Comparison of Numerical Simulation Models and Measured Low-Frequency Behavior of Loudspeaker Enclosures *

Transcription

1 Comparison of Numerical Simulation Models and Measured Low-Frequency Behavior of Loudspeaker Enclosures * MATTI KARJALAINEN, 1 AES Fellow, VEIJO IKONEN, 2 POJU ANTSALO, 1 PANU MAIJALA, 3 AES Member, LAURI SAVIOJA, 1 AES Member, ANTTI SUUTALA, 2 AND SEPPO POHJOLAINEN 2 1 Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing, Espoo, Finland 2 Tampere University of Technology, Department of Mathematics, Tampere, Finland 3 V T T Automation, Machinery Acoustics, Tampere, Finland The vibroacoustic behavior of a set of prototype loudspeakers below 1 khz was studied in detail by comparing measurements and element model simulations. Sound fields were measured using a microphone array of electret capsules designed for the purpose and vibrations using a laser vibrometer and accelerometers. Simulations were carried out using analytical, finite- and boundary-element methods, and finite-difference methods. The enclosure conditions were varied from a fixed wall case buried in sand to a free-standing empty box and a freestanding damped box. The applicability of each modeling technique is discussed. 0 INTRODUCTION * Manuscript received 2001 January 16; revised 2001 August 20. An early version of this paper was presented in the 104th Convention of the Audio Engineering Society, Amsterdam, The Netherlands, 1998 May The aim of this study was to investigate the behavior of loudspeaker enclosures through both measurements and element-based modeling. Both the interior acoustic field and the structural vibrations were studied up to about 1 khz. The goal was to test different element-based modeling techniques in comparison with measured data to determine the applicability and accuracy of existing modeling tools and software. The motivation to reliable modeling of enclosure behavior is that even in a rigidly designed closed box the effect of wall vibrations on radiated sound is clearly measurable and in poor designs easily perceivable. Thus vibroacoustic computational modeling and analysis could be a useful tool for loudspeaker design and prototyping as long as modeling is fast and accurate enough. Based on the successful simulation of internal field and surface vibrations it is in principle straightforward to compute the responses in the external sound field, for example, using the BEM method. However, this presupposes a detailed element model of the driver diaphragm, which was beyond the scope of the present study. The design of a loudspeaker has traditionally been an iterative process based on approximate rules, experience from prior designs, and finally trial and error by constructing and modifying prototypes [1], [2]. Computerbased methods have helped in exploring basic features of driver-to-enclosure matching and crossover network design. However, not very many papers have been published on the use of computer-based methods and tools in the vibroacoustic design of loudspeaker enclosures, although they have been studied for more than 20 years. Examples of such studies are [3] [9]. More often the focus of modeling has been on driver behavior and on radiation, such as in [6], [10] [18]. Other related publications of interest are, for example, [19] [22]. The detailed behavior of a loudspeaker consisting of an enclosure and driver(s) is very complex and escapes fully analytical mathematical solutions. Approximate (semianalytic) approaches may turn out to be useful, however, especially within a limited frequency range and when the geometry of the system is simple enough, such as a shoe J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

2 box design. At low enough frequencies, lumped-element models have been used successfully for both electroacoustic and vibroacoustic subsystems. In more complex cases the system should be considered as a complex electrovibroacoustic system, which is not easily partitioned into simple submodels. The progress in computer-based numerical simulation of complex, spatially distributed systems, using various element methods, has raised the question of how useful they might be for practical loudspeaker design [2]. These modeling techniques include the finite-element method (FEM), the boundary-element method (BEM), and the finite-difference time-domain (FDTD) method. In advanced forms they can be used for simulating any linear and time-invariant (and, with limitations, nonlinear) vibroacoustic systems at low frequencies. Low frequencies means here that the element size in the model mesh, and thus the number of spatially discrete elements, limits the highest useful frequency of simulation. Loudspeaker design at low to mid frequencies is in principle a good application for such methods. Several commercial or experimental tools are available for element-based vibroacoustic modeling and simulation, such as SYSNOISE [23], I-DEAS Vibroacoustics [24], and Comet/Acoustics [25]. Other FEM/BEM tools not specifically tuned to acoustic problems are ABAQUS [26], ANSYS [27], MSC/NASTRAN [28], and FEMLAB [29]. Some problems with using the FEM/BEM programs, at least from the point of view of loudspeaker design, are that they are expensive, they require powerful computers to work fast enough, the construction of the model is tedious, and the availability of material data (acoustic and mechanical parameters) is poor. Experimental programs, such as from academic institutions, often lack documentation and continuing support. Thus, such simulation and design tools are not widely used in loudspeaker design, and information on their usefulness as well as comparisons of their properties are practically nonexisting. As progress in this field is fast, it is important to be prepared to utilize such tools when they turn out to be productive. Potentially, computer-based design tools promise to make the product development period faster whenever the designer can start from an approximate model and rapidly go through variations and experimentations using software modeling up to a prototype which, when actually built, works close to expectations. Even more ambitiously, the computer could automatically run through some optimization steps to search for the best match to given specifications and target criteria. In this study we selected cases that are simple enough, yet practical and realistic. Thus we specified and constructed closed-box enclosures with a single driver element so that it was simple to vary some parameters of interest, such as enclosure size and material, driver position, stiffness of the enclosure walls, and damping material inside the box. Next the actual vibroacoustic behavior of the cases was measured extensively with such parametric variations. A dense microphone array of electret capsules was constructed to measure the sound field inside and outside the loudspeaker boxes in the form of impulse LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES responses to the electric excitation of the driver. A laser vibrometer and accelerometers were used to obtain vibration data of the walls and the cone of the driver so that an almost complete picture of the behavior below about 1 khz was captured. Acoustical and mechanical parameters of the materials (MDF and PVC for construction and mineral wool for damping) were also measured. The next step of the study was to model the loudspeakers using various FEM, BEM, and FDTD software tools. Computational models were built, and measured or estimated material parameters were given to the models. Simulation results from using the models are presented in this paper and compared with the measured behavior. In addition, the models were hand-tuned to match measured data as well as possible. This resulted in material parameters that often work better than the directly measured ones, at least in modeling similar cases, but may not be generalizable to considerably different cases. Results of numerical simulations and measured responses will also be compared with simpler semianalytic modeling. After presenting results, the usefulness of the models and how they could be improved will be discussed. Directions for further studies are touched upon as well. 1 CASE STUDY: CLOSED-BOX LOUDSPEAKERS To study a problem of manageable complexity, yet interesting from a practical point of view, we designed and constructed a set of closed-box prototype loudspeakers. The sizes and materials of the enclosures were varied in two versions. The larger box has inner dimensions of 600 mm mm and the smaller one, 250 mm 200 mm 150 mm. The two material choices were MDF and PVC. The structure of the large enclosure is shown in Fig. 1. The front panel (facing up in the figure) is removable and is built in two variations, one with a driver element symmetrically in the middle and another with asymmetric positioning (rim case), as shown in Fig. 1(a). The loudspeaker element is a 6.5-in driver of type SEAS P17 REX. The large enclosure was made of 20-mm MDF, all panels being rigidly coupled at their edges. In our simulations and measurements the behavior of the box was studied both buried in sand and freestanding to allow the walls to vibrate. In both cases it was studied as an empty box and with damping material (Partek mineral wool) on the back wall, inside the box [Fig. 1(b)]. 1.1 Conceptual Loudspeaker Model From a vibroacoustic point of view the loudspeaker works as follows. The driver element converts its electrical excitation to movement of the diaphragm, which has acoustic loading due to the air outside and inside the cabinet. Another vibroacoustic coupling of interest is from the interior sound field to the walls of the enclosure, causing them to vibrate and, due to this vibration, to radiate an external sound field, in addition to the radiation of the driver diaphragm. In general the wall vibrations should be small enough not to add their resonances to the radiated J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1149

3 KARJALAINEN ET AL. sound. The driver element has also a direct mechanical coupling to the front panel and via it, indirectly, to all other panels of the enclosure. If the walls were rigid, only the vibration of the driver diaphragm would be of interest. In practice, however, the vibration of the enclosure walls is not negligible and should be included in a detailed simulation of the system. Furthermore, the acoustic loading of the cabinet interior on the driver diaphragm movement has an effect on its radiation to the external field. All these effects influence the magnitude and phase responses of the loudspeaker and its directivity pattern. 2 VIBROACOUSTIC MEASUREMENT SYSTEM In order to enable an evaluation of how realistic the numerical simulation results of the loudspeakers are, we decided to construct a system to permit extensive vibroacoustic measurements. It consists of an array of miniature electret microphones to collect acoustic responses and a combination of a laser vibrometer and vibration sensors (accelerometers) to probe the mechanical vibrations in the loudspeaker systems. A computerized measurement system was programmed to collect sound field and mechanical vibration data in the form of impulse responses. 2.1 Microphone Array A microphone array of 90 small electret capsules was constructed so that it fits into the interior of the large loudspeaker cabinets (Figs. 2 and 3). A frame of metal tubing PAPERS was used to support the row and column wires, spaced 40 mm 40 mm, as shown in Fig. 2. At each wire crossing an electret microphone (Hosiden 2823) and a cascaded diode were attached, as depicted in Fig. 3. Digitally controlled analog multiplexers were used to select one of the column wires and one of the row wires at a time. Only a single electret capsule, activated by current through the load resistor R, is functional at a time to capture the sound pressure field and to transduce it to the microphone preamplifier. Thus the multiplexed microphone array can be used to measure the acoustic responses in the spatially distributed mesh positions point by point, both inside and outside the box. The lower cutoff frequency of the microphone amplifier combination was 30 Hz and the response was found to be flat within 1 db in the measurement range of greatest interest for our study (100 Hz to 2 khz) so that only the slightly different gains of individual capsules needed compensation. 2.2 Vibration Measurements For vibration measurements a laser vibrometer (Polytec OFV3001) and acceleration probes were used. A mesh of 40 mm 40 mm was measured point by point, to obtain the vibration responses of all the walls. A number of points in the cone of the driver were also registered. Point mobility measurements were made by applying impact testing to the walls of the enclosure as well as to isolated pieces of MDF plates corresponding to the walls of the enclosure. The following equipment was used: (a) (b) Fig. 1. (a) Mounting of loudspeaker element (asymmetric position). Driver element is centered on front plate. In case of rim loudspeaker, dimensions (in mm) are according to (b). Plate material is 20-mm MDF. (b) Closed-box enclosure with microphone array and damping mineral wool inside. Inner cabinet dimensions are 600 mm, 400 mm, and 250 mm J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

4 impact hammer with Brüel & Kjær 8200 force transducer and B&K 4393 accelerometer with two B&K 2635 charge amplifiers. An HP 3565 S analyzer and STAR software were used to determine modal damping. 2.3 Data Acquisition System and Analysis Tools The measurement system was based on the QuickSig signal processing environment [30], developed in the Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology. Impulse response measurements were carried out using Schroeder phasesequence [31] or random-phase flat spectrum (RPFS, designed by inverse FFT from flat-magnitude and random-phase spectrum) excitation signals of typically 8192 samples at a sampling rate of Hz, typically averaged over 10 repetitions. In practice this is equivalent to the more commonly used maximum-length sequence (MLS) [32] measurements. The frequency range of interest, from the viewpoint of element-based modeling in this study is up to 1 2 khz. The signal-to-noise ratio of the acoustic measurements was under all conditions better than 40 db so that its effect on the magnitude responses, for example, is negligible. The same data acquisition system was also used for vibration measurements (except for impact testing). Further signal analyses of acoustic and vibration data were carried out in MATLAB. LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES 3 MEASURED LOUDSPEAKER BEHAVIOR Typical acoustic responses, as measured inside the large MDF enclosure, are shown in Fig. 4. Fig. 4(a) and (b) shows the impulse responses from driver terminals to the sound pressure in one mesh point, r14c5 (row 14, column 5), for a sand-supported (a) undamped and (b) mineral wool damped enclosure. Fig. 4(c) shows the magnitude responses for the sand-supported and freestanding cases without interior damping, and Fig. 4(d) the corresponding magnitude responses for the case with 100-mm mineral wool at the back panel. The impulse response plotted in Fig. 4(a) reveals long ringing of interior resonances in the undamped case. The corresponding ringing is radically shorter in a damped case, as shown in Fig. 4(b). The same information is presented in the frequency domain in subplots Fig. 4(c) and (d). The former shows the resonances and antiresonances in the undamped enclosure as measured in the sandsupported and the freestanding cases, respectively. The mode frequencies exhibit strong resonances. The difference between these two curves is surprisingly small. Only minor extra effects are introduced in the freestanding case, visible for example around 200 Hz. The same is also true for the damped case of Fig. 4(d), except that air resonances and antiresonances are effectively smoothed out. Examples of enclosure wall vibrations are presented in Fig. 2. Structure and dimensions of electret microphone array. Fig. 3. Principle of electret microphone array, multiplexer, and amplifier circuit. J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1151

5 KARJALAINEN ET AL. Fig. 5, based on three different measurements. Fig. 5(a) illustrates the accelerance (acceleration/force) of an isolated wall panel, excited near a corner, as measured by impact testing. This information can be used to estimate the parameters of MDF for vibroacoustic modeling. Fig. 5(b) shows the corresponding behavior when the sidewall ( mm) is excited at a point 260 mm from the front panel and 297 mm from the top plate. This can be compared further with the velocity of the same point as a response to an electric excitation of the driver element [Fig. 5(c)]. Only the lowest mode (200 Hz) has a prominent effect on external sound field radiation. Fig. 6 illustrates velocity responses measured in different positions on the diaphragm of the driver element SEAS P17 REX [Fig. 6(a)], which was mounted in the cabinet of Fig Fig. 6(b) shows a set of impulse responses (for voltage excitations) of the driver cone. It can be noticed how the responses are more delayed in positions outside and inside the voice coil due to the limited propagation speed of vibrations. This implies that in vibroacoustic modeling, especially while computing the externally radiated sound field and directivity pattern, a piston model is not accurate. Magnitude responses, normalized by the response at the 30-mm position, are plotted 1 The enclosure modes affect the measured cone vibration as a response to electrical excitation, but since applying a detailed driver model was beyond the scope of the study, this effect was not investigated. PAPERS in Fig. 6(c). Up to almost 1 khz the magnitude responses are almost flat, except toward the edge of the diaphragm. 4 ANALYTICAL AND SEMIANALYTICAL MODELING An accurate analytical solution of coupled vibroacoustic equations for a loudspeaker is out of the question. Yet it is possible to try a simplified and approximate solution, especially at relatively low frequencies. In this section we will try this approach since the loudspeakers in our study have a relatively regular shape. 4.1 Analytical Modeling Techniques The first approximation of a closed loudspeaker enclosure is obtained when the walls of the enclosure are considered to be rigid. The driver is modeled as a simple piston with a given velocity. The volume inside the enclosure is denoted by. Its boundary, the walls of the enclosure, are denoted by 1 2, where 2 is the surface of the piston and 1 refers to the other wall surfaces. The spatial variable (three-dimensional position) is denoted by x, and the frequency f is given by the angular frequency ω 2πf. The pressure field p(x, ω) inside the enclosure is given as a function of frequency by the solution of the Helmholtz equation [33, chap. 6], 2 p + k 2 p = 0 (1) (a) (b) (c) (d) Fig. 4. Examples of responses inside enclosure. (a), (b) Impulse responses from electric excitation of driver element to sound pressure at microphone mesh point r14c5 (see Fig. 2), 120 mm from back plane. (a) Empty enclosure, in sand. (b) Damped enclosure with 100- mm mineral wool, in sand. (c), (d) Magnitude responses in same position for enclosure in sand and freestanding. (c) Undamped. (d) Damped. Levels offset for clarity J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

6 LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES (a) (b) (c) Fig. 5. Results of vibration measurements. (a) Accelerance (acceleration/force) of isolated side plate with impact test at plate corner. (b) Accelerance of side plate of freely supported undamped enclosure at point 260 mm from front panel and 297 mm from top plate (driver side). (c) Side plate velocity at same point, as a response to electrical excitation of driver element. (a) (b) Fig. 6. Velocity responses measured on diaphragm surface. (a) Dimensions of diaphragm and measurement points. (b) Velocity impulse responses (voltage excitation) at different points. (c) Magnitude responses at some points normalized to point at 30 mm. J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1153 (c)

7 KARJALAINEN ET AL. PAPERS with the boundary conditions p n = 0, x 1 p n = iρ 0ωv n, x 2 Here k = ω/c is a wave number, c = 343 m/s is the speed of sound in air, and n is the normal of the boundary pointing away from the fluid. In boundary conditions p/ n is the directional derivative of the pressure in the direction of the normal n. In the boundary condition for the piston area, i is the imaginary unit, ρ 0 = 1.21 kg/m 3 is the density of air in the equilibrium state, and v n (x, ω) is the velocity of the piston in the direction of its surface normal n. In this case the velocity v n is considered to be constant in 2. This boundary-value problem can be solved using the Green s function G ω, which is the solution of the equation [33] (2) 2 G ω + k 2 G ω = δ(x x 0 ) (3) with the homogeneous boundary conditions G ω n = 0, x. (4) Here δ(x x 0 ) is the Dirac delta function and point x 0 is considered a source point. Using the eigenfunctions ψ N and the eigenvalues k N of the Helmholtz equation, Eq. (1), with the boundary condition p/ n = 0, the Green s function can be expressed as a series [33, chap. 9.4], G ω (x, x 0 )= N=0 ψ N (x)ψ N (x 0 ) ψ N 2 [k 2 N (ω/c)2 ]. (5) For a rectangular enclosure with dimensions l x l y l z the eigenfunctions are of the form [33] ψ nx n y n z (x) = cos(k x x) cos(k y y) cos(k z z) (6) where k x = n x π/l x, k y = n y π/l y, and k z = n z π/l z. The coefficients n x, n y, and n z are nonnegative integers, creating triplets of numbers, which are used to index the eigenfunctions and the corresponding eigenvalues, ( ) 2 ( ) 2 ( ) 2 kn 2 nx π ny π nz π xn yn z = + +. (7) l x The solution of the boundary-value problem given by Eqs. (1) and (2) can be obtained by using the integral equation [33] p(ω, x) = l y l z simplifies to the form p(ω, x) = iωρ 0 G ω (x, x 0 )v n (ω, x 0 ) ds 0. (9) 2 In order to obtain numerical results from these equations, the Green s function was approximated by using only a finite number of terms in the summation. The form used for the Green s function was G ω (x, x 0 ) = (10) Since only a finite number of terms were used in the summation, the integration in Eq. (9) could be taken inside the summations. The coordinate system was chosen so that the enclosure was in the positive octant of the coordinate space and that the corner of the enclosure was in the origin. The driver element, modeled as a piston, was placed on the wall that lies on the xy plane. The dimensions of the enclosure studied were as shown in Fig. 1. The center of the circular piston element was at the point x e = 125 mm, y e = 125 mm, and z = 0 (the rim position). The radius of the piston was r = 75 mm. The actual numerical computations were made using MATLAB. The integrations over the diaphragm area, considered as a flat piston surface, were computed numerically using MATLAB s quad8 function. The transfer function between the piston velocity v n (ω) and the pressure at some measurement point x m can be computed by dividing computed pressure p(ω, x m ) by velocity v n (ω). In the case studied here, the frequencydependent part of the velocity can be taken out of the integral in Eq. (9), so the transfer function can be obtained from this equation by setting v n (ω) = 1. The measurement point chosen was 120 mm from the back plate, at r12c2 in the microphone mesh. The frequency response of the computed transfer function is compared in Fig. 7 with the corresponding measured response. The overall fit is good, except for the damping of resonances since in the analytical model the walls were assumed totally rigid. Other minor deviations could be reduced by adjustments of interior measures of the enclosure, except at higher frequencies above 700 Hz, where for example the piston assumption of the driver diaphragm movement is no longer valid. (The deviation at very low frequencies is due to the highpass characteristics of the measurement microphone.) [ f (ω, x 0 )G ω (x, x 0 ) dx 0 + G ω (x, x 0 ) p 1(ω, x 0 ) p 1 (ω, x 0 ) G ] ω(x, x 0 ) ds 0. (8) n 0 n n x =0 n y =0 n z =0 ψ nx n y n z (x)ψ nx n y n z (x 0 ) ψ nx n y n z 2 [ k 2 n x n y n z (ω/c) 2 ]. Here the notation ds 0 has been used to emphasize that the boundary integration is also performed with respect to x 0. Since Eq. (1) is homogeneous, the boundary condition for the Green s function, Eq. (4), is homogeneous, and using boundary conditions for pressure [Eq. (2)], this equation 5 VIBROACOUSTIC MODELING TECHNIQUES Numerical modeling is, in principle, a solution to any problem that can be formulated quantitatively and pre J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

8 cisely enough. There are, however, limitations to the computational resources, such as finite memory, accuracy, and computation time, which restrict the applicability of element-based numerical methods. Although they are becoming more and more relaxed with the rapid development of computer hardware, they will nevertheless remain some of the limiting factors. Another and in practice a very important restriction is the accuracy of the available material and structural parameters. The acoustic properties of absorbent materials are seldom known precisely, and even less information is available about the dynamic parameters of enclosure construction materials. These materials are not homogeneous so that the range of variations of the parameters should be known, not only the values from a single sample measurement. In this section we will present the basic principles of the techniques for element-based modeling that have been applied in our study. These methods include the finite-element method (FEM), the boundary-element method (BEM), and the finite-difference time-domain method (FDTD), especially its digital waveguide mesh formulation. 5.1 Finite-Element Method FEM [34] [36] is a popular method for solving partial differential equations (PDEs). A PDE is transformed into an integral equation, the solution domain is discretized with a mesh, and the solution is approximated at the nodes of the mesh by means of element functions Overview of FEM for Acoustic Problems In this chapter FEM is presented for solving the following PDE. The internal acoustic field of a loudspeaker box is modeled by using the inhomogeneous Helmholtz equation 2 p + k 2 p = 0, x. (11) LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES It should be noted that in this notation both Z and v are functions of place and frequency, Z = Z(x, ω) and v = v n (x, ω). Instead of looking for an exact solution of this equation and the associated boundary conditions, an approximate solution is searched for. First the weak formulation of Eq. (11) is derived [34], ( p w + k 2 pw) d ( ) ρiω Z p + ρiωv w dɣ = 0. (15) Here the so-called test function w belongs to a suitable function space V. The original PDE can now be replaced by its weak formulation. An N-dimensional subspace of V, denoted by V N, is chosen and the weak formulation of PDE is projected into this subspace. This means that is divided into finite elements and the geometry of the domain is described with the vertices of the elements. Then a basis function φ i of subspace V N is chosen for each node x i, i = 1,, N, such that φ i (x i ) = 1 and φ i (x j ) = 0, j i. The solution for Eq. (15) is approximated as p N p j (ω)φ j (x). (16) j=1 This is the Galerkin method. Because Eq. (15) is valid for all test functions w, it is also valid for the basis functions φ j. Substituting the trial Eq. (16) and the basis functions into Eq. (15), a system of linear equations to solve the unknowns p j, j = 1,, N, is obtained, N [ ( φ j φ i k 2 φ j φ i )p j d j=1 + 1 ] ρiω Z φ jφ i p j dɣ = ρiωvφ i dɣ. (17) 2 The MDF walls of the box are acoustically very hard, and they have been modeled using an impedance boundary condition. p n = ρiω Z(ω) p, x 1. (12) Because the simulation is carried out at low frequencies, the loudspeaker element can be approximated as a simple piston by means of a velocity boundary condition. p n = ρiωv n, x 2. (13) The notation i is used for the different parts of the boundary of region, also denoted by Ɣ. Here ρ is the fluid density, Z(ω) the acoustical impedance, and v n the normal velocity. From now on the boundary conditions, Eqs. (12) and (13), are combined as p n = ρiω p ρiωv, x. (14) Z Fig. 7. Analytical modeling. Magnitude responses (diaphragm velocity to sound pressure) at point 120 mm from back plate and at r12c2 in microphone mesh in sand-supported undamped enclosure. measured; computed from analytic modeling with rigid walls. J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1155

9 KARJALAINEN ET AL Matrix Representation of FEM Let us define the following acoustic mass, damping, and stiffness matrices M = (M ij ), C = (C ij ), K = (K ij ), the source vector F = (f i ), and the pressure vector P = (p i ) as K ij = φ i φ j d C ij = M ij = 1 ρ Z φ iφ j dɣ 1 c 2 φ iφ j d f i = vφ i dɣ. 2 Eq. (17) can now be expressed in matrix form, (18) KP + iωcp ω 2 MP = iρωf. (19) This is a system of linear equations which can be solved for p i, i = 1,..., N, at all frequencies of interest using standard linear algebra. 5.2 Boundary-Element Method The BEM [37], [38], [6] is another approach to solving PDEs. The PDE is transformed into an integral equation which consists of boundary integrals only. As a consequence, the three-dimensional acoustical problem is reduced to a two-dimensional one. When the problem is discretized, a system of linear equations is obtained. In addition to boundary nodes, BEM can be used to calculate the solution for Eq. (11) at an arbitrary point of region Overview of BEM for Acoustic Problems In this section a direct BEM, or collocation method, for solving the Helmholtz equation [Eqs. (11) (13)] is presented [37], [38]. Assuming that the equation has a solution in, the following integral equation is valid for all functions p* that are regular enough ( 2 p + k 2 p )p d = (p p n p ) n p dɣ. (20) Next p* is chosen to be the Green s function of the differential operator 2 + k 2. This means that p* is the solution of the PDE, 2 p* + k 2 p* = δ(η). (21) In an infinite domain, δ is the Dirac delta function. In the theory of BEM the Green s function is often called the PAPERS fundamental solution. For a three-dimensional Helmholtz operator the fundamental solution associated with point η is p η (x) = e ik η x 4π η x. (22) When some assumptions are made, Eq. (20) can be approximated at points ξ of the boundary Ɣ = as 1 2 p(ξ) + p p n dɣ = p n p dɣ. (23) This equation consists only of values of p and its normal derivative at the boundary Ɣ. It is solved in the same way as in FEM: the boundary is approximated with surface elements and the solution is searched at nodes. Let be divided into N disjoint parts Ɣ i. For simplicity, the following notation is used: Q = p n, Q i = p i n. (24) Here pi is the fundamental solution associated with node x i. How the integrals over the boundary parts Ɣ i are computed depends on the boundary elements selected. If constant elements are used, the following system of linear equations is obtained (i = 1,, N): 1 2 p(x i) + N j=1 ( p(x j ) Qi + ρiω ) Ɣ j Z p i dɣ N = ρiωv pi dɣ. (25) Ɣ j j= Matrix Representation of BEM With the notations ( H ij = Qi + ρiω ) Z p i dɣ, i j H ii = f i = Ɣ j Ɣ i N j=1 p i = p(x i ) ( Qi + ρiω ) Z p i dɣ v pi dɣ Ɣ j i, j = 1,..., N, Eq. (25) can be written in matrix form (26) HP + ρiωf. (27) Using the fundamental solution pη and the values of P, Eq. (11) can be solved at every inner point η, p(η) [ N [ ] ] 1 p(x j ) Q η dɣ + ρiω Ɣ j Z p(x j) + v pη dɣ. (28) Ɣ j j= J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

10 5.3 Coupled FEM/BEM When the coupling between the structures, that is, the loudspeaker box and the acoustic field, is taken into consideration, the situation becomes more complex. The behavior of the walls has been modeled with the following PDE of the thin plate: 4 u 12ρ 1(1 s 2 )ω 2 Ed 2 u = 0 (29) where u is the wall displacement, ρ 1 the wall material density, s Poisson s ratio, E Young s modulus, and d the wall thickness. The coupling between acoustic and structural models can be represented as a coupling boundary condition, p n = ρω2 u. (30) FEM has been used to solve Eq. (29), and this structural model has been coupled with the acoustical FEM/BEM model. The coupling boundary condition [Eq. (30)] is described with a coupling matrix T. Here a structural FEM model has been coupled with an acoustical FEM model, { K s U ω 2 M s U = TP (31) KP + iωcp ω 2 MP = ρiωf+ ρω 2 T T U where K s and M s are the structural stiffness and mass matrices and U is the structural displacement vector. When the structural FEM model is coupled with an acoustical BEM model, the following system of matrix equations is obtained: { K s U ω 2 M s U = TP (32) HP = ρiωf+ ρω 2 T T U. 5.4 Finite-Difference Schemes FDTD methods are found to be a possible solution for acoustic problems such as room acoustics simulations [39], [40]. Here we study their applicability to loudspeaker modeling. The main principle in the finite-difference methods is that derivatives are replaced by the corresponding differences [41]. There are various techniques available, but for the wave equation it is suitable to use the so-called center scheme, such that dp(t) dt p(t + t) p(t t) 2 t. (33) For this purpose the wave equation is presented in the time domain, c 2 2 p = d2 p dt 2. (34) After having applied the discretization technique of Eq. LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES (33) twice, both for space and for time, in a onedimensional case Eq. (34) results in the following form: p(x + x, t) 2p(x, t) + p(x x, t) ( x) 2 c 2 = p(x, t + t) 2p(x, t) + p(x, t t) ( t) 2 (35) where the sound pressure p is a function of both time and place. This scheme can easily be expanded to higher dimensions. Spatial dimensions may be separated and discretized individually. Thus in a three-dimensional case, similar terms, concerning spatial differences y and z, are added to the left-hand side of Eq. (35). The difference scheme in Eq. (35) is explicit. In practice it means that the sound pressure values for the next time step t + t can be calculated purely from the data of time t and earlier discrete time moments. As the finite-difference schemes are often calculated in the time domain, the results can be visualized easily and the propagation of wave fronts in the space under study are clearly seen. Fig. 8 shows an example of visualized time domain simulation. It presents a two-dimensional slice inside the enclosure of Fig. 1, 230 mm from the back plate. The excitation has been a Gaussian pulse and the driver element was located at the rim position. In the figure the primary wave front is approaching the bottom of the cabin, and behind that the first reflections from the sidewalls can be seen. Using the FDTD method it is easy to visualize the temporal evolution of the sound field inside and outside the loudspeaker cabinet. Another advantage of time-domain calculations of impulse responses is the ability to use the results directly for auralization purposes, that is, the simulation results can be listened to easily. There are also drawbacks to the finite-difference schemes. Traditionally the space discretization has been done such that the resulting elements are cube shaped in a rectangular mesh. That causes both dispersion and magnitude errors at higher frequencies. Due to that limitation the valid frequency range of the FDTD method is somewhat lower than that for the corresponding FEM. In practice for an FDTD grid at least 10 nodes per wavelength are needed Digital Waveguide Mesh Method The digital waveguide mesh method is an FDTD scheme. Its background is in digital signal processing. The method was first developed for the physical modeling of musical instruments [42]. The method is computationally efficient, and with one-dimensional systems, such as flutes or strings, even real-time applications are easily possible [43], [44]. A digital waveguide mesh is a regular array of discretespace digital one-dimensional waveguides arranged along each perpendicular dimension, interconnected at their crossing, as illustrated in Fig. 9, which represents a twodimensional digital waveguide mesh. Two conditions must be satisfied at a lossless junction connecting 2N lines of J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1157

11 KARJALAINEN ET AL. equal impedance [45]: 1) The sums of inputs equals the sum of outputs (flows add to zero), i=2n i=1 p + i = i=2n i=1 p i. (36) 2) The signals in each crossing digital waveguide line (delay line) are equal at the junction (continuity of impedance), p i = p j, for all i, j (37) where N is the model dimensionality, p + i represents the incoming signal in the digital waveguide node i, and p i is the outgoing signal in the same digital waveguide node. The actual value of a digital waveguide node is the sum of its input and output, p i = p + i + p i. (38) Since that value is the same in all delay lines connected to the node, it is also the value of the node p. The delay line between two nodes implements a unit delay, such that what goes out from a digital waveguide gets into its opposite end at the next time step, p + i (n) = p i, opposing (n 1). (39) Based on these conditions, a difference equation can be derived for the nodes of an N-dimensional rectangular mesh, p k (n) = 1 N 2N l=1 p l (n 1) p k (n 2) (40) where p represents the sound pressure at a junction at time PAPERS step n, k is the position of the junction to be calculated, and l represents all the neighbors of k. As one can see in this formulation, the incoming and outgoing signals ( p +, p ) have been eliminated and only the actual value p of a node is needed. This digital waveguide mesh equation is equivalent to a difference equation derived from the wave equation by discretizing time and space, as shown in Eq. (35). The discretization is done such that t = 1 time step, x = y = z = 1 grid unit, and the wave propagation speed is c = 1 3 x t. (41) The real update frequency of a three-dimensional mesh is f s = c real 3 (42) dx where c real represents the speed of sound in the medium and dx is the actual unit distance x between two nodes. That same frequency is also the sampling frequency of the resulting impulse response. Boundary conditions are presented as relative impedances normalized to the air such that the value 1 represents the impedance of air. Another choice for setting the boundary conditions is by using digital filters, as presented in [46]. In Fig. 9 the boundaries are filters having a transfer function H(z). When compared to FEM models, this is an advantage, especially in more complex cases where nonlinear or time-variant boundaries are needed. A more detailed study on deriving the mesh equations and boundary conditions is presented in [47]. In the digital waveguide mesh method the error caused by cube-shaped elements (as described in the previous section) can be reduced, for example, by using tetrahedral elements [48] or some interpolation technique [49]. In all Fig. 8. Finite-difference time-domain simulation of wave front inside enclosure. A two-dimensional slice at a height of 230 mm from back plate is visualized. Primary wave front is approaching enclosure bottom; behind that the first reflections from sidewalls can be seen J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

12 these structures there still remains dispersion, which can be compensated to a certain degree by the frequency warping technique presented in [50]. In this study we have used cube-shaped elements. 6 COMPARISON OF MEASURED AND SIMULATED BEHAVIOR In this section we show the results of simulating the behavior of closed-box loudspeakers using element-based numerical modeling tools and compare them with the corresponding measured behaviors. The cases of interest were a large enclosure in freestanding and sand-supported conditions, with empty and damped interiors, using MDF and PVC construction materials, and a smaller enclosure of MDF material. The focus is on the applicability of element-based modeling techniques in vibroacoustic analyses and the design of loudspeakers. The internal sound field of the closed-box loudspeaker was first simulated under various conditions with different element-based methods. 6.1 FEM and BEM The loudspeaker of Fig. 1 was simulated using FEM and BEM. Calculations were carried out using the vibroacoustic software SYSNOISE, revisions 5.3 and 5.4, on an Alpha workstation. It is often required that an element mesh have at least six nodes per wavelength. The models LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES used here should be valid at the frequencies of interest, up to about 1 khz. Both FEM and BEM simulations of an empty enclosure (Fig. 10) show accurate results in comparison with the measured behavior of the loudspeaker, except at frequencies above Hz. Magnitude responses at a point 120 mm from the back plane and in mesh position r12c2 (see Fig. 2) are shown. The enclosure was buried in sand in the case of Fig. 10, which means that the walls are effectively supported and damped. In spite of this, they have a finite impedance (nonzero absorption), and thus minor damping of the interior modes, as can be found in Fig. 4(c). Such high impedance of the supported wall material (MDF) is difficult to measure reliably. A large frequency-independent realvalued impedance was adjusted for element-based simulations to yield proper Q values for the resonance peaks. SYSNOISE simulations using both FEM [Fig. 10(a)] and BEM [Fig. 10(b)] modeling resulted in a relatively good match with measured responses up to about Hz. The driver element was modeled as a round flat piston of the size of the diaphragm, 2 the element velocity being the excitation signal. The next case was FEM modeling the same loudspeaker 2 The diaphragm was assumed to move homogeneously, although Fig. 6 shows that this is not exactly true. When simulating the interior sound field, this assumption was found to be reasonably good in the frequency range of interest. Fig. 9. Two-dimensional digital waveguide mesh consisting of one-dimensional digital waveguides interconnected at their crossings. At boundaries there are filters H(z) implementing reflection characteristics of each surface. J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1159

13 KARJALAINEN ET AL. with absorption material inside the interior of the enclosure. A 50-mm-thick layer of mineral wool was attached to the back plate of the cabinet. Although it is possible to use FEM modeling of absorbent materials in SYSNOISE, it did not yield satisfactory results, and instead the damping material has been modeled using frequency-dependent impedance boundary conditions. Fig. 11 plots the magnitude response of diaphragm velocity to sound pressure at a single interior point of the large MDF enclosure as measured and by FEM simulation with measured absorber impedance and hand-adjusted impedance to have a better match at low frequencies. The hand-adjusted case (dashed line) shows a useful fit to the measured one (dash-dot line) up to about 700 Hz, whereas simulation with the measured impedance (solid line) does not yield as good a match. Note that the absorbent was modeled as an equivalent impedance on the back plate but also as a 50-mm area on the sidewalls. When a 100-mm absorbent was used, the results with equivalent impedance boundary conditions did not work as well as in Fig. 11. This demonstrates that for enclosures with relatively much filling a simulation model is needed where each partial medium is simulated as a volume coupled to other volumes. In another study, reported in [51], the interior sound field of the same loudspeaker was modeled with the I-DEAS vibroacoustics software. FEM and BEM simulations were carried out for hard walls and coupled wall interior interaction without damping material, as (a) PAPERS well as a damped case with FEM. The results were comparable to the performance of the SYSNOISE software. 6.2 Digital Waveguide Mesh The simulations made in this study used a threedimensional digital waveguide mesh covering the interior space of the loudspeaker. Currently this method is capable of simulating only uncoupled systems, and thus only the inside sound field of the loudspeaker was simulated. The loudspeaker was modeled as a rectangular cabinet, and the element was a cylinder acting like a piston sound source. The simulations were made with a 10-mm spatial discretization, resulting in approximately mesh nodes. The simulations were carried out on an SGI Octane workstation. Fig. 12 shows the results of digital waveguide mesh simulations of the loudspeaker interior at the same point used in Section 6.1 for FEM and BEM. Boundary conditions were varied so that in Fig. 12(a) the walls of the empty loudspeaker enclosure were assigned a frequency-independent relative impedance of value 100. The results match the measured magnitude response fairly well up to about 550 Hz. Modal frequencies are well matched up to 1000 Hz. When absorption material was added, the relative impedance of the back plate of the enclosure was changed to be close to 1.0, depending on the thickness of the absorbing material. At the same time the location of the back plate was also changed such that it was at the height of the surface of the mineral wool. Fig. 12(b) shows the simulation result when a 50-mm mineral wool layer was at the back wall. The response curve shows a useful match with the measured response up to about 350 Hz. The general form of the response is similar even above this, but the computed modal frequencies tend to be remarkably higher than in the measured data. 6.3 Structural Vibrations In the next experiment the loudspeaker was modeled as a coupled vibroacoustic system of air volume and vibrating walls. The coupling effect between structure and fluid (air) was not completely modeled the vibration of a side plate in an undamped enclosure due to an interior sound (b) Fig. 10. FEM and BEM modeling of empty enclosure. Magnitude responses (diaphragm velocity to sound pressure) at point 120 mm from back plate and at r12c2 in microphone mesh as measured in sand-supported undamped enclosure. (a) FEM. (b) BEM. computed; measured response. Fig. 11. FEM modeling of damped enclosure. Magnitude responses (diaphragm velocity to sound pressure) at point 120 mm from back plate and at r12c2 in microphone mesh as measured in damped enclosure with 50-mm mineral wool. computed with measured impedance; computed with handtuned impedance; measured response J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

14 field was simulated using SYSNOISE. It was assumed that the joints of the enclosure walls are rigid (clamped). Fig. 13 illustrates the simulated and measured magnitude responses from the diaphragm velocity to the velocity at a specific point of a side panel. It can be concluded that the fit between measured and computed behaviors is reasonable but not very precise compared with interior sound field simulation. Especially above 600 Hz the model is not useful. Below this the modal frequencies are estimated quite well, but the levels of the peaks deviate more. Since the higher frequency modes do not radiate effectively, one or more of the lowest resonances are most important in practice. In this sense the modeling result can be considered useful. One serious problem in the accurate modeling of structural vibrations is to obtain accurate material parameters since often they are not available without difficult measurement procedures, and the materials may not be homogeneous. Another factor that makes modeling problematic is that the properties of the joints between wall plates, for example, when glued together, may vary and can be difficult to estimate. 6.4 Effect of Construction Materials In order to determine the effects of cabinet construction materials, a similar enclosure to Fig. 1 was constructed of PVC, then measured and modeled. Fig. 14 shows the pressure response inside the cabinet computed by BEM modeling, at a point comparable to Fig. 10(b), for the MDF LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES case. The response structure shows more details for the PVC case and the BEM model does not have as good a fit to the measured curve as in the MDF case. 6.5 Effect of Enclosure Size One more closed-box loudspeaker, smaller in size, was constructed of MDF material to study whether the modeling works also with smaller cabinet designs. The inner dimensions of the box were 250 mm 200 mm 150 mm. Fig. 15 plots the measured and computed sound pressure responses inside the cabinet at a point 50 mm from the front plate, the right-hand plate, and the bottom plate in an undamped small MDF enclosure. Modeling was carried out by the coupled BEM technique. The computed response is accurate up to about 1100 Hz. Above that it starts to deviate, probably due to the unrealistic modeling of a driver element as a simple piston. 6.6 External Sound Field To the user of a loudspeaker only the radiated sound field response is of interest. If the vibration behaviors of the driver diaphragm and the cabinet walls have been carefully simulated, the radiated response can be solved, for example, by applying BEM techniques. To obtain accurate results, the role of diaphragm modeling is critical, since the directivity of radiation should be (a) Fig. 13. Vibration modeling. Magnitude responses (diaphragm velocity to sidewall velocity) at point 260 mm from front plate and 297 mm from top plate (driver side) in undamped enclosure. computed; measured response. (b) Fig. 12. Waveguide mesh modeling. Magnitude responses (diaphragm velocity to sound pressure) at point 120 mm from back plate and at r12c2 in microphone mesh. (a) In sandsupported undamped enclosure. (b) In damped enclosure with 50-mm mineral wool. computed; measured response. Fig. 14. Coupled BEM modeling of PVC enclosure. Magnitude responses (diaphragm velocity to sound pressure) at point 120 mm from back plate and at r14c5 in microphone mesh as measured in undamped PVC enclosure. computed; measured response. Compare with MDF case in Fig. 10(b). J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1161

15 KARJALAINEN ET AL. realistic at all frequencies of interest. Typically the walls radiate efficiently only at the lowest modes, and thus they are less critical from the point of view of the external field. The measured behavior of the diaphragm velocity response in Fig. 6 reveals that the piston model is not accurate, and this was also found in preliminary simulations of external responses. The development of element models for the diaphragm and the external sound field was left for future studies. 7 DISCUSSION AND CONCLUSIONS The aim of this study was to apply element-based vibroacoustic simulations to the modeling of a closed-box loudspeaker in order to test the applicability of these methods in loudspeaker design. Simulation results are validated by comparing them with the measured behavior of a real loudspeaker. Good agreement of modeled and measured as well as semianalytic results in simple configurations, such as in Figs. 7 and 10, confirms the general validity of the approach and the measurements. First the measurement system, especially the electret microphone array, was found very useful for obtaining extensive and reliable data. These data have been stored on a CD-ROM for further experiments and analysis. The first observation is that semianalytic models may yield surprisingly accurate simulation results if the enclosure is simple enough, as shown in Fig. 7. Even more simplified approaches, such as in [52], not based on the threedimensional wave equation, are accurate in practice for the lowest frequencies. The second finding was that all element-based methods yielded accurate enough internal sound field simulations at low frequencies (below Hz) for an undamped enclosure. The anomalies at higher frequencies may be due to inaccuracies in the model parameters, the nonpiston behavior of the driver element, the relatively small size of the FEM/BEM meshes (about 50-mm spatial discretization), and the fact that the magnet of the driver was not included in the interior air space models. FEM- and BEM-based models outperformed the accuracy of the digital waveguide mesh (difference method) in these simulations. In principle the digital waveguide mesh Fig. 15. Coupled BEM modeling of small enclosure. Magnitude responses (diaphragm velocity to sound pressure) at point 50 mm from front plate, right side plate, and bottom plate as measured in undamped small MDF enclosure. computed; measured response. PAPERS should be equally accurate up to frequencies of about 5 khz. One problem was the regular mesh required by the latter method with 10-mm discretization, whereby points of computation (including the point of observation) were discretized to the nearest available point in the mesh. The simulation of a more realistic case, that of a damped enclosure [Figs. 11 and 12(b)] was not as accurate. We did not succeed in using SYSNOISE (versions 5.3 and 5.4) in a straightforward way to simulate the coupling of the air space and the damping mineral wool. The mineral wool was given as an equivalent impedance condition and its placement on only one wall required some hand-tuning of the model parameters to obtain a fairly good match to the measured sound field. The FEM/BEM yielded better results than the digital waveguide mesh method since it applied a frequency-dependent complex impedance of the damping material, whereas the digital waveguide mesh technique used a simple real-valued impedance. The vibration behavior of the enclosure plates due to driver excitation was simulated by the FEM/BEM in SYSNOISE. The simulations were encouraging, although the results are not always accurate enough and the model only partially included the vibroacoustic couplings within the system. This problem is important since the wall vibrations at low frequencies are of interest to the overall radiation of the loudspeaker. Especially the lowest mode(s) should be simulated accurately enough according to frequency and level. The effect of different cabinet construction materials (MDF versus PVC) was not large, although the more complex behavior and stronger acoustic-mechanic coupling of PVC made it more difficult to simulate accurately [Figs. 10(a) versus 14]. For a smaller MDF cabinet the simulation was accurate up to 1 khz but not above, maybe because of nonpistonic behavior of the driver diaphragm. In this study we paid attention to the vibroacoustics primarily within the closed-box loudspeaker, although the final user will be interested in the radiated sound field only. Computation of the external field is possible if an accurate model of the vibrating surfaces is available. The difficulty with wall vibrations is to make a mechanical model accurate enough, in particular knowing the parameters and couplings at joints. Since higher modes of walls do not radiate efficiently, wall vibration simulations are critical at the lowest modes only. For frontal (external) sound field computation the driver element should be modeled accurately at all frequencies of interest, and the piston model was found problematic well below 1 khz due to unrealistic directional patterns. To probe further the realism of our study, several other questions were encountered. The effect of the supporting structures, such as bracing to make the box more rigid, can be simple or complex. Some cases, such as bars between opposite walls, can be included easily in most models. The thin-plate assumption of the walls used in our models makes adding braces more complicated, although theoretically it should be possible to simulate any structure. Another issue that we found difficult was the damping material that fills major parts of the interior. It should not 1162 J. Audio Eng. Soc., Vol. 49, No.12, 2001 December

16 be too hard to include such portions of the space routinely, but we did not manage to do this with SYSNOISE (versions 5.3 and 5.4). The next large set of questions was, what are the problems with vented-box loudspeakers or other solutions of complex vibroacoustic couplings. In this study we did preliminary experiments with a bass-reflex enclosure made of the box of Fig. 1. A combination of FEM and BEM was used to enable the acoustic coupling through the port. The simulations yielded a valid general shape of response, but in detail they deviated too much to be useful in practice so that further studies are needed. As can be concluded from our experiments, there remains much work to be done to make element-based modeling a really accurate and useful tool in complex loudspeaker design. If a model is well tuned and the designer understands the complexities of its behavior, it may help, for example, with rapid checking of alternative prototypes as far as they remain within the validity range of the model, including all important parameters. Especially material data should be known accurately enough. In our project, as often happens in this kind of a study, the most important outcome was a more thorough understanding of the behavior of the loudspeakers under study. This knowledge can be utilized in further design work, even without complicated computational tools. The final question to be discussed here is the state of the art and the future of the software tools. None of the programs checked (SYSNOISE, I-DEAS) was specially tuned to loudspeaker modeling problems. Thus it required much effort to obtain the first useful results, even in simple cases. The modeling software is also very expensive, thus out of reach for potential users outside well-financed companies and university groups. The programs are not particularly easy to learn and use. They sometimes exhibit unstable or unpredictable behavior. Due to their proprietary nature, they are not open and the user may not have the possibility to know the detailed principles and assumptions the programs are built on. Computation times may also sometimes turn out to be impractical. The computational efficiency of our simulations was, however, fairly good: a typical run time was about 15 minutes for the uncoupled FEM/BEM problems and a few minutes for digital waveguide methods using an efficient workstation of the year Although even faster simulation would be desired for rapid experimentation, such a speed is fairly tolerable. As hardware and software development in general is rapid, we may expect to see faster, more user-friendly, less expensive, and more problem-domain-tailored tools in the not too distant future. Two kinds of benefits are to be expected more rapid prototyping or design of new or modified models with less physical construction, and a deeper understanding of the underlying mechanisms in loudspeakers through computational modeling. 8 ACKNOWLEDGMENT Special thanks for support and help are due to Aki Mäkivirta and Ari Varla, Genelec Oy, Iisalmi, Finland; LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES Jorma Salmi, Gradient Oy, Järvenpää, Finland; Kaarina Melkas, Nokia Research Center, Tampere, Finland; Juha Backman, Nokia Mobile Phones, Espoo, Finland; Antti Järvinen, Finnish Broadcasting Company (YLE); Jukka Linjama, VTT Manufacturing Technology, Espoo, Finland, presently with Nokia Mobile Phones; and the Technology Development Centre of Finland (TEKES). 9 REFERENCES [1] M. 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18 [47] L. Savioja, M. Karjalainen, and T. Takala, DSP Formulation of a Finite Difference Method for Room Acoustics Simulation, in Proc IEEE Nordic Signal Processing Symp. (Espoo, Finland, 1996 Sept.), pp [48] S. Van Duyne and J. O. Smith, The Tetrahedral Digital Waveguide Mesh, in Proc IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (New Paltz, NY, 1995 Oct.). [49] L. Savioja and V. Välimäki, Improved Discrete- Time Modeling of Multi-Dimensional Wave Propagation Using the Interpolated Digital Waveguide Mesh, in Proc. Int. Conf. On Acoustics, Speech and Signal Processing, vol. 1 (Munich, Germany, 1997 Apr ), pp. LOW-FREQUENCY BEHAVIOR OF LOUDSPEAKER ENCLOSURES [50] L. Savioja and V. Välimäki, Reducing the Dispersion Error in the Digital Waveguide Mesh Using Interpolation and Frequency-Warping Techniques, IEEE Trans. Speech Audio Process., vol. 8, pp (2000 Mar.). [51] A. Järvinen, Vibro-acoustic Modeling of a Loudspeaker, in Proc. Nordic Acoust. Mtg. (NAM 98) (Stockholm, Sweden, 1998 Sept. 7 9), pp [52] J. Backman, Computing the Mechanical and Acoustical Resonances in a Loudspeaker Enclosure, presented at the 102nd Convention of the Audio Engineering Society, J. Audio Eng. Soc. (Abstracts), vol. 45, p. 414 (1997 May), preprint THE AUTHORS V. Ikonen P. Antsalo P. Maijala L. Savioja A. Suutala S. Pohjolainen Veijo Ikonen was born in Ruovesi, Finland, in He is a research assistant at Tampere University of Technology, Institute of Mathematics, and is currently involved in a project of modeling the acoustics of vehicle cabins by using finite and boundary-element methods. ikonen@alpha.cc.tut.fi Poju Pietari Antsalo was born in Helsinki, Finland, in He studied electrical and communications engineering at the Helsinki University of Technology, and obtained the master s degree in acoustics and audio signal processing in He has carried out research on room acoustics and audio reproduction, particularly at low frequencies. antsalo@cc.hut.fi Panu Maijala was born in Kuopio, Finland, in He received the M.Sc. degree in 1999 from Helsinki University of Technology (HUT). He works at the Finnish Research Center, VTT, on the topics of aeroacoustics, electroacoustic transducers, outdoor sound propagation, and product sound quality. His research interests include binaural technology, digital signal processing, measurements in acoustics, musical acoustics, and psychoacoustics. He is a semiprofessional musician and plays some string, woodwind, brass, percussion, and keyboard instruments. Mr. Maijala is a member of the Audio Engineering Society and the Acoustical Society of Finland. panu.maijala@vtt.fi Lauri Savioja was born in Turku, Finland, in He studied computer science and acoustics and received the degrees of master of science in technology J. Audio Eng. Soc., Vol. 49, No.12, 2001 December 1165