PAPERS Journal of the Audio Engineering Society Vol. 64, No. 5, May 2016 ( C  2016) DOI: http://dx.doi.org/10.17743/jaes.2016.0011 Simulating Axisymmetric Concave Radiators Using Mode Matching Methods BJØRN KOLBREK, AES Student Member (bjorn.kolbrek@iet.ntnu.no) , KAREN BRASTAD EVENSEN † , AND U. PETER SVENSSON, AES Member Department of Electronics and Telecommunications, NTNU–Norwegian Universityof Science and Technology, Trondheim, Norway Currently, simulation of cone loudspeaker directivity is usually performed using Finite Element or Boundary Element methods, which can be slow and demanding on memory. This paper investigates the use of a mode matching method for simulating concave, axisymmetric geometries. The method is based on a method presented by Pagneux et al. [1] for simulation of horns but extended so that vibrating walls can be taken into account. The method is compared to a Boundary Element solution, and the results are shown to be in excellent agreement. Finally, examples of use are given. 0 INTRODUCTION The directivity of cone loudspeakers, as well as the ra- diation impedance, are topics that have been of interest to audio engineers for a long time. It is common to use the approximation of a rigid plane piston for the study of these quantities, but it is also known that, in particular for the directivity, this is not a good approximation [2]. Quite early, Stenzel made extensive investigations of pis- tons and membranes during his time at AEG [3–5]. In 1941, Brown [6] gave expressions for the radiation from rigid and non-rigid cones, using a Kirchhoff approximation. The non-rigid cones were characterized by the sound speed in the cone material. Carlisle [7] provided a similar analysis, where the directivity was computed from the sum of the pressures radiated from annular rings. Another approxima- tion was given by Geddes [8], who computed the velocity distribution at the mouth of the device, converted this veloc- ity distribution into a modal representation, and used this to compute directivity. The radiation impedance was not computed. Ando et al. [9] developed a method based on mode match- ing for calculating the radiation from a horn or loudspeaker cone mounted in the end of a semi-infinite circular tube. Ando’s method appears to be extremely complex, and the paper requires a detailed study simply to figure out which equations to solve and in what order. A similar method was † Presently at COWI AS, Grenseveien 88, 0605 Oslo, Norway presented by Oie et al. [10] for a radiator in an infinite baf- fle. While the method for non-vibrating walls is described in detail, the transition from a horn structure to a struc- ture with vibrating walls is not clear, and this method, too, appears rather complex. Later analyses usually depend on Finite Element or Boundary Element methods [11–15], but an interesting departure from this was demonstrated by Murphy [16], who used a method similar to that of Carlisle. Murphy modeled the cone as a set of concentric rings and delay lines. The radiation impedances of the rings (considered separately) were used as radiation loads for a distributed mechanical model of the cone, where each ring was con- nected to the next by masses and springs to take sound propagation in the cone material into account. The to- tal radiation impedance was not given explicitly, and it seems that trying to calculate it from the pressures and volume velocities in the circuit would also integrate the cone mass and flexural compliance into this impedance value. A mode matching method based on ideas similar to those of Ando and Oie has been presented by Pagneux et al. [1] and used by Kemp and others [17–19]. This method is well suited to simulate horns: it is fast, uses little mem- ory, and is easily scalable. It is, however, not directly suited for simulating structures like a loudspeaker cone, since it is based on working one’s way from the mouth of the horn back to the throat, given a radiation impedance specified at the mouth. If any surface inside the structure J. Audio Eng. Soc., Vol. 64, No. 5, 2016 May 311