John
For a speaker in a flat open back baffle the directivity can be modeled fairly well as the product of the dipole cosine pattern and the normal rigid piston directivity. But when the speaker is say in the end of a tube open at the other end, then this simple model will fail because the path lengths in the plane of the speaker from the front and the back are not equal. Hence the directivity will need to widen from the flat baffle case.
Do you know of a good model for this? Have you ever measured this effect and compared the two situations? Any ideas?
For a speaker in a flat open back baffle the directivity can be modeled fairly well as the product of the dipole cosine pattern and the normal rigid piston directivity. But when the speaker is say in the end of a tube open at the other end, then this simple model will fail because the path lengths in the plane of the speaker from the front and the back are not equal. Hence the directivity will need to widen from the flat baffle case.
Do you know of a good model for this? Have you ever measured this effect and compared the two situations? Any ideas?
Hi Earl,
This is sort of like my u-frame woofer. One source is at the end of the u-frame and the other end is opened. The length of the tube is 18". But here is the thing, if you measure the near filed response of the source and the response in the plane of the rear opening you will find that they are both minimum phase. If you look at the impulse response for the open end the delay associated with the propagation distance of the length of the tube is clearly evident. But the duct resonances (1/4 wave and higher frequencies) introduce a negative group delay which seems to have the effect of canceling the propagation distance. The result is that at low frequency, where the front and rear amplitude response are the same, the response is dipole. As the frequencies rises the front and rear amplitude responses deviate, and being minimum phase, they do not form a dipole but tend more to a monopole.
I have some discussion of the here.
And also some specific discussion of the U-frame woofer system I designed which shows measurements at the front and rear, and the roll of damping (forming an acoustic low pass filter, and resistive damping of the 1/4 wave resonance) in restoring the internal delay and allowing quasi-cardioid operation.
This is sort of like my u-frame woofer. One source is at the end of the u-frame and the other end is opened. The length of the tube is 18". But here is the thing, if you measure the near filed response of the source and the response in the plane of the rear opening you will find that they are both minimum phase. If you look at the impulse response for the open end the delay associated with the propagation distance of the length of the tube is clearly evident. But the duct resonances (1/4 wave and higher frequencies) introduce a negative group delay which seems to have the effect of canceling the propagation distance. The result is that at low frequency, where the front and rear amplitude response are the same, the response is dipole. As the frequencies rises the front and rear amplitude responses deviate, and being minimum phase, they do not form a dipole but tend more to a monopole.
I have some discussion of the here.
And also some specific discussion of the U-frame woofer system I designed which shows measurements at the front and rear, and the roll of damping (forming an acoustic low pass filter, and resistive damping of the 1/4 wave resonance) in restoring the internal delay and allowing quasi-cardioid operation.
I forgot the link for the measurements. They are here.
And a more direct answer to your question is that at low frequency the same dipole modle should work. The problem come in with the lack of symmerty between the front and rear sources due to the duct resonances and how one would model the directivity of the open ended tube. I guess a first approximation would be to model it as a flat piston. So you would have a doublet for which the two sources would have different amplitude response at higher frequency and some form of directivity.
And a more direct answer to your question is that at low frequency the same dipole modle should work. The problem come in with the lack of symmerty between the front and rear sources due to the duct resonances and how one would model the directivity of the open ended tube. I guess a first approximation would be to model it as a flat piston. So you would have a doublet for which the two sources would have different amplitude response at higher frequency and some form of directivity.
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John
Thanks
That would be doable except for the distance between the front and rear adds some complexity. I suppose that I could calculate the volume velocities of the front and rear using a tube model and then use these in a spherical enclosure model which would get the spacing right, but the added volume of the sphere would be wrong.
When I was trying to model this some years ago I was looking at the solution for the response of a flat piston at the end of a long pipe. I figured that would serve as the model for directivity and all I had to do then was superimpose the amplitude and phase response onto that solution. Then I was going to sum the two sources with the correct path length differences to get the polar response. Never actually got around to it.
A flat piston at the end of a pipe does not have a simple solution - not that I know of anyway - so that isn't the way that I would think to go. If it did then what you suggest would be quite doable. There is a "sort of" solution to the pistion in the end of the pipe in Morse as I recal and maybe I'll look that up. I'll let you know.
Ok, for those of you who do follow Morse - who is the king of theoretical acoustics IMO - there is an expression for the sound radiation from the end of a pipe given in Methods of Theoretical Physics on page 1459. It is quite complicated and I am not sure how useful it will be, but it is an interesting start.
There is also a 1947 paper in Physical Review on Radiation of sound from an unflanged circular pipe. V73, No 4, p383 by Levine anf Schwinger.
John
Thanks again for the paper.
I was seriously humbled by the math and then I noted one of the authors was Julianne Swinger. Dr. Swinger is renown as one of the greatest mathematicians of the 20th century. He and Dr. Richard Feynman had competing theories of Quantum Electrodynamics (QED), a major break through in quantum theory. In the end it was shown, as it often is, that the two were equivalent, but Feynman's was more direct, less intricate math (a typical Feynman characteristic.) It is Feynman's formulation that has survived.
At any rate, the analysis is wide of the topic since so much of it is interested in the amount of the duct wave is reflected from the tube end. Little of it applies to the actual directivity problem that interests us. Now it is certain that the directivity answer is buried in there somewhere, but it is neither obvious, nor apparantly easy, to make the required simplifications.
Thanks again for the paper.
I was seriously humbled by the math and then I noted one of the authors was Julianne Swinger. Dr. Swinger is renown as one of the greatest mathematicians of the 20th century. He and Dr. Richard Feynman had competing theories of Quantum Electrodynamics (QED), a major break through in quantum theory. In the end it was shown, as it often is, that the two were equivalent, but Feynman's was more direct, less intricate math (a typical Feynman characteristic.) It is Feynman's formulation that has survived.
At any rate, the analysis is wide of the topic since so much of it is interested in the amount of the duct wave is reflected from the tube end. Little of it applies to the actual directivity problem that interests us. Now it is certain that the directivity answer is buried in there somewhere, but it is neither obvious, nor apparantly easy, to make the required simplifications.
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