Earl, please send me a PM. I would imagine that your in-box is much, much more crowded than mine ...
Regarding the spaced point sources, this is a trivial problem if the sources are true point sources, but not so easy if they have directivity. If they are true point sources then the final directivity will be the product of the sources and a spaced line of sources of distance x.
This will then be Cos( k * x * sin(theta_z)), i.e. there are lobes in the vertical response that depend on the distance x, so this array and the infinite array do not act the same at all.
The one thing that is important to consider is that ONLY the infinite line array has the -3 dB/dd characteristic. Any finite array has a far field falloff at -6 dB/dd. In the near field a finite cylindrical array falls at 3 dB but then changes to -6 dB depending on the frequency. Hence the 3 dB fall is so idealized that it is unlikely to ever occur in any real situation.
In my book I show a detailed discussion of the finite cylinder in section 3.6 (this book is free at my website, so there is no excuse for not reading it if this stuff interests you.) The near field is looked at in some detail in section 3.7 with plots of this very complex situation. Basically the near field is so highly variable for the finite source that one would not want to be in the near field. And the far field behaves very much like a spherical source (all finite sources appear as spherical sources at a sufficient distance.)
This will then be Cos( k * x * sin(theta_z)), i.e. there are lobes in the vertical response that depend on the distance x, so this array and the infinite array do not act the same at all.
The one thing that is important to consider is that ONLY the infinite line array has the -3 dB/dd characteristic. Any finite array has a far field falloff at -6 dB/dd. In the near field a finite cylindrical array falls at 3 dB but then changes to -6 dB depending on the frequency. Hence the 3 dB fall is so idealized that it is unlikely to ever occur in any real situation.
In my book I show a detailed discussion of the finite cylinder in section 3.6 (this book is free at my website, so there is no excuse for not reading it if this stuff interests you.) The near field is looked at in some detail in section 3.7 with plots of this very complex situation. Basically the near field is so highly variable for the finite source that one would not want to be in the near field. And the far field behaves very much like a spherical source (all finite sources appear as spherical sources at a sufficient distance.)
YES, the impulse- and step-responses haves tails ... as we developed and explained quite thoroughly in this thread
High-pass filters at 20Hz have impulse tails, low-pass filters at 20kHz have impulse tails, all-pass filters at 1kHz have impulse tails ... and yet, our hearing is remarkably insensitive to all of these. Why? Because we are much more sensitive to magnitude response, than (absolute) phase response (phase tends to matter, a lot, when two signals combine ... at crossover, for example, or when listening to more than one channel ... but that's because the relative phase of two signals in summation will directly impact the resultant magnitude).
The impulse tail of the Infinite Line Source directly corresponds to the "-3dB per octave" frequency response. Once the magnitude response of the line source is equalized to something close to "flat" (over ~20kHz bandwidth), i wouldn't worry too much about the impulse- or step- response anymore ...
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I agree that the impulse having a tail is not an issue. What I think that our ears are sensitive to is a non-uniform/asymptotic falling of the impulse, i.e.a fall in level and then a rise and fall again. That kind of thing appears to be an issue as the energy transfer of a speaker should be monotonically falling, otherwise its multi-path like.
makes sense to me!
Just for clarification ... it would be incorrect to say that our ears are completely deaf to absolute phase, of course. I can construct a very high-order all-pass filter, with perfectly flat magnitude response, but with enormous phase & group delay aberrations at, say 1kHz, and surely at some point it would become audible (delaying the treble a few seconds past the midrange, for example!). But, our sensitivity to magnitude response is just so much greater ...
The key point is that the time-domain impulse response reflects BOTH the magnitude AND phase response. BOTH of these have to be VERY well-behaved, for the impulse response to "look pretty". But, as far as our ears are concerned, it's MUCH more important to get the magnitude part "right".
Just for clarification ... it would be incorrect to say that our ears are completely deaf to absolute phase, of course. I can construct a very high-order all-pass filter, with perfectly flat magnitude response, but with enormous phase & group delay aberrations at, say 1kHz, and surely at some point it would become audible (delaying the treble a few seconds past the midrange, for example!). But, our sensitivity to magnitude response is just so much greater ...
The key point is that the time-domain impulse response reflects BOTH the magnitude AND phase response. BOTH of these have to be VERY well-behaved, for the impulse response to "look pretty". But, as far as our ears are concerned, it's MUCH more important to get the magnitude part "right".
First I want to thank @werewolf for this great and entertaining lesson. Not the first lesson from you I have followed closely, there are many more to thank you for.
One of the things I particularly like about (floor to ceiling) arrays is how they can perform inside a (normal living) room. I have looked at many options before I finally decided to go with arrays.
I had some restrictions when I opted to build myself a new pair of speakers, forced upon me by my other half. Lucky for me the line arrays were acceptable to her. No big horns were in my future, she made that very clear to me.
The arrays have a small footprint which made them acceptable, despite their height and visual presence. That, coupled with the fact that they were meant to sit quite close to the wall. As another one of her strict demands was not to have speakers out in the room away from the walls.
Let me explain that room behaviour I mentioned. As I have arrays consisting of a row of (relatively) small sources, most of the stuff usually found inside a real living room had way less of a detrimental effect on the sound I experienced in the room than I was used to from a pair of multi-ways.
Each driver to ear has a slightly different reflection point compared to the next. It will seldom match the next driver in the line exactly. This is the reason why it 'seems' the line is a lot longer than it really is (the floor to ceiling array acts more like an infinite array over a large part of it's output due to those floor end ceiling reflections).
But it also means that smaller objects don't even show up in measurements. They get averaged out by the drivers that don't share that particular reflection surface.
It acts much like the array of microphones in that way.
The biggest things that do show up at the listening spot are parallel planes to the arrays. 2 sidewalls and the wall behind the listener in my case (to be exact).
It took me 3 (2 of them rather high at side walls) damping panels to tame those early reflections. My objective was to absorb reflections in the first ~20 ms as much as I could (or was allowed to by my other half).
The absorption part was rather easy though. I studied a lot of indoor measurements to determine the frequency area's at which I had excessive energy. Comparing gated results to the ungated sweeps and looking at measurements taken along the listening area of interest. Those results determined the thickness of the absorption panels I planned on using.
In real life, even the parallel walls each cause slightly different reflections, with each driver contributing only a small part of that reflection. In the case of a (non directive) point source, you'd have a bigger or stronger reflection to battle. And the floor and ceiling. It goes without saying my girl didn't want me to make our living room look like a studio, so no floor and ceiling gimmicks allowed to favour sound reproduction. Only panels that I could hide or disguise were acceptable. The 2 big ones are behind curtains (that were there already, phew) and one panel is disguised as a poster.
The FIR filtering I use is based on an ever changing frequency dependent window that keeps in mind the arrival times at all frequencies. So I gate more cycles at, say 15 kHz than I have to use at mid frequencies. The goal was to get away with the shortest varying frequency dependent window I could use. Always checking the final processing at other spots beside the sweet spot. The average of multiple spots, taken along the couch is a spitting image of the sweet spot. Coincidence? No, the reflective energy would normally be responsible for abrupt changes. That part is absorbed.
If I move up or down, left or right the sound does not change. As mentioned earlier, when standing up, it takes the imaging queues up with you. But it does not sound any different, tonal balance remains the same.
In a graph, taken from both line arrays playing at once this shows the result at the listening spot:
In a graph like this we can see the reflections, but most of them are spread out and attenuated.
The IR (after FIR correction) of both left and right channels in one graph:
The shape of the IR is largely determined by my preference for a room curve which is slightly tilted from low to high frequency.
By the way, I do realise I do get combing at higher frequencies with my full range based array, I have never denied its existence. You can't really fix that with DSP except maybe at one small spot. That's no good at all, at least I would not recommend that.
Most people won't realize they will get combing at the ear too, from our array (2 in total) of ears. That will happen much lower in frequency and is covered in another thread in more detail.
@werewolf, wouldn't that ('flaw' of Stereo) make a great subject for analysis one day?
If we move further from the array the comb pattern shifts up in frequency. Ra7 has some interesting raw graphs in his line array thread. These measurements at varying distances, using the same level for each distance can also be used to check if we really do get the 3 dB fall off for each doubling of distance out in our rooms.
There's much more info in my thread as well. Even some reviews (links in first post) from a few members of this forum that dropped by for a listen. (can we even call it 'dropping by' if they came from the US and Denmark all the way to the Netherlands?) Their reviews will probably underline the wide sweet spot werewolf mentioned.
I shared as much as I could in my journey, most of the time showing actual measurements at the listening spot. I would love to see more people do that, it would help to get people acknowledge room influence and it's role in perception.
I use my array full range, covering the entire spectrum. The only reason to get away with it is because most music does not hit that hard below 40 Hz. So the EQ is set to flat to ~30 Hz from which it gradually rolls off to 20 Hz and a bit steeper below 20 Hz. (sometimes there is a surprisingly amount of energy there, below 20 Hz in songs)
Frequency dependent window of 6 cycles applied here
In due time I will see if adding some subs can gain me some headroom. No hurry yet, I'm not missing anything.
Disclaimer: By no means will I call this system superior to any other solutions. It has been a marvellous ride for me to get where I am. It was the best compromise I could think of, for me. Focussing on making the room and speaker work together to try and get the best result at the listening area, not one single sweet spot.
One of the things I particularly like about (floor to ceiling) arrays is how they can perform inside a (normal living) room. I have looked at many options before I finally decided to go with arrays.
I had some restrictions when I opted to build myself a new pair of speakers, forced upon me by my other half. Lucky for me the line arrays were acceptable to her. No big horns were in my future, she made that very clear to me.
The arrays have a small footprint which made them acceptable, despite their height and visual presence. That, coupled with the fact that they were meant to sit quite close to the wall. As another one of her strict demands was not to have speakers out in the room away from the walls.
Let me explain that room behaviour I mentioned. As I have arrays consisting of a row of (relatively) small sources, most of the stuff usually found inside a real living room had way less of a detrimental effect on the sound I experienced in the room than I was used to from a pair of multi-ways.
Each driver to ear has a slightly different reflection point compared to the next. It will seldom match the next driver in the line exactly. This is the reason why it 'seems' the line is a lot longer than it really is (the floor to ceiling array acts more like an infinite array over a large part of it's output due to those floor end ceiling reflections).
But it also means that smaller objects don't even show up in measurements. They get averaged out by the drivers that don't share that particular reflection surface.
It acts much like the array of microphones in that way.
The biggest things that do show up at the listening spot are parallel planes to the arrays. 2 sidewalls and the wall behind the listener in my case (to be exact).
It took me 3 (2 of them rather high at side walls) damping panels to tame those early reflections. My objective was to absorb reflections in the first ~20 ms as much as I could (or was allowed to by my other half
).The absorption part was rather easy though. I studied a lot of indoor measurements to determine the frequency area's at which I had excessive energy. Comparing gated results to the ungated sweeps and looking at measurements taken along the listening area of interest. Those results determined the thickness of the absorption panels I planned on using.
In real life, even the parallel walls each cause slightly different reflections, with each driver contributing only a small part of that reflection. In the case of a (non directive) point source, you'd have a bigger or stronger reflection to battle. And the floor and ceiling. It goes without saying my girl didn't want me to make our living room look like a studio, so no floor and ceiling gimmicks allowed to favour sound reproduction. Only panels that I could hide or disguise were acceptable. The 2 big ones are behind curtains (that were there already, phew
) and one panel is disguised as a poster.The FIR filtering I use is based on an ever changing frequency dependent window that keeps in mind the arrival times at all frequencies. So I gate more cycles at, say 15 kHz than I have to use at mid frequencies. The goal was to get away with the shortest varying frequency dependent window I could use. Always checking the final processing at other spots beside the sweet spot. The average of multiple spots, taken along the couch is a spitting image of the sweet spot. Coincidence? No, the reflective energy would normally be responsible for abrupt changes. That part is absorbed.
If I move up or down, left or right the sound does not change. As mentioned earlier, when standing up, it takes the imaging queues up with you. But it does not sound any different, tonal balance remains the same.
In a graph, taken from both line arrays playing at once this shows the result at the listening spot:
In a graph like this we can see the reflections, but most of them are spread out and attenuated.
The IR (after FIR correction) of both left and right channels in one graph:
The shape of the IR is largely determined by my preference for a room curve which is slightly tilted from low to high frequency.
By the way, I do realise I do get combing at higher frequencies with my full range based array, I have never denied its existence. You can't really fix that with DSP except maybe at one small spot. That's no good at all, at least I would not recommend that.
Most people won't realize they will get combing at the ear too, from our array (2 in total) of ears. That will happen much lower in frequency and is covered in another thread in more detail.
@werewolf, wouldn't that ('flaw' of Stereo) make a great subject for analysis one day?
If we move further from the array the comb pattern shifts up in frequency. Ra7 has some interesting raw graphs in his line array thread. These measurements at varying distances, using the same level for each distance can also be used to check if we really do get the 3 dB fall off for each doubling of distance out in our rooms.
There's much more info in my thread as well. Even some reviews (links in first post) from a few members of this forum that dropped by for a listen. (can we even call it 'dropping by' if they came from the US and Denmark all the way to the Netherlands?
) Their reviews will probably underline the wide sweet spot werewolf mentioned.I shared as much as I could in my journey, most of the time showing actual measurements at the listening spot. I would love to see more people do that, it would help to get people acknowledge room influence and it's role in perception.
I use my array full range, covering the entire spectrum. The only reason to get away with it is because most music does not hit that hard below 40 Hz. So the EQ is set to flat to ~30 Hz from which it gradually rolls off to 20 Hz and a bit steeper below 20 Hz. (sometimes there is a surprisingly amount of energy there, below 20 Hz in songs)
Frequency dependent window of 6 cycles applied here
In due time I will see if adding some subs can gain me some headroom. No hurry yet, I'm not missing anything.
Disclaimer: By no means will I call this system superior to any other solutions. It has been a marvellous ride for me to get where I am. It was the best compromise I could think of, for me. Focussing on making the room and speaker work together to try and get the best result at the listening area, not one single sweet spot.
Alright Earl ... i read what you wrote about the impulse response from even versus odd dimensions, and of course i didn't believe it to be true, in general
not that i doubted you but i just needed to understand it with my "engineer" brain.So i quickly calculated the impulse response from an infinite plane, using the approach and math in this thread
Sure enough ... the impulse response from an Infinite Plane Source is indeed impulsive, with no "tail".Yes, there's another wolfie 'sidebar' coming ...
Yes, I did Physics/Maths at uni and I understand the integration.
The problem was that the line source seemed anomalous when a point source and a plane source are both non-dispersive.
However, Earl's reply provides an explanation and stirs up a dim memory.
At the time the theory of waves in an arbitrary number of dimensions seemed a bit academic and I didn't really take it in.
Mea culpa to my uni teachers, I really should have paid more attention.
Thanks to you both.
Best wishes
David
Cross posted with "Werewolf", relieved to learn it wasn't just me.
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Are gravitational waves in fact dispersive?
My not-well-informed belief was they are not, based on the "chirp" that was played in some of the publicity around the first detection, and the idea that they travel at velocity = c.
Best wishes
David
Water gravity waves, ie ordinary ocean waves, are dispersive.
I never considered if there was a deeper reason than just the detail of how the forces worked out. Hmm.
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A good question. I am not exactly sure as I only remember this being discussed "in general" not specifically to gravitational waves. It could be that since gravitational waves travel at speed c that they are not traveling in time (like light waves, although light waves do not perturb space-time like gravity waves,) and as such they are 3D waves and are not dispersive. This is not my expertise, although I have studied General Relativity. But gravity waves are fairly new by comparison. Even Einstein had his doubts about them and they weren't completely resolved in his lifetime.
I believe Einstein's doubts were mainly that they would ever be detectable by achievable instruments.
To do so was an extraordinary feat, chapeau to forum member Scott Wurcer who was very chuffed that his brainchild, the AD797, was used in the low noise instrumentation.
Some people had doubts about whether the waves were detectable even in principle.
Despite the fact I studied the subject too, it's not my area of expertise either.
But I still retain an interest, hence my question.
Best wishes
David
Guess wesayso's build are considered a tall array and for me it was impossible to point any subjective fingers at treble transients, think had close to total of 10 hours listening and guess thought to myself before the visit that combing at higher frequencies would show its existence someway, maybe other ears a oscilloscope or owner himself can point it out but i couldn't at the time.
Is subjective follower alright : ) those fabulous acoustic graphs taken at listening position play over average fantastic sound and for ears think problematic bouncing area say 400Hz and down is to die for and could be because of the the averaged out thing using high numbers of 50 drivers and each their stroke is so less that its not to notice with eye. No matter it can look ugly for other system setups agree wesayso it would be nice see more acoustic measurement actual taken at listening spot.
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Closest thing to tall array I've heard is Acoustat panels. Nothing to complain about the sound of treble there. Actually I am not bothered by ragged treble response of coaxials and horns either, but there ripples are at over 10kHz. Tall lines start showing mutliple interferences soon over 1kHz.
The major difference is that these interferences are born at the source, are "heard" and measured during the first few milliseconds, instead of reclections in the room appearing after several (tens) of milliseconds. Psycho-acoustics...
The major difference is that these interferences are born at the source, are "heard" and measured during the first few milliseconds, instead of reclections in the room appearing after several (tens) of milliseconds. Psycho-acoustics...
I don't see any signs of combing until I'm past 7 kHz. This is with FIR correction applied and at the sweet spot.
With an approximate IIR correction things are looking a little different. Even then it's hard to tell by listening, FIR correction sounds smoother though, to my ears.
In comparison the combing due to having two ears spaced apart starts at about ~1800 Hz in a common Stereo triangle. (created by an average 0.270 ms delay) This won't depend on the speaker system used but will be softened at higher frequencies by head shading.
For another view at each driver in my particular array, here's a table of distances of each driver to the ear (measurements in mm):
Obviously, this does not include the floor and ceiling mirror images
With an approximate IIR correction things are looking a little different. Even then it's hard to tell by listening, FIR correction sounds smoother though, to my ears.
In comparison the combing due to having two ears spaced apart starts at about ~1800 Hz in a common Stereo triangle. (created by an average 0.270 ms delay) This won't depend on the speaker system used but will be softened at higher frequencies by head shading.
For another view at each driver in my particular array, here's a table of distances of each driver to the ear (measurements in mm):
Obviously, this does not include the floor and ceiling mirror images
Appendix (aka Earl's fault )
Let's develop (or at least outline) the time-domain impulse response from an Infinite Plane Source
Consider an infinite plane in the y-z axes. We'll develop the impulse response along the x-axis, at a distance "r" from the plane. Since the plane is infinite in extent, the response we develop will apply to any point at a distance "r" from the plane.
Consider that our planar source is excited by a signal of uniform volume acceleration (as we did with our point source and line source) per unit area, call it "Aa". More on this later ...
We'll proceed by following from POSTS #17 & #18, where we formulated the impulse response to be a continuous summation (integral) of the impulse response from all of the infinitesimal "elements" that make-up the source. In the case of a planar source, the easiest "elements" to pick are elemental circular "rings", because symmetry dictates that the response from any point on a ring will be the same at our measurement point (centered above the ring at some distance "r") ... and this observation allows us to do the integral in one variable only (told ya i was lazy).
The area of a "ring" is simply the circumference of the ring, multiplied by the infinitesimal "width". We'll be integrating along the z-axis (although we could pick the y-axis as well), so the infinitesimal width of our ring is simply "dz", and the circumference is simply "2pi*z". The total "source strength" of our elemental ring is then given by : Aa*2pi*z*dz.
We'll ignore the "Aa" term ... for now ... because i've been assigning that to the "source", rather than to the "transfer function".
Now we're ready to write the impulse response :
h(t) = (rho/2pi)*INT[ (1/R)*delta[t - R/c]*2pi*z ]dz, from z= 0 to +infinity
where:
R = sqrt[r^2 + z^2] = distance from elemental "ring" to measurement point
And we note that the integral is simply the summation (integral) of the weighted (by 1/R) and delayed (by R/c) impulse responses from all of our ring-shaped "elements"
To solve the integral, we'll perform a familiar substitution of variables to exploit the "sampling property" of the delta function:
v = R/c = {sqrt[r^2 + z^2]}/c
from which it immediately follows that :
z = sqrt[(cv)^2 - r^2]
dv/dz = z/{c*sqrt[r^2 + z^2]}
dz/R = (c/z)*dv
and the integration limits in the new variable "v" are :
z = 0 ======> v = r/c
z = infinite ===> v = infinite
Our time-domain impulse response for the Infinite Plane Source is now given by this new integral in "v" :
h(t) = (rho)*INT[ c*delta[t - v] ]dv, from v=r/c to v=+infinity
which reduces to :
h(t) = (rho*c)*u[t - Td]
where:
u[t] = unit-step function
Td = r/c = time delay from infinite plane to measuring point
Yes, the impusle response from an Infinite Planar Source is ... a step function! Why? Because, as we add-up all the contributions from each of the "element rings" that form our planar source, each elemental ring gets farther way from the measuring point ... but they also get stronger due to increased circular area. So their 'contributions' to the impulse response remain the same, as time goes on
Wait one second! Weren't we led to believe that the impulse response from a planar source is impulsive, because the source is an even dimension (in this case, 2D)? Yes we were ... and we weren't misled
See, i've been assigning the source a constant volume acceleration, whereas in acoustics it's common to assign the source a constant volume velocity. Since acceleration is the first derivative (wrt time) of velocity, a source with constant volume acceleration (wrt frequency) must display a volume velocity that falls, in frequency, as ~(1/w). Therefore, in order to determine the time-domain response for a source with constant volume velocity from a source with constant volume acceleration ... we must multiply (in frequency) by ~w. In other words, we must differentiate (in time) the response we just developed, in order to determine the impulse response from a source with constant volume velocity.
What do you get when you differentiate a step function? You get an "impulse", of course
that's all i got
)Let's develop (or at least outline) the time-domain impulse response from an Infinite Plane Source
Consider an infinite plane in the y-z axes. We'll develop the impulse response along the x-axis, at a distance "r" from the plane. Since the plane is infinite in extent, the response we develop will apply to any point at a distance "r" from the plane.
Consider that our planar source is excited by a signal of uniform volume acceleration (as we did with our point source and line source) per unit area, call it "Aa". More on this later ...
We'll proceed by following from POSTS #17 & #18, where we formulated the impulse response to be a continuous summation (integral) of the impulse response from all of the infinitesimal "elements" that make-up the source. In the case of a planar source, the easiest "elements" to pick are elemental circular "rings", because symmetry dictates that the response from any point on a ring will be the same at our measurement point (centered above the ring at some distance "r") ... and this observation allows us to do the integral in one variable only (told ya i was lazy).
The area of a "ring" is simply the circumference of the ring, multiplied by the infinitesimal "width". We'll be integrating along the z-axis (although we could pick the y-axis as well), so the infinitesimal width of our ring is simply "dz", and the circumference is simply "2pi*z". The total "source strength" of our elemental ring is then given by : Aa*2pi*z*dz.
We'll ignore the "Aa" term ... for now ... because i've been assigning that to the "source", rather than to the "transfer function".
Now we're ready to write the impulse response :
h(t) = (rho/2pi)*INT[ (1/R)*delta[t - R/c]*2pi*z ]dz, from z= 0 to +infinity
where:
R = sqrt[r^2 + z^2] = distance from elemental "ring" to measurement point
And we note that the integral is simply the summation (integral) of the weighted (by 1/R) and delayed (by R/c) impulse responses from all of our ring-shaped "elements"
To solve the integral, we'll perform a familiar substitution of variables to exploit the "sampling property" of the delta function:
v = R/c = {sqrt[r^2 + z^2]}/c
from which it immediately follows that :
z = sqrt[(cv)^2 - r^2]
dv/dz = z/{c*sqrt[r^2 + z^2]}
dz/R = (c/z)*dv
and the integration limits in the new variable "v" are :
z = 0 ======> v = r/c
z = infinite ===> v = infinite
Our time-domain impulse response for the Infinite Plane Source is now given by this new integral in "v" :
h(t) = (rho)*INT[ c*delta[t - v] ]dv, from v=r/c to v=+infinity
which reduces to :
h(t) = (rho*c)*u[t - Td]
where:
u[t] = unit-step function
Td = r/c = time delay from infinite plane to measuring point
Yes, the impusle response from an Infinite Planar Source is ... a step function! Why? Because, as we add-up all the contributions from each of the "element rings" that form our planar source, each elemental ring gets farther way from the measuring point ... but they also get stronger due to increased circular area. So their 'contributions' to the impulse response remain the same, as time goes on
Wait one second! Weren't we led to believe that the impulse response from a planar source is impulsive, because the source is an even dimension (in this case, 2D)? Yes we were ... and we weren't misled
See, i've been assigning the source a constant volume acceleration, whereas in acoustics it's common to assign the source a constant volume velocity. Since acceleration is the first derivative (wrt time) of velocity, a source with constant volume acceleration (wrt frequency) must display a volume velocity that falls, in frequency, as ~(1/w). Therefore, in order to determine the time-domain response for a source with constant volume velocity from a source with constant volume acceleration ... we must multiply (in frequency) by ~w. In other words, we must differentiate (in time) the response we just developed, in order to determine the impulse response from a source with constant volume velocity.
What do you get when you differentiate a step function? You get an "impulse", of course
that's all i got
That was very educational, thank you, but it raises new questions.
For an impulse input of volume acceleration, the point source has a (delayed) impulse response, the line source has a "step" that decays and the plane source a step that doesn't.
That is a consistent trend, not some kind of oscillation like odd and even numbers.
In fact it looks to be an example of maths that I don't see often, "fractional integration".
So why do we discuss the response of the point source in terms of volume acceleration but the plane source in volume velocity?
Best wishes
David
As i see it ... we can analyze any source with constant volume velocity OR constant volume acceleration. I chose constant volume acceleration for the Point Source to begin with, because : first, it represents the behavior of most transducers (speakers) with which we're familiar (above resonance) when driven by a voltage source with constant voltage (wrt frequency); and second, it achieves the "flat" frequency response we've all come to associate with an ideal point source.
I also analyzed the Line Source and Plane Source with constant volume acceleration, for consistency in the thread
But, if i'm correct, it's true to conclude that the impulse response of a source with even dimensions (0 for the point, and 2 for the plane) will be "tail-less", when we consider the source to be of constant volume velocity. Finally, one can find the impulse responses for sources with constant volume velocity by simply differentiating (wrt) the impulse responses we've already developed in this thread
That's all assuming i'm right, of course
we need Earl to verify cuz i can be mad as a hatter sometimes - Status
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