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Accurate Calculation of Radiation and Diffraction from Loudspeaker Enclosures at Low Frequency

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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 3-dimensional 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 high-frequency models can then be smoothly joined.

JAES Volume 61 Issue 6 pp. 356-365; June 2013
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Claus Futtrup

Comment posted August 6, 2013 @ 16:35:42 UTC (Comment permalink)

Hi Jeff Candy.

Thank you for writing an AES article related to loudspeakers. I received the JAES for June 2013 today and immediately read your article and I find your article very interesting, including the approach to handling diffraction as well as the multitude of valuable information gained (including off-axis response + the radiation impedance on the transducer itself).

At low frequencies it is well known that the SPL doesn't drop as much as the commonly mentioned 6 dB. Furthermore it is well known that room gain isn't as much as you would think either. Although I haven't tested your method yet, it seems that the information gained at low frequencies provide good insight into room gain aspects as well (I'm well aware that the situation in a specific room is dominated by a specific vibration mode) and could be helpful for loudspeaker system designers.

P.S. Only a few software packages exist that I know of, which is commercially available and which will simulate radiation behind the speaker and also document the influence from rear edges of the box onto the on-axis response. One such software is LEAP EnclosureShop V5. The manual can be found on the internet (at The exact method in EnclosureShop is not publicly available. I appreciate that you describe clearly with a recipe how to complete the calculations and thereby provide a publicly available method for such simulations.

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Jeff Candy

Comment posted August 10, 2013 @ 23:55:29 UTC (Comment permalink)

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 well-known.  I am somewhat of a neophyte here, and it must show in the manuscript. Regarding the low-frequency 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 low-frequency limit.  Regarding LEAP, I suspect that the low-frequency limit is also not correct as a consequence of the limitations of the ray-tracing 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.

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Claus Futtrup

Comment posted August 12, 2013 @ 15:38:14 UTC (Comment permalink)

Hi Jeff

Just to clarify; when I said it is well known that the drop is less than 6 dB, related to the lowest frequencies, I didn't imply it was quantified. It is more a common experience among designers from the everyday life of trimming loudspeaker systems. Besides such practical experience is heavily "smeared" by room interaction. To the contrary I said your paper is interesting because it quantifies this.

Do you agree that gain from nearby walls can be analyzed with the MFS method, or is it out of scope?

Regarding LEAP I can follow you (and I found the discussion on the Parts Express forum from 2012). We'll need feedback from Chris N. Strahm (LinearX) to get to the bottom.

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Claus Futtrup

Comment posted August 12, 2013 @ 19:56:27 UTC (Comment permalink)

Hi Jeff

I have received the information that LEAP does not use ray-tracing techniques, instead it uses the Biot-Tolstoy-Medwin diffraction model. You can find more information about this model here:

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U. Svensson

Comment posted August 13, 2013 @ 15:17:31 UTC (Comment permalink)

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 low-order diffraction method I used at that time, which in turn was based on the Biot-Tolstoy-Medwin 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. 3681-3691. 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 non-convex geometries, as discussed by Jason Summers in the same JASA issue, pp. 3673-3676. I have developed a Matlab toolbox for edge diffraction calculations, and anyone interested is welcome to contact me to get a copy.


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Author Response
Jeff Candy

Comment posted August 16, 2013 @ 15:47:52 UTC (Comment permalink)

Hello Claus,

Thanks for the clarifications and comments.  Yes, nearby reflecting surfaces can be treated with MFS.  A trivial example is woofer floor-bounce, which can be solved using the method of images (i.e., place a mirror-image 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 high-frequency 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 up-to-date algorithm.  I did not know the BTM method had been implemented.  This is certainly not a high-frequency 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 well-conditioned 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.

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Jaime Ramis

Comment posted January 25, 2014 @ 15:00:59 UTC (Comment permalink)


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,

916-927, 2012.

PS: I can send the paper by e-mail. I am a co-author





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Author Response
Jeff Candy

Comment posted January 30, 2014 @ 12:44:55 UTC (Comment permalink)

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.

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Claus Futtrup

Comment posted January 30, 2014 @ 12:45:09 UTC (Comment permalink)

Dear Jamie Ramis

I am interested. Please see my profile for contact info.

Best regards,

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