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Page 112 ï~~Super-Spherical Wave Simulation in Flaring Horns David Berners and Julius 0. Smith, III Center for Computer Research in Music and Acoustics (CCRMA) Dept. of Music, Stanford University, Stanford, CA 94305 email@example.com, firstname.lastname@example.org Abstract: The flared horn is modeled according to Webster's equation. A change of variables transforms the equation into the form of the one dimensional Schrodinger wave equation. The Schrodinger form facilitates specification of arbitrary axisymmetric wavefronts for the acoustic disturbance within the horn. To provide a physically motivated choice of wavefront shape, Poisson's equation is solved inside the horn subject to the boundary condition that the normal component of the potential gradient is zero at the boundary of the horn. Since the disturbance within the horn must satisfy the wave equation, the velocity potential satisfies Poisson's equation when viscous effects and losses are ignored. Physical data from brass instrument bells are used to model musical horns using the Poisson solution, and results are compared to those obtained by traditional models which assume spherical wavefronts. Acknowledgment This work was made possible by the generous cooperation of Mr. Greg Hilliard of Frank Holton and Company who provided us with precise physical measurements of a test bell. 1 Introduction In [Berners and Smith, 1994] a method has been proposed to determine the acoustical properties of flared bores using a form of Webster's horn equation developed in [Benade and Jansson, 1974]. The method allows any choice of acoustic wavefronts propagating in the bore as long as axial symmetry is preserved. Here we discuss an application of that method using Poisson's Equation to predict the acoustic wavefronts within a bore. 2 Wavefront Computation Because of similarities between the acoustic wave equation and the classical field equation for electrostatics, the isophase pressure wave surfaces at relatively low frequencies are equivalent to equipotential surfaces found within an insulator having the shape of the horn wall with an axial potential drop. An insulating boundary with a high relative dielectric constant causes equipotential planes to be perpendicular to the boundary. This is appropriate in comparison to the velocity potential within an acoustic horn, which also must have equipotential lines which are perpendicular to the boundary due to the fact that --=0 On at the boundary wall. 112 [ C M C P R OC E E D I N G S 1995
Page 113 ï~~Egqulalent B Raci Horn Axis Axial rosnlro, m Figure 1: a) Calculated Isophase Surfaces for the Exponential Horn, b) Equivalent Bell Radii Wavefronts within a typical brass bell were computed using the electrostatic Poisson solver RELAX3D which was developed at the TRIUMF Meson Facility in Vancouver, Canada (triumf.ca). The bell boundary was defined by sampled measurements taken from a bell mandrel. Figure la shows equipotential surfaces of the solved grid along with the horn boundary. The surfaces become less spherical in regions of greater flare. For a given assumed acoustic wavefront shape, we define the equivalent radius as r(z) = l- VS(Z), where S(z) is the surface area of the wavefront crossing the horn axis at z. Figure lb shows equivalent radii for plane waves, spherical waves, and waves obtained by the Poisson solver within the test horn. It can be seen that the equivalent radius for the Poisson solution falls between those for the planar and spherical wavefronts. Sphedoal and Poron Weveorn lemr Rleflection Mqria deor Pomon and Spherical Warfrore 700 0.9 0.s em-Sphericad Barrier 5007 0.6 o.00 I, R 2OOO.0 0i~ ~rr.2o~nlkk ~ r00r10.4 00 0.02 0.0+ 0.06 006e 0.1 0.12 0.14 0.16 0.18 00 500 1000 150O0 2000 2500 3000 3500 4000 Axial Po.doe, mcaWaeFr Rfey. Hz Figure 2: a) Barrier Functions, b) ReflectionMagnitudes Figure 2a shows the barrier functions for the spherical and Poisson solution wavefronts. As defined in [Berners and Smith, 1994], the barrier is the normalized second derivative of the equivalent radius, --5. The spherical wavefront model produces a higher, narrower barrier which results in a slightly higher cutoff frequency in the reflection coefficient shown in Figure 2b. However, the difference between the two models is minimal in terms of response, and it is likely that the spherical model would be acceptable for this test case. References [Berners and Smith, 1994] D. P. Berners, J. 0. Smith. On the Use of Schrodinger's Equation in the Analytic Determination of Horn Reflectance, ICMC Proceedings 1994: 419-422. [Benade and Jansson, 1974] A. H. Benade, E. V. Jansson. On Plane and Spherical Waves in Horns with Nonuniform Flare I. Theory of Radiation, Resonance Frequencies, and Mode Conversion, Acustica 31(2): 80-98, 1974. I C M C P R O C EE D I N G S 199511 113