody]

Maybe I didn't explain it very well. When the diameter of the radiating area is comparable to the wavelength then that size and shape will dominate the directivity pattern from there upwards. An infinite baffle will free the response of low frequency diffraction but it is that diffraction on a finite sized baffle and box that will determine the low frequency directivity below the point where the driver sets the response.

To me it seems more useful to consider the whole mechanism rather than abstract it back to only the driver but as frequency rises the driver becomes dominant so in that sense it is not meaningless to consider separately if that is all that is possible.

What I had said, in essence, is that given the way in which the question was asked here, an implied assumption is that the baffle is large enough such that its affect on directivity is negligible. To me, it is difficult to fathom that anyone would look at the question and at the manner in which it was asked and not infer that this is an implied assumption of the question. How else would it make sense to ask about the correctness of a simplistic formula that relates directivity, frequency, and piston diameter, except by taking this as an assumption? It is fair and proper to point out that this assumption is a great oversimplification of the directivity behavior of most any real speaker. But I draw the line at the suggestion that this assumption about the baffle was not implicit in the question, as it was stated, by the OP. I say again as plainly as possible: the question as stated is not meaningful except by way of this implied assumption.

I have a tremendous appreciation for the use of digital simulation in acoustical analysis. It is a tremendously powerful tool, and I have tons of respect for people who know how to use this tool, and especially for skilled users who are happy to share their knowledge and skill for the benefit of others. Sometimes, though, I find myself wanting to remind everyone that acoustics was a highly developed discipline well before the advent of digital simulation. In that earlier era, insight was obtained by tediously developing analytical formulas and by tediously performing experiments to fine-tune the analytical formulas so that they accurately describe physical reality. I am very highly impressed by what people like Harry Olson were able to do. Olson's "Elements of Acoustical Engineering" was originally published in the early 1940s. The final version was finished in the late 1950s. The text is packed full of formulas and graphs of every imaginable sort. I was able to take a few excerpts from the PDF file, which serve as examples of what may be found within Olson's text.

 

[fluid]

It is fair and proper to point out that this assumption is a great oversimplification of the directivity behavior of most any real speaker. But I draw the line at the suggestion that this assumption about the baffle was not implicit in the question, as it was stated, by the OP. I say again as plainly as possible: the question as stated is not meaningful except by way of this implied assumption.

In case it is not clear I'm not trying to disagree with this but pointing out that enforcing an infinite baffle has consequences that may not be obvious. I have overlaid two of ctrl's graphs to show what I mean.

You can see in this graph that the infinite baffle simulation differs markedly from any real life practical baffle the driver would be placed on. The purple line I have added is the approximate frequency where the wavelength is the same as the diameter of the radiating surface. Any likely finite baffle will converge at this point but the infinite baffle does not. Edit: Graph changed due to incorrect scales, thanks to ctrl

Sometimes, though, I find myself wanting to remind everyone that acoustics was a highly developed discipline well before the advent of digital simulation.

All of the software is based on the brilliant work that has gone before and wouldn't be possible without that effort. I'm sure if Olsen had it at his disposal he would have used it. The fact he got as far as he did without it is a testament to him.