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A model that accurately computes the sound pressure field of a loudspeaker would be an efficient tool for designing a real transducer system. Although there are many tools for calculating radiation and diffraction from loudspeaker cabinets, the results are only valid for high frequencies; traditional approaches for modeling diffraction produce significant errors at frequencies below 500 Hz. This research describes an approach to solve the 3dimensional Helmholtz equations of a piston radiator in a rectangular solid enclosure using the Method of Fundamental Solutions. This method enables accurate calculation of sound pressure, including an exact representation of diffraction. The radiation impedance of a piston in a finite enclosure can also be computed. In practice, there is a maximum frequency that depends on the cabinet size. The low and highfrequency models can then be smoothly joined.
Author:
Candy, Jeff
Affiliation:
San Diego, CA, USA
JAES Volume 61 Issue 6 pp. 356365; June 2013
Publication Date:
July 8, 2013
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Claus Futtrup 
Comment posted August 6, 2013 @ 16:35:42 UTC
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Hi Jeff Candy.
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Author Response Jeff Candy 
Comment posted August 10, 2013 @ 23:55:29 UTC
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Hello Claus, Thanks very much for the comments and kind words. You are quite probably correct in suggesting that the deviation from 6.02dB is wellknown. I am somewhat of a neophyte here, and it must show in the manuscript. Regarding the lowfrequency limit, the real challenge is of course to compute it correctly as it is critical for some modeling tasks, as explaned in the manuscript. I was motivated to write the paper when I found, to my dismay, that the modeling tools I had access to were really "off" in the lowfrequency limit. Regarding LEAP, I suspect that the lowfrequency limit is also not correct as a consequence of the limitations of the raytracing approach. However, in the absence of a careful benchmark, I don't have solid justification for this suspicion. Incidentally, the MFS method may also applicable to the problem of horns and waveguides, and in the case of azimuthal symmetry, is probably practical to implement on a PC. If there is interest I could probably work on that. 
Claus Futtrup 
Comment posted August 12, 2013 @ 15:38:14 UTC
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Hi Jeff

Claus Futtrup 
Comment posted August 12, 2013 @ 19:56:27 UTC
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Hi Jeff

U. Peter Svensson 
Comment posted August 13, 2013 @ 15:17:31 UTC
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Hi Jeff and Claus, It is great to see more tools in the toolbox for studying the sound radiation from loudspeakers. Even if we have the Boundary Element Method as a reference method since decades, that method is probably not accessible to everyone. I can confirm that I emailed with Chris Strahm a zillion times before the release of Leap 5 and he continued the development of the loworder diffraction method I used at that time, which in turn was based on the BiotTolstoyMedwin diffraction model. In those days, I couldn't compute high enough orders to study the VLF response, but a recent integral equation formulation permits the calculation of arbitrarily high orders. It was published by Asheim and myself in JASA this year, volume 133, pp. 36813691. For convex rigid scatterers it seems to give very, very accurate results regardless of frequency, even down to 0 Hz  and we could confirm that we get exactly 6.02 dB at 0 Hz. Or, 0 dB at 0 Hz and +6.02 dB in the HF limit, if one prefers. There are still some unanswered questions regarding the accuracy for nonconvex geometries, as discussed by Jason Summers in the same JASA issue, pp. 36733676. I have developed a Matlab toolbox for edge diffraction calculations, and anyone interested is welcome to contact me to get a copy. 
Author Response Jeff Candy 
Comment posted August 16, 2013 @ 15:47:52 UTC
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Hello Claus, Thanks for the clarifications and comments. Yes, nearby reflecting surfaces can be treated with MFS. A trivial example is woofer floorbounce, which can be solved using the method of images (i.e., place a mirrorimage of the loudspeaker under the floor). I have tried this and it appears to work correctly. A more challenging and general approach is also possible, in priciple, whereby sources can be placed not only inside the loudspeaker but also just outside the walls of the room to enforce the boundary conditions (zero normal velocity) on all reflecting surfaces. I have not tried this, and my guess is that it would have more severe highfrequency limitations (due to matrix condition number problems) than the case of an isolated loudspeaker. But yes, in principle, the effect of walls, cone geometry, bizzare enclosure shapes, are treatable. Before I would embark on these more advanced applications, however, I would improved the algorithm for source placement so as to reduce the matrix condition problems. Source placement is critical to the success of the method. Another alternative would be to use quad precision arithmetic.
Regarding LEAP, I am sorry if I mischaracterized the most uptodate algorithm. I did not know the BTM method had been implemented. This is certainly not a highfrequency asymptotic method like ray tracing. My limited understanding of BTM is that solutions are exact only for certain geometries, but I am afraid I cannot comment further. The litmus test of course is to carry out a benchmark, and for that I suggest Chris S. might simulate the case shown in Fig. 2 of the paper with LEAP. Here the gain is much less than 6dB.
Finally, thanks very to Peter for his clarifications. I would be happy to spend some time on a benchmark of MFS versus the integral formulation of BTM you describe. I have the feeling that a hybrid method might be optimal: MFS at low frequency where the method is wellconditioned and BTM at higher frequency where it must be far more efficient than MFS.
Shall we continue this discussion via email? I would prefer that. 
Jaime Ramis 
Comment posted January 25, 2014 @ 15:00:59 UTC
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Hello, Very good paper!. Congratulations! For an example of application to horns you can see L. Godinho, J. Ramis, W. Cardenas, J. Carbajo, P. Amado Mendes  A numerical MFS model for computational analysis of acoustic horns. Acta Acustica united with Acustica 98, 916927, 2012. PS: I can send the paper by email. I am a coauthor

Author Response Jeff Candy 
Comment posted January 30, 2014 @ 12:44:55 UTC
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Hi Jaime, Thanks. Sure, if you don't mind, please send me a copy if the paper via my email address, which should be visible on my member profile page. 
Claus Futtrup 
Comment posted January 30, 2014 @ 12:45:09 UTC
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Dear Jamie Ramis

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