John
I don't think that this is true, but I don't have the time or interest to delve into it any further. I will say this, the modes in an OS waveguide are dispersive in the near field and they can be obtained as a finite series of the Spherical wave Functions (the Spherical Hankel Functions or Bessel Functions of the 2nd kind). Clearly in the cylindrical case shown, the lowest mode was not dispersive but the higher ones were. I believe that the same is true in the spherical case as well as the planar case.
According the Dash and Fricke paper for a spherical wave, "For the n = 0 spherical wave, cp is constant. For n = 1, 2 etc, cp increases below kr = 1."
For the cylindrical wave all mode are dispersive unless r > wave length/2Pi (kr>1).
Are you sure spherical waves are dispersive? The AES article I linked describes the zero-order spherical wave as possessing a constant phase and group velocity. Of course, N>1 order spherical wave functions will be dispersive.
Perhaps your observation (nearfield dispersion) can be explained due to the fact that we encounter sources of finite dimensions, not theoretically perfect (infinitely small) point-sources. I'm not sure how the concept of the nearfield applies to the point source. The perfect theoretical point source shouldn't have a transition from nearfield to farfield, as it is simply the expansion of a sphere.
How closely did you look at the AES article I cited? If you had, you will see that the zero-order cylindrical wave function is indeed dispersive. Phase and group velocity are non-constant for all orders of cylindrical waves.
By the way Earl, here is a look at my latest OB system. Can be driven with a single amp. Has and F3 of between 25 and 30 Hz (depending on how the bass eq is set). Nominal sensitivity 90dB and max SPL for single speaker is better than 100 dB at 30 Hz.
Hey John, and sorry to jump in here guys, but I was wondering if you've got some of the details of this system somewhere - looks interesting.
Okay, you can continue now. Looking forward to the results of your thermal simulation...
John
For the spherical case, that's what I said and you said I was wrong. In the cylindrical case I misread the diagram. To be non-dispersive the cp must be flat. I noted that it went down while all the higher modes went up but thats still dispersive - that was MY mistake.
What does the order, n, of the wave refer to? Does n>0 indicate nodes in the radiation pattern? I've done a bit of searching for an answer to this so I wouldn't have to bother others, but I haven't come up with it yet. (No doubt someone will demonstrate that it's one click from where I was looking...)
If waves are not dispersive when kr>1, does that imply that for typical listening distances, and for frequencies in the midbass region and above, dispersion isn't an issue?
Few
n> zero does not imply radiation patten. It is simple that there are different order solutions which mathematically satify the wave equation. However, if you read through the paper and hten the conclusion you will read the statement,"The anomalous dispersion at small kr values causes propagation velocities to rapidly become thermodynamically unsupportable." The anomalous dispersion only occurs for higher order modes. Translated this loosely means higher order modes, which mathematically possible, are physically unsustainable. Or, only the n= 0 modes are of interest with regard to acoustic waves in air. Perhaps Earl has a different take on this?
Regarding your second question, all this means is that once the wave propogates past a certian distance formthe source (depending on frequency) the dispersive behavior stops. For a given frequency there is decreasing dispersion with distance from the source, but the integrated effects of dispersion remain. Think of it like you runners running together for 200 yards. One runner in on a paved surface for the full 200 yards. The other has to run the first 100 yards on loose sand, and the last 100 on pavement. For the first 100 yeards the runner on sand will fall behind further and further. Once he gets to the pavment he will be able to run at the same speed as the second runner, but he will remain behind the other runner and never catch up. In sand the runner has a different "phase velocity".
What does the phase velocity look like for different loudspeakers with gradually changing dispersion? I was sort of thinking about this with just musical instruments the other day. Don't musical instruments produce some weird omega vs K curves?
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Thanks for digging into this Few, good questions.
So in theory for a line source we get a cylindrical wavefront. What conditions are required for a transducer to act as a line source and make this happen? Is it possible to derive some rudimentary expression for the dimensions to predict this?
I've written a simple simulator for electrostatic panels that calculates the frequency dependent pressure at a given location. It calculates the surface integral shown below. This expression was derived by applying the superposition principle, which says we can get the total pressure by summing the (complex) output of infinitesimal area's of the diaphragm. If dispersion takes place, does the superposition principle still hold? (I assume the calculated result will be wrong since we don't account for dispersion, but does the principle still hold?)
Finally, if we assume a planar speaker of say 6' produces cylindrical waves, what would be a good way to verify this? Is it as simple as measuring group delay?
So in theory for a line source we get a cylindrical wavefront. What conditions are required for a transducer to act as a line source and make this happen? Is it possible to derive some rudimentary expression for the dimensions to predict this?
I've written a simple simulator for electrostatic panels that calculates the frequency dependent pressure at a given location. It calculates the surface integral shown below. This expression was derived by applying the superposition principle, which says we can get the total pressure by summing the (complex) output of infinitesimal area's of the diaphragm. If dispersion takes place, does the superposition principle still hold? (I assume the calculated result will be wrong since we don't account for dispersion, but does the principle still hold?)
An externally hosted image should be here but it was not working when we last tested it.
Finally, if we assume a planar speaker of say 6' produces cylindrical waves, what would be a good way to verify this? Is it as simple as measuring group delay?
Actually since n represents azimuthal modes I guess it does say something about radiation patten. But again, n>0 isn't of interest (or shouldn't be) for an acoustic radiator. Or may I should say that n>0 could be though of a breakup of a cylindrical radiator.
John K: Thanks for the very helpful response.
Few
You're welcome, arend-jan. Always glad to put my ignorance to good use.
Few
Good thing you corrected the first part. Want to rethink your answer on the second part because its wrong as well.
I came across a paper called 'Dynamics Distortion: Loudspeaker Sensitivity
Modulation Generated by Audio Signals' by Carlo Zuccatti and Marco Bandiera. Perhaps of interest are the thermal parameters mentioned for three different sized drivers.
Modulation Generated by Audio Signals' by Carlo Zuccatti and Marco Bandiera. Perhaps of interest are the thermal parameters mentioned for three different sized drivers.
Well worth reading, but a little disappointing in the end. I remember glancing at the article this summer and thinking "nothing new there" and then forgot about it. Thanks for pointing it out.
Their consclusions are exactly what I have been saying "this could be a factor" but they say it with even less data than I already have. The paper is far from conclusive and doesn't even deal with the issue of tweeters versus woofers at all - the bigger issue in my mind. That the HFs compress far more than the LFs do for, all the reasons that John's analysis showed, is likely to be a very inportant aspect of the problem.
Its Interesting that they didn't write of the effect as "unimportant", yet others who haven't studied it nearly as in-depth have.
They point out the obvious solutions and then also point out how impracticle those are, but then they failed to mention that the problem is exacerbated by low efficiency designs, which makes a very logical solution quite evident.
I have no problem being corrected is I post an incorrect comment. If C/Co is less than 1 (n = 0) for kr<1 then for two waves of different frequency the lower frequency wave travels further with C < Co, and C for the lower frequency wave will always be less that that of the higher frequency wave
An externally hosted image should be here but it was not working when we last tested it.
so the lower frequency wave will fall behind the higher frequency wave, further and further, until both waves have propagated past the point where C = Co. If that interpretation is incorrect, please enlighten me.
By the way, with regard to thermal compression, realize I am talking about the direct effect of compression on amplitude. I am not implying that such a change in amplitude does not introduce other distortions, which may or may not be audible, just that the change in amplitude is not likely to be audible.
Hi John
Its this quote that is incorrect
The higher terms account for all directivity, and are most probably more interesting than the n=0 term (at least to me). The n= 0 term (lets talk the spherical case for now) is an omnidirectional term. It will alsways dominate at LF since it is the only term with a significant LF level. But as the frequency goes up, the other terms begin to become important each in succession, each peaking at a different value of kr, but all leveling off at 1.0.
For example, a point source on a sphere, like an impulse in time, will contain all terms in n. As the frequency goes up the terms come into play successively creating the directivity pattern. Like all "impulse" responses, or Green's functions in the spatial case, any arbitrary source can be described as a sum (or integral) over the Greens function for each point in the source. Hence ALL sources contain ALL orders of radiation unless by some chance some of them cancel.
Another way of saying the above is that the Green's function can always be defined as an infinite sum over any complete orthonormal set of functions. This result is key to all theoretical physics from sound radiation to Quantum mechanics (which are similar to identical calculations in many cases)
The n=0 case is a monopole, with equal radiation in all directions. The n=1 case is a dipole with a single nodal surface through the origin. The n=1 case is actually degenerate and there are three solutions in the three orthogonal directions, but this can always be reduced to a single mode in a single direction defines by the two Euler angles. A sum of a monopole and a dipole creates a cardiod. A piston in a sphere contains all orders and it is the presence of the higher orders that creates the directivity pattern.
The general case for a sphere, being 3 dimensional, involves three seperation constants, of which n defines the logitudinal case (using the familiar global coordinate notaion), an l define the latitudes. If the source is axisymmetric with its origin at a pole then only the n terms result. And elliptical source would involve both n and l. The final characteristic value is kr with r the radial location, which is not periodic and hence not an integer.
A identical concept is true for ALL forms of radiation from a square piston in a baffle(cosines), a circular piston (Bessel functions), Elliptic piston (Nueman functions) ... to Oblate Spheroidal Wave functions, and the most general case the Elliptic Spheriodal wave function, of which all finite size sources are special cases (the functions of which have never been generalized to my knowledge).
The cylindrical case is a bit different since the problem is infinite in both the r and z coordiantes so there is only a single integer value n which defines the polar radiation in theta - around the cylinder. The vertical polar is defined by a kz value where the kz and kr terms are orthogonal, but coupled, such that the total k must be the vector sum of the two spatial k's. This leads to a serious complication for calculations for cylinders with finite sources and in general the solutions involve complex concepts such as Saddle point integration and the like, since no closed form solutions are possible as they are for the spherical and infinite baffle cases.
Chemists will recognize the n an l numbers in the spherical case as defining the shells in which the electrons must reside - its an identical formulation.
So, yes, the n>0 terms are important.
Its this quote that is incorrect
The higher terms account for all directivity, and are most probably more interesting than the n=0 term (at least to me). The n= 0 term (lets talk the spherical case for now) is an omnidirectional term. It will alsways dominate at LF since it is the only term with a significant LF level. But as the frequency goes up, the other terms begin to become important each in succession, each peaking at a different value of kr, but all leveling off at 1.0.
For example, a point source on a sphere, like an impulse in time, will contain all terms in n. As the frequency goes up the terms come into play successively creating the directivity pattern. Like all "impulse" responses, or Green's functions in the spatial case, any arbitrary source can be described as a sum (or integral) over the Greens function for each point in the source. Hence ALL sources contain ALL orders of radiation unless by some chance some of them cancel.
Another way of saying the above is that the Green's function can always be defined as an infinite sum over any complete orthonormal set of functions. This result is key to all theoretical physics from sound radiation to Quantum mechanics (which are similar to identical calculations in many cases)
The n=0 case is a monopole, with equal radiation in all directions. The n=1 case is a dipole with a single nodal surface through the origin. The n=1 case is actually degenerate and there are three solutions in the three orthogonal directions, but this can always be reduced to a single mode in a single direction defines by the two Euler angles. A sum of a monopole and a dipole creates a cardiod. A piston in a sphere contains all orders and it is the presence of the higher orders that creates the directivity pattern.
The general case for a sphere, being 3 dimensional, involves three seperation constants, of which n defines the logitudinal case (using the familiar global coordinate notaion), an l define the latitudes. If the source is axisymmetric with its origin at a pole then only the n terms result. And elliptical source would involve both n and l. The final characteristic value is kr with r the radial location, which is not periodic and hence not an integer.
A identical concept is true for ALL forms of radiation from a square piston in a baffle(cosines), a circular piston (Bessel functions), Elliptic piston (Nueman functions) ... to Oblate Spheroidal Wave functions, and the most general case the Elliptic Spheriodal wave function, of which all finite size sources are special cases (the functions of which have never been generalized to my knowledge).
The cylindrical case is a bit different since the problem is infinite in both the r and z coordiantes so there is only a single integer value n which defines the polar radiation in theta - around the cylinder. The vertical polar is defined by a kz value where the kz and kr terms are orthogonal, but coupled, such that the total k must be the vector sum of the two spatial k's. This leads to a serious complication for calculations for cylinders with finite sources and in general the solutions involve complex concepts such as Saddle point integration and the like, since no closed form solutions are possible as they are for the spherical and infinite baffle cases.
Chemists will recognize the n an l numbers in the spherical case as defining the shells in which the electrons must reside - its an identical formulation.
So, yes, the n>0 terms are important.
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I agree with everything you say. When you said second part I though you were referring to the dispersion. I see you were not.
With regard to n>zero, yes there are interesting things happening, but form the point of view of designing a speaker based on a cylindrical wave front, a line source, we would not want to use the source at frequencies where it supports n>0 modes in the audio range for the exact reasons you state, non constant directivity or lobing. n> 0 is breakup, i.e. where the cylinder starts expanding and contracting with angular dependence. In designing the transducer we would need to investigate the behavior of the higher order modes and, hopefully, make sure they were heavily damped, or limit the use of the transducer to frequencies well below undamped modes, much as we limit the use of metal cone drivers to frequencies well below the nasty breakup peaks often seen. My comment was obviously too brief.
We are on the same page, I believe. Perhaps I should have expressed this more clearly.
Sound Radiation from Cylindrical Radiators
Hi John
No, that is still not correct. Lets consider a true line source, a finite cylinder, infinite in z (because its easier). If it is a "line" source on the cylinder then it takes, once again, an infinite number of "modes", n>0 to "expand" this source in terms of the eigen-functions in the theta directions (sines and cosines, i.e. a Fourier transform). Unless the source goes all the way around the cylinder with equal magnitude and phase, there will always be terms for n > 0, i.e. if the theta function differs from unity anywhere then terms n>0 (cosine(0) = 1.0) are required.
So even a rigid piston source in a cylinder will have terms > 0, in fact an infinite number of them. All the "breakup" does is to vary the contributions of the modes, just as the terms in a Fourier transform would vary if the function varies, but, in general, the same number of terms are required.
So, a line source will have significant terms in n>0 anytime there is any directivity in the angular direction, and I would submit that this is always the case.
What I believe that you are thinkning of is the problem of a finite source (line or otherwise) in an infinite baffle. Then the n=0 term does contain directivity and does represent the rigid piston case, but this is not true of any problem where the boundary is curved. So your explaination is correct for a line source in an infinite baffle but not for a line source in the cylindrical coordinates that we have been talking about. Further, I don't believe that the infinite baffle case has dispersion in its solution even in the near field. That is unique to the curved boundary problems.
For an infinite baffle case, the directivcity is found quite simply as the 2-dimensional fourier transform of the source distribution (the source being assumed to be described as two independent distributions in x and y. In general this is not the case. For example, an ellipse cannot be described as a product of an x function times a y function, so it cannot be evaluated by this technique.)
Ok Earl, I see where I was not thinking correctly. The azimuthal modes say whether or not the surface is moving radially out/inward in phase or with some azimuthal variation in phase. It says nothing about the frequency of the radial motion. That still leaves the ideal pulsating cylinder as one with pulsates uniformly with all modes but n=0 suppressed.
I guess I was thinking about how the cylinder would be excited and the possibility that by exciting the pulsation at a frequency with wave length equal to an integer fraction of the circumference these azimuthal modes could be excited, hence "break up" where as excitation at frequencies with wave length less the the circumference would not excite these modes.
Anyway, I want to get back the VC heating. Line source speakers just don't interest me very much.
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