Yes, your analysis in terms of volume acceleration seems totally reasonable to me, for the reasons you stated.
But in those terms the plane source has a hell of a tail, it's infinite!
So to say the point source has no tail with (constant) volume acceleration, the line source has a tail in volume acceleration and the plane source has no tail in terms of volume velocity seems inconsistent.
Best wishes
David
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Only when we consider the sources to be constant volume velocity, do the even-numbered dimensions (for sources) create tail-less impulse responses ... that's my story and i'm sticking to it
Happy to be proven wrong, of course !!
Doesn't this contradict your own excellent reasons to chose constant volume acceleration?
Like you, I am interested in Earl's take on this.
Best wishes
David
No contradiction at all!
One can choose to analyze sources with constant acceleration, velocity or displacement. No selection among these is "wrong" ... i just think that one choice is more directly connected to the way we typically operate transducers. Doesn't make the other ways "wrong".
All i've done, is develop some math to prove, or disprove, Earl's contention that sources in even dimensions have no impulse response tail. And i've found that this IS true, if sources are analyzed with constant volume velocity ... which is, i think, the preferred choice for physicists
That's all
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But if we do this for the point source then isn't the response the "derivative" of an impulse rather than an impulse?
(I realize that to "differentiate" a Dirac Delta strictly we need hyperfunction theory, or maybe distributions will do, so I won't even try to be formally correct.)
So it would not preserve waveshape in that domain?
And it looks that as you add dimensions you would next have to use constant volume displacement, and so on, it's not an odd/even alternation.
My concern is not whether a constant volume velocity or constant volume acceleration is the "correct" analysis, as you say, we can chose to suit our purpose.
My concern is consistency.
No fair to compare a CVA analysis of a point source with a CVV of a plane
Have you done an analysis of a point source with CVV?
That would be better than my shonky differentiation of an impulse
Best wishes
David
American spellcheck doesn't know "shonky" : untrustworthy, maybe crooked, underhand, devious.
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In electrical engineering, we differentiate Dirac impulses all the time ... the result is an impulsive "doublet"
A positive-going impulse, immediately followed a negative-going impulse, all at t=0 (most easily seen by constructing a single impulse, in the limit, as a finite pulse getting "narrower" and "taller" ... and watch the derivative as you go). Still ... no tail!
yeah i got nuthn ...
Exercises left to the reader ... please feel free to pursue, even in this threadMy concern is not whether a constant volume velocity or constant volume acceleration is the "correct" analysis, as you say, we can chose to suit our purpose.
My concern is consistency.
No fair to compare a CVA analysis of a point source with a CVV of a plane
Have you done an analysis of a point source with CVV?
That would be better than my shonky differentiation of an impulse
Best wishes
David
American spellcheck doesn't know "shonky" : untrustworthy, maybe crooked, underhand, devious.
but fair warning ... typing all those equations (with selective bolding) is no small task A point source radiates from 0 dimensions (a point has no dimension).
A line source radiates from 1 dimension (a line has only one dimension).
A planar source radiates from 2 dimensions (a plane has two dimensions).
Earl made the comment that sources which radiate from "even" dimensions have no "tail" to their impulse responses. I found that to be true (at least, for dimensions of 0 and 2) ... IF the source radiates with a constant velocity, rather than constant acceleration
Sorry guys, I feel asleep in class! Because it was 3 am!
One cannot compare Volume Velocity with Volume Acceleration.
In 1D the response to an impulse in VV, which is what I always use, contrary to this thread, is an impulse, in 2D it has a tail and in 3D it is a doublet. The doublet is easy to see since an impulse in VV will have zero static pressure change over time and an impulse has a finite static change, a doublet has zero.
Hope this clears things up.
One cannot compare Volume Velocity with Volume Acceleration.
In 1D the response to an impulse in VV, which is what I always use, contrary to this thread, is an impulse, in 2D it has a tail and in 3D it is a doublet. The doublet is easy to see since an impulse in VV will have zero static pressure change over time and an impulse has a finite static change, a doublet has zero.
Hope this clears things up.
It's a mathematical abstraction that doesn't exist in physical reality. It can be useful for making calculations used to model reality, however. But, if you are not going to do mathematical modeling of reality, its probably not going to be of any use to you. On the other hand, if you do want to learn some math including about doublets, then you should probably pick up with math wherever you left off in school and eventually you will get to using doublets.
Start with a pulse, what the physicists call a Dirac Delta.
It is instantaneous but the total amount (mathematicians call this the "integral") adds up to one unit of whatever it happens to be a pulse of, acoustic volume say.
There is a mathematical issue there, because "instantaneous" means happens in 0 time and so we end up with divide-by-zero type problems.
Physicists don't care much, and EEs even less, and in any case it is possible to do it mathematically strictly if we consider it as a limit of a series of narrower and narrower but taller and taller pulses.
Now a 'doublet' is the slope (mathematicians call this the 'derivative') of that infinitely narrow, infinitely tall pulse.
Similar problem here, to define a slope of a function that happens in zero time and is infinitely steep.
"Werewolf" says he does this all the time! and he is correct for practical purposes, it can even be done mathematically strictly as a limit of steeper and steeper slopes, one up and then one down, a 'Doublet'.
The strict version is called a distribution or a hyperfunction, which sounds much cooler, but I don't want to even try to be formally correct, as I commented earlier
Best wishes
David
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If i may, don't get discouraged.
Please remember ... if we consider idealized point sources and line sources as "models" for some (if not all) of the speakers we construct, and if we consider these sources to be operating with constant volume acceleration to most closely match transducers operating with constant voltage (above resonance) ... then the time-domain STEP responses tend to avoid the crazy "impulse" abstractions, at both the source and measuring/listening points (as explained earlier).
You'll just need to differentiate the step response, in time ... which is represented by multiplying by "w", in frequency ... to find the frequency response and time-domain impulse response.
In short, if you don't like impulses much
use their integrals instead SPL vs displacement advantage of floor to ceiling arrays
Consider a woofer with Sd= 300 sq.cm and Xmax= 12mm
and a floor to ceiling arrays of 25 3.5inch drivers with Sd= 36 sq.cm and Xmax=4mm.
Both the system has same air volume displacement capability.
Given the fact that SPL only decreases 3dB per doubling distance from a line source against 6dB for point source, an array obviously has advantages when it come to SPL at listening distance vs air displacement. This is very obvious at mid frequencies and up but never established for bass frequencies in a practical listening room situation and is difficult to test.
If the above mentioned systems generate 50Hz at same net air volume displacement and the listening position is 3m from the source, will the array have any SPL advantage?
I assume the nearfield to farfield transition point for the 10 inch woofer to be 0.5m from source ( I am just pulling this number out of nowhere) and that of the 2.3m high array to be 2.4m ( this is based on Griffin's paper on arrays ([f/1000]*[3H]^2)
so, we get an extra 1.9m nearfield transition after 0.5m from the source in case of the arrays. Hence the SPL advantage for same displacement at any listening distance greater than 2.4m from source is 10log(1.9/0.5) ~ 5.8dB.
So our array acts like a single woofer with Sd 300 sq.cm and 23.4mm Xmax???
*********
I have heard people say many times that in practical room listening situations spl decrease from woofer (point sources) per doubling listening distance is less than 6dB. So that brings them close to 3dB figure in case of an array. What they forget is that an array will get the same advantage if placed in the same room, hence SPL will decrease less than 3dB per doubling distance.
or Maybe not. Maybe an array deals differently with room gain ( not talking about vertical infinite line source extension here ). But that requires an answer or argument addressing that specifically, not something silly and simple as the above paragraph.
Hoping someone helps with this. Thanks
Consider a woofer with Sd= 300 sq.cm and Xmax= 12mm
and a floor to ceiling arrays of 25 3.5inch drivers with Sd= 36 sq.cm and Xmax=4mm.
Both the system has same air volume displacement capability.
Given the fact that SPL only decreases 3dB per doubling distance from a line source against 6dB for point source, an array obviously has advantages when it come to SPL at listening distance vs air displacement. This is very obvious at mid frequencies and up but never established for bass frequencies in a practical listening room situation and is difficult to test.
If the above mentioned systems generate 50Hz at same net air volume displacement and the listening position is 3m from the source, will the array have any SPL advantage?
I assume the nearfield to farfield transition point for the 10 inch woofer to be 0.5m from source ( I am just pulling this number out of nowhere) and that of the 2.3m high array to be 2.4m ( this is based on Griffin's paper on arrays ([f/1000]*[3H]^2)
so, we get an extra 1.9m nearfield transition after 0.5m from the source in case of the arrays. Hence the SPL advantage for same displacement at any listening distance greater than 2.4m from source is 10log(1.9/0.5) ~ 5.8dB.
So our array acts like a single woofer with Sd 300 sq.cm and 23.4mm Xmax???
*********
I have heard people say many times that in practical room listening situations spl decrease from woofer (point sources) per doubling listening distance is less than 6dB. So that brings them close to 3dB figure in case of an array. What they forget is that an array will get the same advantage if placed in the same room, hence SPL will decrease less than 3dB per doubling distance.
or Maybe not. Maybe an array deals differently with room gain ( not talking about vertical infinite line source extension here ). But that requires an answer or argument addressing that specifically, not something silly and simple as the above paragraph.
Hoping someone helps with this. Thanks
They won't act the same. Bring in the reflections to your considerations for both the array and the single woofer, as we are in a room with walls, floors and a ceiling.
Are we talking about a single woofer vs an array of smaller woofers, both EQ-ed to a certain curve? Without EQ the smaller woofers would drop off below their Fs, which probably is higher up in frequency than the Fs of the bigger woofer.
Much of the advantage I see in that woofer array will be in the way it spreads out the reflections(*), compared to the single big woofer. With a reflective floor and ceiling it will even 'act' as a longer array, moving the nearfield/farfield transition further back. But without EQ it probably won't be able to compete with a big woofer made for this purpose.
Each small woofer makes up a small part of the total SPL output and this may give a distortion advantage. Together they can take quite some juice.
(*) = Each woofer in the array will reflect of near boundaries as judged from the listening position. The next woofer in line will have slightly different reflections, averaging out the reflections of the first woofer and so on. Parallel planes will have those reflections closer together though. So placement does matter. The array will average out the negative effect of the floor (and ceiling) reflections. (the usual floor dip we often see in measurements at listening positions)
Are we talking about a single woofer vs an array of smaller woofers, both EQ-ed to a certain curve? Without EQ the smaller woofers would drop off below their Fs, which probably is higher up in frequency than the Fs of the bigger woofer.
Much of the advantage I see in that woofer array will be in the way it spreads out the reflections(*), compared to the single big woofer. With a reflective floor and ceiling it will even 'act' as a longer array, moving the nearfield/farfield transition further back. But without EQ it probably won't be able to compete with a big woofer made for this purpose.
Each small woofer makes up a small part of the total SPL output and this may give a distortion advantage. Together they can take quite some juice.
(*) = Each woofer in the array will reflect of near boundaries as judged from the listening position. The next woofer in line will have slightly different reflections, averaging out the reflections of the first woofer and so on. Parallel planes will have those reflections closer together though. So placement does matter. The array will average out the negative effect of the floor (and ceiling) reflections. (the usual floor dip we often see in measurements at listening positions)
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The question is about SPL at a particular listening distance for same air volume displacement by the drivers at a particular low frequency ( say 50Hz for example).
This does not concern with anything electrical ie EQ, efficiency or sensitivity of either system. You have to get the same physical air displacement at source ( Sd x excursion) by whatever electrical means necessary and compare SPL. This is not a question on distortion and bass quality either.
EQ and curve management are requirements for practical usability of an array and that is a different subject altogether.
Thank you for sharing your views.
Your thread is a big hit and shares a ton of knowledge.
But it has to my surprise, never been established/ claimed or clearly suggested even once that arrays like yours has advantage in low frequency SPL (like 50Hz) for same volume displacement.
Maybe because it is hard to test and/or we have to deal with farfield nearfield transition at low bass frequencies which might be confusing.
I think this is a good place to ask a question like this.
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