Did a quick simulatin on the Uppsala-contour as polars looked so good and I wondered how well it will behave outside the two decades of bandwidth shown:
Not that bad I'd say !
Please note that the performance may even be better as its only based on the not so perfect contour I got from the small pix below, which translates to some wiggles in the plots.
Comparing to my Gauss-JMLC :
I get the feeling that the Uppsala contour is based on / or could be seen as a modified LeCleach contour as well – exactly as mine is.
Looking at both horns above, we see that the CD / non CD behaviour is very simmilar to each other. Mine being a littel bit less directive and going down a little bit lower (also being slightly shorter and wider ).
Petty interesting find, whgeiger !
Michael
Not that bad I'd say !
Please note that the performance may even be better as its only based on the not so perfect contour I got from the small pix below, which translates to some wiggles in the plots.
Comparing to my Gauss-JMLC :
I get the feeling that the Uppsala contour is based on / or could be seen as a modified LeCleach contour as well – exactly as mine is.
Looking at both horns above, we see that the CD / non CD behaviour is very simmilar to each other. Mine being a littel bit less directive and going down a little bit lower (also being slightly shorter and wider ).
Petty interesting find, whgeiger !
Michael
1) Here is where Geddes "OS Waveguide" comes from, and there it is called a HORN, circa 1940.
Title: The Acoustical Impedance of an Infinite Hyperbolic Horn
Author: J. E. Freehafer
Publication: ASA-J, Vol. 11, No. 4, p. 467-476 (Apr-1940)
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Abstract: In discussing the acoustical properties of horns, it is in general a mathematical neceessity to assume plane waves. In the case of a horn in the form of an hyperboloid of one sheet, however, it is possible to avoid this assumption and to obtain an exact solution to the problem. The analysis, carried through with the aid of the differential analyzer, leads to curves representing the acoustical resistance and reactance as functions of the ratio of the radius of the throat to the wave length. Comparison with the conical horn shows that the hyperbolic is superior.
2) Horn shape optimization studies, which address the tradeoff between efficiency (the ratio of acoustic energy that exits/enters the horn) and dispersion uniformity (the degree to which, frequency response is of constant magnitude accross a specified coverage angle in the far field), have been recently conducted by faculty and students of Uppsala University, Sweden. These studies, described in the referenced articles, are yielding horn shapes that are not like those addresses by Freehafer and a lot later by Geddies. . A typical horn profile and related dispersion patterns are shown in the attached diagrams. The reference horn is conical. The Freehafer horn is asymptotically conical as well.
Regards,
WHG
References: Horn Shape Optimization
Title (1): Shape Optimization for Acoustic Wave Propagation Problems
Title (2): Doctoral Thesis, Comprehensive Summary
Author: Udawalpola, Rajitha
Affiliation: Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science
Publication: Acta Universitatis Upsaliensis, 14-Jan-2010
Abstract(1): Boundary shape optimization is a technique to search for an optimal shape by modifying the boundary of a device with a pre-specified topology. We consider boundary shape optimization of acoustic horns in loudspeakers and brass wind instruments. A horn is an interfacial device, situated between a source, such as a waveguide or a transducer, and surrounding space.
Abstract(2): Horns are used to control both the transmission properties from the source and the spatial power distribution in the far-field (directivity patterns).
Abstract(3): Transmission and directivity properties of a horn are sensitive to the shape of the horn ?are. By changing the horn ?are we design transmission efficient horns. However, it is difficult to achieve both controllability of directivity patterns and high transmission efficiency by using only changes in the horn ?are. Therefore we use simultaneous shape and so-called topology optimization to design a horn/acoustic-lens combination to achieve high transmission efficiency and even directivity.
Abstract(4): We also design transmission efficient interfacial devices without imposing an upper constraint on the mouth diameter. The results demonstrate that there appears to be a natural limit on the optimal mouth diameter.
Abstract(5): We optimize brass wind instruments with respect to its intonation properties. The instrument is modeled using a hybrid method between a one-dimensional transmission line analogy for the slowly flaring part of the instrument, and a finite element model for the rapidly faring part.
Abstract(6): An experimental study is carried out to verify the transmission properties of optimized horn. We produce a prototype of an optimized horn and then measure the input impedance of the horn. The measured values agree reasonably well with the predicted optimal values.
Abstract(7): The finite element method and the boundary element method are used as discretization methods in the thesis. Gradient-based optimization methods are used for optimization, in which the gradients are supplied by the adjoint methods.
Title: Shape Optimization of an Acoustic Horn
Author (1): Erik BÄangtsson
Author (2): Daniel Norelan
Author (3): Martin Berggre
Affiliation: Uppsala University, Department of Information Technology
Publication: Uppsala University, Tech. Report 2002-019, 8-May-2002
URL: http://www.it.uu.se/research/publications/reports/2002-019/2002-019-nc.pdf
Abstract(1): Shape optimization of an acoustic horn is performed with the goal to minimize the portion of the wave that is reflected. The analysis of the acoustical properties of the horn is performed using a finite element method for the Helmholtz equation.
Abstract(2): The optimization is performed employing a BFGS Quasi-Newton algorithm, where the gradients are provided by solving the associated adjoint equations. To avoid local solutions to the optimization problem corresponding to irregular shapes of the horn, a filtering technique is used that applies smoothing to the design updates and the gradient. This smoothing technique can be combined with Tikhonov regularization.
Abstract(3): However, the use of smoothing is crucial to obtain sensible solutions. The smoothing technique we use is equivalent to choosing a representation of the gradient of the objective function in an inner product involving second derivatives along the design boundary. Optimization is performed for a number of single frequencies as well as for a band of frequencies. For single frequency optimization, the method shows particularly fast convergence with indications of super-linear convergence close to optimum.
Abstract(4): For optimization on a range of frequencies, a design was achieved providing a low and even reflection throughout the entire frequency band of interest.
Title: Topology Optimization of an Acoustic Hrorn
Author(1): Eddie Wadbro (1)
Author(2): Martin Berggren (1),(2)
Affiliation(1): Uppsala University
Affiliation(2): Swedish Defence Research Agency
Publication: Uppsala University, Tech. Report 2006-009, 8-May-2002
Abstract(1): We present a method for topology optimization of an acoustic horn with the aim of radiating sound as efficiently as possible. Using a strategy commonly employed for topology optimization of elastic structures, we optimize over a scalar function indicating presence of material. The Helmholtz equation modeling the wave propagation is solved using the finite element method and the associated adjoint equation provides the required gradients.
Abstract(2): Numerical experiments validates that the result of the optimization provides horns with the desired acoustical properties. The resulting horns are very efficient in the frequency span subject to optimization.
Title: Topology Optimization for Acoustic Wave Propagation Problems
Author: Eddie Wadbro
Affiliation: Uppsala University, Department of Information Technology
Publication: Uppsala University, Dissertation 2006-009, 1-Oct-2006
URL: http://www.it.uu.se/research/publications/lic/2006-009/2006-009.pdf
The aim of this study is to develop numerical techniques for the analysis and optimization of acoustic horns for time harmonic wave propagation. An acoustic horn may be viewed as an impedance transformer, designed to give an impedance matching between the feeding waveguide and the surrounding air. When modifying the shape of the horn, the quality of this impedance matching changes, as well as the angular distribution of the radiated wave in the far field (the directivity).
Abstract(2): The dimensions of the horns considered are in the order of the wavelength. In this wavelength region the wave physics is complicated, and it is hard to apply elementary physical reasoning to enhance the performance of the horn. Here, topology optimization is applied to improve the efficiency and to gain control over the directivity of the acoustic horn.
Title: Shape and Topology Optimization of an Acoustic Horn-Lens Combination
Author (1): Eddie Wadbro
Author (2): Rajitha Udawalpola,
Author (3): Martin Berggren
Affiliation(1): Uppsala University
Publication: Journal of Computational and Applied Mathematics (ISSN 0377-0427), Vol. 234, No.6, pg. 1781,Jul-2010
Abstract: Using gradient-based optimization combined with numerical solutions of the Helmholtz equation, we design an acoustic device with high transmission efficiency and even directivity throughout a two-octave-wide frequency range. The device consists of a horn, whose flare is subject to boundary shape optimization, together with an area in front of the horn, where solid material arbitrarily can be distributed using topology optimization techniques, effectively creating an acoustic lens.
Super!
Hi Michael,
I suspected this would be the finding. What is obvious from this body of work, is that any horn shape that approaches a conical profile is patently sub-optimal. This fact is clearly demonstrated by the reflectance spectra and dispersion patterns presented. I am still studying the articles, and will have further comments later. In the past I have used Salmon necks terminated by tractrix bells to arrive at specific loudspeaker horn designs.
Regards,
WHG
P.S. Note that the cylindrical segment only used in the simulations is refered to as a "waveguide" by the authors. ;-)
Hi Michael,
I suspected this would be the finding. What is obvious from this body of work, is that any horn shape that approaches a conical profile is patently sub-optimal. This fact is clearly demonstrated by the reflectance spectra and dispersion patterns presented. I am still studying the articles, and will have further comments later. In the past I have used Salmon necks terminated by tractrix bells to arrive at specific loudspeaker horn designs.
Regards,
WHG
P.S. Note that the cylindrical segment only used in the simulations is refered to as a "waveguide" by the authors. ;-)
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