Comparison of Numerical Simulation Models and Measured Low-Frequency Behavior of a Loudspeaker 4722 (P3-3)

MattiKarjalainen VeijoIkonenHelsinki University of Technology Espoo Finland Tampere University of Technology Tampere Finland

AnttiJdrvinen PanuMaijalaHelsinki University of Technology Espoo Finland Helsinki University of Technology Espoo Finland

LauriSavioja AnttiSuutalaHelsinki University of Technology Espoo Finland Tampere University of Technology Tampere Finland

JuhaBackman SeppoPohjolainenNokia Mobile Phones Salo Finland Tampere University of Technology Tampere Finland

Presented at AUDIOthe 104th Convention1998 May 16-19Amsterdam

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Thispreprinthas been reproducedfrom the author'sadvancemanuscript,withoutediting,correctionsor considerationbythe ReviewBoard. TheAES takesno responsibilityfor thecontents.

Additionalpreprintsmay be obtainedby sendingrequestandremittanceto the AudioEngineeringSociety,60 East42nd St.,New York,New York 10165-2520, USA.

Al/rights reserved.Reproductionof thispreprint, orany portionthereof, is not permittedwithoutdirectpermissionfrom theJournalof the Audio EngineeringSociety.

AN AUDIO ENGINEERING SOCIETY PREPRINT

Comparison of Numerical Simulation Modelsand Measured Low-Frequency Behavior

of a Loudspeaker

Matti Karjalainen 1, Veijo Ikonen 2, Antti J_irvinen l,

Panu Maijala l, Lauri Savioja l, Antti Suutala 2,Juha Backman 1,3, and Seppo Pohjolainen 2

1Helsinki University of TechnologyEspoo, Finland

2 Tampere University of TechnologyTampere, Finland

3Nokia Mobile Phones, $alo, Finland

matt i. karj alainen_hut, f i, i77309_cc, tut. f i, antt i. j arvinen_hut, fi

panu.maij ala_hut, fi, lauri, savioj a_hut, fi, s 129948_alpha. cc. tut. f i

j uha. backman_nmp, nokia, eom, seppo, pohj olainen©cc, tut. f i

http://acoustics, hut .fi/

ABSTRACT

The vibroacoustic behavior below lkHz of a prototype closed-box loud-

speaker has been studied in detail by comparing measurements and elementmodel simulations. Sound fields were measured using a microphone array

of 90 electret capsules and vibrations using a laser vibrometer and accel-erometers. Simulations have been carried out using analytical, finite and

boundary element, and finite difference methods. The enclosure conditions

were varied from fixed wall case buried in sand to free-standing empty box

and to free-standing damped box. Two positions of the driver in the front

plate were examined (only the 'rim' position is documented here). Theapplicability of each modeling technique is discussed 1.

1Special thanks for support are due to'.Kaarina Melkas, Nokia Research Center, Tampere, FinlandJorma Salmi, Gradient Oy, J_rvenpg_&, FinlandAki M&kivirta and Ari Varla, Genelec Oy, Iisalmi, FinlandJukka Linjama, VTT Manufacturing Technology, Espoo, FinlandTechnology Development Centre Finland (TEKES)

1

INTRODUCTION

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. Computer-basedmethods have helped in exploring basic features of driver to enclosure

matching and crossover network design. However, not much have been

published on the use of more advanced computer-based methods and tools

in loudspeaker design.

The detailed behavior of a loudspeaker consisting of an enclosure and

driver(s) is very complex and escapes analytical mathematical solutions.Approximate (semianalytic) approaches may turn out to be useful, how-ever, especially within a limited frequency range and when the geometry

of the system is simple enough, e.g., a shoebox design. At low to mid fre-

quencies, lumped element models may be used both for the electroacousticand vibroacoustic subsystems. In more complex cases the system should

be considered as a complex electro-vibro-acoustic system that is not easily

partitioned to any simple submodels.

The progress in computer-based numeric simulation of complex spatially

distributed systems using various element methods has raised the question

of how useful they might be for practical loudspeaker design [1]. 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 systemsat 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 the 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 [2], I-DEAS

Vibroacoustics [3] and Comet/Acoustics [4]. Other FEM/BEM tools that

are not specifically tuned to acoustic problems are ABAQUS [5], ANSYS

[6], and MSC/NASTRAN [7].The problem of using the FEM/BEM programs, at least from the point

of view of loudspeaker design, is that they are expensive, need powerfulcomputers to work fast, the construction of the model is tedious, and the

availability of material data (acoustic and mechanical parameters) is poor.

Experimental programs, e.g., from academic institutions, often lack docu-mentation and continuing support. Thus, such simulation and design tools

are not widely used in loudspeaker design, and information on their useful-

ness as well as comparison of their properties is practically non-existing.

As the progress in this field is fast, it is important to be prepared toutilize such tools whenever they turn out to be productive. Potentially,

computer-based design tools promise to make the product developmenttime faster whenever the designer can start from an approximate model and

rapidly go through variations and experimentations using software model-

ing up to a prototype which, when actually built, works closely enough as

expected. Even more ambitiously, the computer may automatically run

through some optimization steps to search for the best match to givenspecifications and targe t criteria.

In this study our interest was focused to the applicability of the element-based simulation and modeling tools to basic loudspeaker design. We se-

lected a case that is simple enough, yet practical and realistic. Thus we

specified and constructed a closed box enclosure with a single driver ele-ment so that it was simple to vary some interesting parameters such as the

driver position, the stiffness of the enclosure walls, and the damping mater-ial inside the box. Next the actual vibroacoustic behavior of our case was

measured extensively with such parametric variations. A microphone array

with 90 electret capsules was constructed to measure the sound field insideand outside the loudspeaker box as impulse responses of electric excitationof 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 in thisdata. The acoustical and mechanical parameters of the materials (MDF for

construction and fiber wool for damping) were also measured.

The next step of the study was to model the loudspeaker using variousFEM, BEM, and FDTD software tools. The computational models werebuilt first and the measured or estimated material parameters were given tothe models. The simulation results of the models are shown in this article

and compared with the measured behavior. Finally, the models were handtuned to match better to the measured data. This yields new material

parameters that work better than the measured ones in modeling similarcases, but may not be generalizable to very different cases. The results

of simulations and measurements are also compared to simpler analytic or

semianalytic models of the same loudspeaker.

After presenting the results we will discuss the usefulness of the models

and how they could be improved. Directions of further studies are shownas well.

I CASE STUDY: A CLOSED-BOX LOUDSPEAKER

To study a problem of reasonably low complexity, yet interesting from a

practical point of view, we designed and constructed a closed-box proto-

type loudspeaker of medium size (600 x 400 x 250 mm 3) and easy enoughto modify and measure. The structure of the enclosure is shown in the

drawings of 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 one with an asymmetric speaker positioning (rim

case), as shown in the top drawing of Fig. 1. The loudspeaker element wasa 6.5 inch driver of type SEAS P17 REX.

The enclosure was made of 20 mm MDF, all panels being rigidly coupledat their edges. In our simulations and measurements the behavior of the

box was studied both as buried in sand and as freestanding to allow wallsto 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).

1.1 Conceptual Model of the Loudspeaker

h'om a vibroacoustic point of view the loudspeaker works as follows. Thedriver element converts its electrical excitation into the movement of the

diaphragm. This has coupling to the air outside and inside the cabinet,transmitting a wave to both parts of the system. Another vibroacoustic

coupling of interest is from the interior sound field to the walls of the

enclosure, making them to vibrate and, due to this vibration, to radiateexternal sound field in addition to the radiation of the driver diaphragm.

The driver element has also a direct mechanical coupling to the front panel

and through it indirectly to all other panels of the enclosure. If the wallswere rigid, only the driver diaphragm vibration would be of interest. In

practice, however, the vibration of enclosure walls is not negligible andshould be included in detailed simulation of the system. Furthermore, the

acoustic loading of cabinet interior on the driver diaphragm movement hasan effect on its radiation to the external field. All these effects influence

the magnitude and phase response of the loudspeaker and its directivity

4

pattern. We have assumed, however, that the transmission of transversal

waves through walls that is important in sound insulation, is not prominent

in loudspeaker enclosures.

2 VIBROACOUSTIC MEASUREMENT SYSTEM

In order to be able to evaluate how realistic the numeric simulation results

of the loudspeaker case are, we decided to construct a system for extensivevibroacoustic measurements. It consists of an array of miniature micro-

phones to collect acoustic responses and a combination of a laser vibrometer

and vibration sensors (accelerometers) to probe the mechanical vibrationsof the loudspeaker system. A computer system was programmed to col-lect the sound field and mechanical vibration data in the form of impulse

responses of the loudspeaker element excitation.

2.1 Microphone Array

A microphone array of 90 small electret capsules was constructed so thatit fits to the interior of the loudspeaker cabinet, see Figs. 2 and 3. A fi'ame/

of metal tube was used to support row and column wires, spaced by 40

mm x 40 mm, as shown in Fig. 2. At each wire crossing an electret mi-

crophone (Hosiden 2823) and a cascaded diode was attached as depicted in

Fig. 3. Digitally controlled analog multiplexers were used to select one ofthe 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 micro-phone preamplifier. Thus the multiplexed microphone array can be used

to measure 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 flat within 1 dB in the most interesting

measurement range 100 Hz - 2 kHz of our study so that only the slightly

varying gains of individual capsules needed compensation.

2.2 Vibration Measurements

For vibration measurements a laser vibrometer (Polytec OFV3001) andacceleration probes were used. A mesh of 40 mm was measured, point by

point, to obtain the vibration responses of all walls. A number of points inthe 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 isolated pieces of MDF plates cor-

responding to the walls of the enclosure. Following equipment were used:impact hammer with Brfiel &: Kjeer 8200 force transducer, B&:K 4393 ac-

celerometer with two B&K 2635 charge amplifiers. HP 3565 S analyzer and

STAR-software were used for modal damping determination.

2.3 Data Acquisition System and Analysis Tools

The measurement system was based on the QuickSig signal processing en-

vironment [8], developed in the Laboratory of Acoustics and Audio SignalProcessing, Helsinki University of Technology. Impulse response measure-

ments were carried out using random phase fiat spectrum (RPFS) excitationsignal of typically 8192 samples at a sampling rate of 22050 Hz, averaged

typically over 10 repetitions. This is in practice equivalent to the more

commonly used MLS (maximum length sequence) measurements. The fre-

quency range of interest, from the viewpoint of element-based modelingin this study, is only up to 1-2 kHz. The signal-to-noise ratio of acousticmeasurements was in all conditions better than 40 dB so that its effect to,

e.g., magnitude responses is negligible.

The same data acquisition system was also used in vibration measure-

ments (except in impact testing). Further signal analysis of acoustic andvibration data was carried out in MATLAB.

3 MEASURED LOUDSPEAKER BEHAVIOR

Typical acoustic responses, as measured inside the enclosure, are shown

in Fig. 4. Subplots (a) and (b) show the impulse response from driverterminals to sound pressure in one mesh point, r14c5 (row 14/column 5),

fol' a sand-supported, (a) undamped vs (b) mineral wool damped, enclosure.

Subplot (c) shows the magnitude responses for the sand-supported and free-standing cases without interior damping, and subplot (d) the corresponding

magnitude responses for the case with 10 cm wool at the back panel.The impulse response in Fig. 4a illustrates a long ringing of interior

resonances in the undamped case. The corresponding ringing is radically

shorter in a damped case, see Fig. 4b. The same information is presented in

6

the frequency domain in subplots 4c and 4d. The former one shows the res-

onances and antiresonances in the undamped enclosure as measured in the

sand supported and free-standing case, respectively. The mode frequenciesexhibit strong resonances. The difference between these two curves is sur-

prisingly small. Only minor extra effects are introduced in the free-standing

case, such as seen around 200 Hz. The same is true also for the damped

case of Fig. 4d except that resonances and antiresonances are effectivelysmoothed out.

The vibration of an enclosure wall is characterized in Fig. 5 based on

three different measurements. Figure 5a illustrates the accelerance (accel-eration/force) of an isolated wall panel near corner, as measured by impact

testing. This information can be used to estimate the parameters of MDF

for vibroacoustic modeling. Figure 5b shows the corresponding behavior

when the side wall (600 mm x 400 mm) is excited at point 260 mm fromfront panel and 297 mm from top plate. This can be compared further withthe velocity of the same point as response to an electric excitation of the

driver element (Fig. 5c). Only the lowest mode (200 Hz) has prominenteffect to external sound field radiation.

4 ANALYTICAL AND SEMIANALYTICAL MOD-ELING

An accurate analytical solution of coupled vibroacoustic equations for a

loudspeaker is out of question. Yet it is possible to try a simplified and ap-proximate solution, especially at relatively low frequencies. In this section

we will try this approach since the loudspeaker in our study has a relativelyregular 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 element is modeled

as a simple piston with given velocity. The volume inside the enclosure

is denoted by fL Its boundary, the walls of the enclosure, are denoted by

c912-- 0_iUc0f_2, where 0f22 is the surface of the piston and cO_1 refers to theother wall surfaces. The spatial variable is denoted by x and the frequency

f is given as angular frequency w -- 2_r/. The pressure field p(x,w) insidethe enclosure as a function of frequency is given by the solution of the

7

Hehnholtz equation [14, Chapter 6]

V2p + k2p = 0 (1)

with boundary conditions

- 0, x & cqf_lOn --

___ (2)On -- -ipowVn, x E Of_2

Here k = _- is a wave number, c = 343 m/s is the speed of sound in air andc

n is the normal of the boundary pointing away from the fluid· In boundary

conditions _ is the directional derivative of pressure in the direction of the

normal n. Ill the boundary condition for the piston area, i is the imaginary

unit, po = 1.21 kg/m 3 is the density of air in equilibrium state and Vn(X,w)is the velocity of the piston in the direction of its surface's normal n. In

this case the velocity vn is considered to be constant in 0fl2. This boundaryvalue problem can be solved using Green's function Gw, which is the solution

of the equation [14]

v_a_+ k_a_= _(x- x0) (3)with homogenous boundary conditions

OG_On =0' xeOm (4)

Here 5(x - xo) is the Dirac delta function and point x0 is considered as a

source point. Using the eigenfunctions _N and eigenvalues kN of the Helm-

holtz equation (1) with boundary condition _ = 0, the Green's functioncall be expressed as a series [14, Chapter 9.4]

a_(_,xo)= _ _N(_)_(x0)

For a rectangular enclosure with dimensions l_ × ly x l, the eigenfunctionsare of tlle form [14]

_n_._n.(x)=cos(k_x)cos(kyy)cos(kzz) (6)

where k_ = '*-/_-,ky = _-_-_and k_ = _f. The coefficients %, ny and n, are· ? y · . z

non-negative integers, creating triplets of numbers, that are used to index

the eigenfunctions and corresponding eigenvalues

k,2 n_n. (nJ_ 2 (ny_r_ 2 (nz_r_ 2=\t_/ +kty] +\_; (7)

The solution of the boundary value problem given by Eq. 1 and Eq. 2 can

be obtained by using the integral equation [14]

=f /( ,xo)a (x,xo)dxo+

pl( ,x0j ]ds0 (8)Since Eq. 1 is homogenous, the boundary condition for the Green's func-

tion (4) is homogenous and using boundary condition for pressure (2), thisequation simplifies to the form

p(w, x) =/orb -iwpoGw(x, xo)vn(w, xo)dso (9)

In order to obtain numerical results from these equations, the Green's func-

tion was approximated by using only a finite number of terms in the sum-mation. The form used for the Green's function was

G (x,xo)= E 2 _ 2

Since only a finite number of terms was used in the summation, the in-

tegration in Eq. 9 could be taken inside the summations. The coordinatesystem was chosen so that the enclosure was in the positive octant of the

coordinate space and that the corner of the enclosure was in origin. The

element, or the piston, was placed on the wall that lies on xy-plane. Thedimensions of the studied enclosure were as show in Fig. 1. The circu-

lar piston element's centre was at point x_ = 125 mm, Ye = 125 mm and

z = 0 (the 'rim' position). The radius of the element was r = 75 mm. Theactual numerical computations were made using MATLAB. The integra-

tions over the elements, considered as a flat piston surface, were computed

numerically using MATLABs 'quad8' function.

The transfer function between the piston velocity vn(w) and pressureat some measurement point xm can be computed by dividing computed

pressure p(w, Xm) by velocity vn(w). In the case studied here, the frequencydependent part of the velocity can be taken out of the integral in the Eq.9, so the transfer function can be obtained from this equation by setting

vn(w) = 1. The chosen measurement point was 120 mm from the back

plate and r12c2 in the microphone mesh. The frequency response of the

computed transfer function is represented in Fig. 6 in comparison with

the corresponding measured response. The overall fit is good except the

damping of resonances since in the analytical model the walls were assumed

totally rigid. Other minor deviations could be reduced by adjustmentsof enclosure interior measures, except at higher frequencies above 700 Hz

where for example the piston assumption of driver diaphragm movement

is not valid anymore. (The deviation at very low frequencies is due to the

high-pass characteristics of the measurement microphone.)

4.2 Weakly Coupled Heuristics

A full description of the coupled vibroacoustic problem of an loudspeaker

enclosure using a Green's function expansion for both the acoustical wave

and the mechanical bending waves is numerically very inefficient due to the

large number of terms needed. However, some heuristic deductions enableus to achieve a very efficient approximation for the effect of coupling of the

lowest modes, which typically are also the only ones which are of importance

when studying the sound radiation by the loudspeaker enclosure [9, 10].Here we take as our aim to develop a first-order approximation of the

vibration, which implies the following assumptions about the mechanisms:

· The starting points for the calculations are the acoustical modes in a

rigid enclosure with locally reacting walls and mechanical modes of anenclosure in a vacuum.

· The vibrational properties of the driver (diaphragm mass, suspensioncompliance, losses, etc.) can be taken into account as a variation of

the impedance of the surface, as included in the boundary conditionsin the discussion above.

· The driving mechanism for the acoustical modes is a volume velocitysource corresponding to the driver and for the mechanical modes a

point force corresponding to the recoil of the driver.

· The coupling between acoustical and mechanical vibrations is assumed

to be weak, so only first-order coupling (acoustical --->mechanical,mechanical -+ acoustical) is taken into account. The mechanical vi-

brations of adjacent panels must be assumed to be strongly coupled.

· The field outside the enclosure can be ignored for the following reas-

ons: outside sound pressure on the surfaces is at least one order of

10

magnitude smaller than the pressure inside at the eigenfrequencies,and there is less frequency and place dependence in the field.

The bending wave equation applicable to the enclosure walls, assumedto consist of thin plates, is of the form

O2u EK 2 c94u

69t2 = P OX4 (11)

where E is the modulus of elasticity and K 2 a geometry-dependent coeffi-cient.

In a finite rectangular plate the boundary condition which is of interest

when analysing the acoustically excited vibrations is the simply supportededge which can rotate, but cannot have transverse displacement; the other

possible boundary condition, clamped edge, where the edge cannot haveeither rotation or transverse displacement has a significantly higher mech-

anical impedance, so neither the mechanical force nor the sound pressure

excite these modes as efficiently.

The simply supported boundary condition, which yields the lowest res-

onance frequency, corresponds to a situation where adjacent enclosure sur-faces move to opposite directions, thus enabling the rotation of the edge,

and there is no torque on the edge. A clamped boundary condition requires

that there is something to provide the torque needed to prevent the edge

from rotating, which is possible only if the adjacent surfaces moves in the

same direction (Fig. 7).The modes corresponding to the clamped boundary conditions also have

significantly higher frequencies than the simply supported modes, the lowest

modal frequency of completely clamped plate being about twice that of asimply supported plate [11]. The situation where clamped modes were

the most appropriate description would be the one where the enclosurevibration is driven by a homogeneous pressure field or by a purely one-

dimensional standing wave, but this situation arises only at low frequencieswhere there are no clamped eigenfrequencies.

In the following discussion we assume the enclosure to be rectangular.

For simply supported edges the modal shapes are given by equation

= A sin (m + X)_x sin (_ + 1)_y (12)L_ Ly

where z is the displacement and Lx and Ly are plate dimensions and m and

11

n non-negative integers. The eigenfrequencies are defined by equation

0.453 cLh[\ L_ / +, Ly } L_Ly

where h is the thickness of plate and cz is the speed of longitudinal wavein the plate given by equation

i E (la)CL= p(1--Y2)

where l_ is the Poisson coefficient. Thus, even if the modal shapes of thesimply supported plate are similar to those of the acoustical modes of a

rectangular cavity, the relationships between modal frequencies are funda-

mentally different, and the modal density at low frequencies (i.e., in the firstoctaves above the lowest mode) is very low as compared to the acousticalmode density.

The assumptions made of the modal shape have implications on thecoupling between various modes. If losses are small, thc vibration can be

efficiently transmitted from one enclosure surface to another at the edge

only if modal indices corresponding to the wave component along the edges

are equal. Similar orthogonality is valid also for the coupling of mechan-ical modes corresponding to the simply supported edges and the acoustical

modes. Bending modes corresponding to other edge boundary conditions

need also hyperbolic functions for their description, so their inner productwith the acoustical modes is non-zero also for unequal modal indices, so all

the modes have some, although small, coupling. However, as stated earlier,

the practical significance of these modes is small.The plane wave decomposition for the sound field inside the enclosure

enables the enclosure surface vibrations to be described as a superposition

of vibrations excited by plane waves. To achieve this we must determine the

impedance of each mode and the force distribution caused by the incident

wave. The vibration velocity can be then formally written as their quotient

[12]:

m Zm

The modal impedances can be determined by writing the standing wave

as a superposition of travelling waves in opposite directions, and using thisdecomposition to describe the modal impedance as a sum of travelling-wave

impedances. The exiting force is given by the sound pressure.

12

The importance of the orthogonality of the coupling is that instead ofusing the full Green's function expansion for both the acoustical and mech-

anical modes, it is necessary to only determine the modal frequencies of

the bending wave modes with the modal indices corresponding to the mostsignificant acoustical modes.

Another interesting problem, discussed here only briefly, is the effect

of the coupling to the modal frequencies. An analysis of losslessly coupled

simple vibrating systems [13] indicates that coupling increases the frequencydifference between the two resonators. Similar results hold also for systems

consisting of coupled acoustical and mechanical standing waves. The coup-

ling strength, which determines the amount of change in the resonancefrequencies, can be determined from the bandwidths and initial frequencydifferences of the resonances.

5 VIBROACOUSTIC MODELING TECHNIQUES

Numerical modeling is, in principle, a solution to any problem that can

be formulated precisely enough. There are, however, limitations of compu-tational resources such as finite memory, accuracy, and computation time

that restrict the applicability of element-based numerical methods. Al-

though they are becoming more and more relaxed with rapid development

of computer hardware, yet they will remain one of the limiting factors.Another and in practice a very important restriction is the accuracy of

available material (and structure) parameters. Acoustic properties of ab-sorbent materials are seldom known precisely, and even less information is

available about the dynamic parameters of enclosure construction materi-

als. These materials are not very homogeneous so that the variation range

of parameters should be known, not only the values from a single samplemeasurement.

In this chapter we will present the basic principles of techniques forelement-based modeling that have been applied in our study. These meth-

ods include the finite element method (FEM), the boundary element method

(BEM), and the finite difference time domain method (FDTD), especiallyits waveguide mesh formulation.

13

5.1 Finite Element Method in Acoustics

The finite element method (FEM) is a popular method for solving partial

differential equations (PDEs). A PDE is transformed into an integral equa-tion, the solntion domain f is discretized with a mesh and the solution is

approximated at the nodes of the mesh by means of element functions.

5.1.1 Derivation of FEM

In this chapter a finite element method for solving the following PDE ispresented. The internal acoustic field of a loudspeaker box is modeled by

using inhomogeneous Helmholtz equation (16). The MDF walls of the boxare acoustically very hard and they have been modeled using an impedance

boundary condition (17). Because the simulation is carried out at low

frequencies, the loudspeaker element can be modeled as a simple piston by

means of a velocity boundary condition (18).

V2p + k2p = O, x _ f (16)Bp -piw

On - Z(w) p' x _ Of! (17)BpOn -- piwv,, x E 092 (18)

Here p is fluid density, Z(w) acoustical impedance and v, normal velocky.

From now on, boundary conditions (17) and (18) are combined as

Bp -piwcon-- Z 'p- pi_vv' p e Of (19)

It should be noted that in this notation both Z and v are functions of place

and frequency, Z = Z(x,_v) and v = v,(x,_v). Instead of looking for anexact solution of this equation and the associated boundary conditions an

approximate solution is searched for. First the so called weak formulation

of Eq. 16 is derived [15]: both sides of the equation are multiplied with an

arbitrary test function w _ V, where V is a suitable function space, in fact

H _(fi) [17], and then the equation is integrated over _:

+ = 0 (20)After Green's theorem [17] is applied

14

The boundary condition can now be taken into account, which leads to anintegral equation

/pice )fa(-Vp. Vw+k2pw)d_-]oa_-_-p+icov wdF=0 (22)

The original PDE can now be replaced by its weak formulation. An N-

dimensional subspace of V, VN, is chosen and the weak formulation of PDE

is projected into this subspace. This means that f_ is divided into finite

elements and the geometry of the domain is described with the vertices of

the elements. Then a basis function _biof subspace VN is chosen for each

node xi, i = 1,..., N, such that _i(xi) = 1, ¢i(xj) = 0, j _ i. The solutionfor Eq. 22 is approximated as

N

p _ Z pj(ce)_j(x) (23)j=l

This is the Galerkin method. Because Eq. 22 is valid for all the test

functions v, it is also valid for basis functions qbj. Substituting the trial Eq.23 and basis functions into Eq. 22, a system of linear equations to solve

the unknowns pi, (j = 1,..., N) is obtained:

N

Z [fo(-V(pj,j)vi+k2(pj j) ,)dig]j=l

- \zpj_j + =j=l

and after simplification

j=l_ [f_(V_j. _i--k2(/_j_i)pjdig "]- fo_l Piceq_JqJiPjdPz ]

= fOa2 icevq_i dr (25)

5.1.2 Matrix representation of FEM

Let us define the following acoustic mass, damping and stiffness matrices

M = (M/j), C = (C/j), K = (Kij), source vector F = (fi) and pressurevector P = (Pi) respectively as:

= fa V4i. V¢j digKij

15

Cij = _al P2_i_j dF (26)

fi = faa2 v_i dF

Eq. 25 can now be expressed in a matrix form

NP + iwCP - co2MP = ipwF (27)

This is a system of linear equations which can be solved for pi, i = 1,..., N

on all frequencies of interest using standard linear algebra.

5.2 Boundary Element Method

The boundary element method (BEM) is another approach to solving PDEs.The PDE is transformed into an integral equation which consists of bound-

ary integrals only. As a consequence, the three-dimensional acoustical prob-

lem 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. 16 in an arbitrary pointof region a.

5.2.1 Derivation of BEM for Acoustic Problem

In this chapter a direct boundary element method, or collocation method,

for solving the Hehnholtz equation (16-18) is presented [18], [19].

Assuming that the equation has a solution in a, the following integralequation is valid for all functions p* regular enough:

fn(V2p+ k2p)p*da = 0 (28)

Applying Green's theorem leads to

f (v"p + k2p)p*da

Opp,_-/ok2p ,da+ c r-fow w*dx (29)

Applying Green's theorem once more results in

_o..O*

_ rjaVp. Vp* da = - j_oap_ n dP +/apV2p ' da (30)

16

Based on Eq. 29 and Eq. 30 the basic integral equation of BEM can bewritten as:

_,(POP*_nnOnnpop ,\)fa (V2p* + k2p*)pd_ = _a dF (31)

Next p* is chosen to be the Green's function of the differential operatorV 2 + k2. This means that p* is the solution of the PDE

V2p * + k2p* = -507 ) (32)

in an infinite domain, 5 is the Dirac delta function. Because of the proper-ties of the delta function, Eq. 31 can be used to calculate a value for p, at

any point r/E _h

_,(POP*_nn_nnOpp*hp(v)= - / (33)In the theory of BEM the Green's function is often called the fundamentalsolution. For three-dimensional Helmholtz operator the fundamental solu-

tion associated with point r/is

e-lkll_-xll

p;(x) - 4_11v _ xl[ (34)

It can be assumed that p is almost constant in a small neighbourhood of the

boundary point _, Uc(_), and the left side of Eq. 31 can be approximated:

ffi(Vp*( )+ k2p*(_) )p(_) df_ _ C(_) (35)

where

¢(_) = lim_0Ju_/"(e)(V2p* (_) + k2p*(_))p(_) d_ (36)

If the boundary is smooth enough, it can be proved that C(_) = -½p(_).At points _ of boundary I' = 09 can Eq. 31 be formulated as:

Op*l_p(_) + _P_n dI'= _ _-_PnP*dI' (37)

This equation consists only of values of p and its normal derivative atthe boundary I'. This equation is solved in the same way as in FEM:

the boundary is approximated with surface elements and the solution issearched at nodes. Let 0_2 be divided into N disjoint parts:

N

=Zr, (38)i=1

17

For simplicity, the following notation is used:

ap,Q, Op* (39)At every boundary point xi Eq. 37 is

12p(xi) + _ /r pQ* dF = _ fr. Qp* dF (40)j=l J j=l J

Here p* is the fundamental solution associated with node xi and Q* =op.[ respectively. It depends on the selected boundary elements how theOn

integrals over boundary parts Fi are computed. If constant elements areused, Eq. 40 becomes simpler:

N N

l_p(xi) + Zp(xJ)/r Q;dF= Z Q(x,)/r p;dF (41)j=l J j=l -/

Finally, the boundary condition (19) is substituted into Eq. 41:

N N (_pi w ),l_p(xi)+ _ p(xj)fr Q? dF = _ --_-p(xj)- piwv frjpdr (42)j=l J j=l

and the system of linear equations is obtained (i = 1,..., N):

_p(xi) N ( piw ,\ Ar+ Y_.p(xJ) fr Q_ +-_-pij dC= _-piWV/r P_ dP (43)j=l J j=l J

5.2.2 Matrix representation of BEM

With the notations (i,j = 1,..., N)

piw ,\Hij -- fr i (Q* +-_-pij dP, i _ j

Hii fPi (Q* piw ,\ 1 (44)= + -Fp,) ar +N

fi = E-vk p'dFj=l at j

Pi = p(xi)

Eq. 41 can be written in matrix form

HP = piwF (45)

Using the fundamental solution Pl and the values of P Eq. 16 can be solvedat every inner point r/6 _:

N 1 v_Ofp:dP]f'r ]P(tl) _-' - r [p(xJ) /r Q: + Piw (_p(xj) + (46)j=l

18

5.3 Coupled FEM/BEM

When the coupling between the structure, i.e., the loudspeaker box andthe acoustic field is taken into consideration, the situation becomes more

complex. FEM can be used to simulate the vibration of walls under acous-

tical excitation and this structm'al model can be coupled with acoustical

FEM/BEM model. In practice the coupling is described with a couplingmatrix T. Here a structural FEM model has been coupled with an acous-tical FEM model

/fsU - w2MsU = -TP (47)KP + iwCP - w2MP = piwF + pw2TTU

Ks and /VI_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 equationsis obtained:

KsU - w2MsU --- -TP (48)HP piwF + pw2TrU

5.4 Finite difference schemes

Finite difference time domain (FDTD) methods are found a possible solu-

tion for acoustic problems such as room acoustics simulation [20, 21]. Here

we study its applicability to loudspeaker modeling.The main principle in the finite difference methods is that derivatives

are replaced by corresponding differences [22]. There are various techniquesavailable but for the wave equation it is suitable to use the so called center-

scheme, such that

dp(t) _ p(t + At) - p(t - At) (49)dt - 2At

For this purpose the wave equation is presented in the time domain:

a 2P (50)c2V2P=After applying the previous discretization technique (shown in Eq. 49)

twice both for space and time, in a one-dimensional case Eq. 50 results

into the following form:

p(x + Ax, t) - 2p¢, t) + p(x - Ax, t) cZ = (51)Ax 2

p(x, t + At) -- 2p¢, t) + pC, t - at)At 2

19

where the sound pressure p is a function of both time and place. This

scheme call easily be expanded also to higher dimensions. Spatial di-mensions mw by separated and discretized individually. Thus in a three-

dimensional case, to the left-hand side of Eq. 51 similar terms are addedconcerning spatial differences Ay and Az.

The difference scheme in Eq. 51 is explicit. In practice it means that

the sound pressure values for the next time step t + At can be calculatedpurely from the data of time t and earlier.

As the finite difference schemes are often calculated in the time domain

the results can be visualized easily and the propagation of wavefronts in

the space under study are clearly seen. Another advantage of time domain

calculation of impulse responses is the ability to use the results directly for

auralization purposes, i.e., the simulation results can be easily listened to.

There are also drawbacks in the finite difference schemes. Traditionallythe space discretization has been done such that resulting elements are cube

shaped in a rectangular mesh. That causes both dispersion and magnitude

error at higher frequencies. Due to that limitation the valid frequency range

of the FDTD method is somewhat lower than in the corresponding FEM.In practice for an FDTD grid at least 10 nodes per wavelength are needed.

5.4.1 Waveguide Mesh Method

The waveguide mesh method is an FDTD scheme. Its background is in

digital signal processing. The method was first developed for physical mod--eling of musical instruments [23]. The method is computationally efficien_

and with one-dimensional systems, such as flutes or strings, even real-time

applications are easily possible [24, 25].

A waveguide mesh is a regular army of discrete space digital 1-D wave-guides arranged along each perpendicular dimension, interconnected at

their crossings as illustrated in Fig. 8 which represents a two-dimensional

waveguide mesh. Two conditions must be satisfied at a lossless junction

connecting 2N lines of equal impedance [26]:

1. the stun of inputs equals the sum of outputs, (flows add to zero),/=2N /=2N

E = E p;- (52)i=1 i=1

2. the signals in each crossing waveguide are equal at the junction, (con-tinuity of impedance).

Pi = Pi, Vi, j (50)

20

where p/+ represents the incoming signal in the digital waveguide i and pi isthe outgoing signal in the same waveguide. The actual value of a waveguide

is the sum of its input and output.

Pi = P+ + Pi (54)

Since that value is the same in all waveguides connected to the node this

value is also the value of the node p. The digital waveguide between two

nodes implements a unit delay, such that what goes out from a waveguide

gets in to its opposite end at the next time step.

P_ ('_) = P_,opposin_(n - 1) (55)

Based on these conditions a difference equation can be derived for the nodes

of an N-dimensional rectangular mesh:

1 2N

pk( )= - l) - - 2) (56)where p represents the sound pressure at a junction at time step n, k is the

position of the junction to be calculated and I 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 isneeded.

This waveguide mesh equation is equivalent to a difference equationderived from the wave equation by discretizing time and space as shown in

Eq. 51. The discretization is done such that At = 1 time step, Ax = Ay =

Az = 1 grid unit and the wave propagation speed:

1 Ax

c- at (57)The real update frequency of a three-dimensional mesh is:

CrealV_

A- dx (5s)where c_.,at represents the speed of sound in the medium and dx is the

actual unit distance Ax between two nodes. That same fi'equency is also

the sampling frequency of the resulting impulse response.Boundary conditions are presented as relative impedances to the air

such that impedance 1 represents the impedance of air corresponding to an

opening. Another choice for setting the boundary conditions is by using

21

digital filters as presented in [27]. In Fig. 8 the boundaries are filters

having a transfer function H(z). When compared to FEM models this isan advantage especially in more complex cases where non-linear or time-variant boundaries are needed.

A more detailed study on deriving the mesh equations and boundary

conditions is presented in [28].In the waveguide mesh method the error caused by cube-shaped ele-

ments (as described in the previous section) can be reduced, e.g., by using

tetrahedral elements [29] or some interpolation technique [30]. In this studywe have used cube-shaped elements.

6 COMPARISON OF MEASURED AND SIMULAT-

ED BEHAVIOR

In this section we show the results of simulating the behavior of the closedbox loudspeaker using element-based numerical modeling tools.

6.1 FEM and BEM simulations

The internal acoustic field of the loudspeaker box has been simulated byusing both finite and boundary element methods. All calculations have

been carried out using vibroacoustic software SYSNOISE, rev. 5.3, and a

300 MHz DEC Alpha workstation. FEM models have 702 nodes and BEMmodels have 394 nodes. Both models use the same mesh but for BEM

all internal nodes have been removed. It is often required that an elementmesh should have at least six nodes per wavelength. In this case the models

used are valid up to 1100 Hz.

Both FEM and BEM simulations of an empty enclosure, Figs. 9 and10, show accurate results in comparison with measured behavior of the

loudspeaker, except at frequencies above 600-700 Hz. Magnitude responses

in a point 120 mm from back plane and mesh position r12c2 (see Fig. 2)are shown.

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 impedance boundary condition (Eq. 17).

It was necessary to use frequency-dependent impedances, which were tunedby hand. Figure 11 shows that the simulation is fairly successful with the

5 cm wool case, but the stronger damping effect of the 10 cm wool (not

22

shown here) might need another approach. In addition to the backplane ofthe cabin, the absorbing boundary condition was used on the side walls to

the height of the mineral wool.

The coupling effect between structure and fluid was not completelymodeled. The vibration of a side plate in an undamped enclosure due to

interior sound field was simulated using SYSNOISE. Three kinds of plate

support principles were tried: clamped, simply supported, and free edges.

None of these yielded results accurate enough. In section 4.2 simple support

was suggested but our simulations matched best to measurements (see Fig.

12) when the free-edge assumption was applied, which is not easily mo-tivated. The damping of structural vibrations was also found inadequate.

Thus the boundary conditions and the material parameters for the struc-tural FEM model as well as mechanical coupling from the driver element

need further attention. However, it seems possible to use coupled element

modeling techniques with these simulations.

6.2 Waveguide mesh simulations

The simulations made in this study used a three-dimensional waveguide

mesh covering the interior space of the loudspeaker. Currently this method

is capable of simulating only uncoupled systems, and thus only the insidesound 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 weremade with a 10 mm spatial discretization resulting in ca. 65.000 meshnodes. The simulations were carried out on an SGI Octane workstation.

In Fig. 13 an example of visualized time-domain simulation is shown.

The figure presents a two-dimensional slice inside the enclosure 230 mmfrom 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

wavefront is approaching the bottom of the cabin and behind that the firstreflections from the side walls 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.

Figures 14 and 15 show the results of waveguide mesh simulations of the

loudspeaker interior in the same point that was used in section 6.1 for FEMand BEM. Boundary conditions were varied so that in Fig. 14 the empty

loudspeaker was given relative impedance of the walls set to frequency-

23

independent value of 100. The results match the measured magnituderesponse fairly well up to about 550 Hz.

When absorption material was added the relative impedance of the back-plane of the enclosure was changed to be close to 1 depending on the thick-

ness of the absorbing material. At the same time also the location of the

backplane ;vas changed such that it was at the height of the surface of the

mineral wool. Figure 15 shows the simulation result when 50 mm thickmineral wool was at the back wall. The response curve shows a useful

match with measured response up to about 350 Hz.

7 DISCUSSION AND CONCLUSIONS

The aim of this study has been to apply element-based vibroacoustic sim-

ulations to the modeling of a closed-box loudspeaker in order to test theapplicability of these methods in loudspeaker design. Simulation results

are validated by comparing them with measured behavior of a real speaker.

Good agreement of modeled and measured as well as semianalytic results

in simple configurations, such as in Figs. 6, 9, and 10, confirms generalvalidity of the approach and the measurements.

First the measurement system, especially the electret microphone array,

was found very important for obtaining extensive and reliable data. Thisdata is stored on a CD-ROM for further experiments and analysis work.

The first observation is that semianalytic and simplified models may

yield surprisingly accurate simulation results if the enclosure is simple

enough, as shown in Fig. 6 and ref. [10]. The second finding was that allelmnent-based methods yielded accurate enough internal sound field simu-

lations at low frequencies (below 500-600 Hz) for an undamped enclosure.The anomalies at higher frequencies may be due to inaccuracies of model

parameters, the non-piston behavior of the driver element, the relatively

small size of FEM/BEM-meshes (5 cm discretization), and the fact thatthe magnet of the driver was not included in the models.

FEM and BEM based models outperformed the accuracy of the wave-

guide mesh (difference method) in these simulations. In principle the wave-guide mesh should be as accurate up to frequencies of about 5 kHz. One

problem with this method was the regular mesh with 1 cm discretization

whereby points of computation (including the point of observation) werediscretized to the nearest available point in the mesh.

The sinmlation of a more realistic case, that of a damped enclosure

24

(Figs. 11 and 15) were not as accurate. We did not succeed to use SYS-NOISE in a straightforward way of modeling the coupling of the air space

and the damping wool. The wool was given as an equivalent impedancecondition and its placement only on one wall required some hand4uning

of the model parameters to obtain a fairly good match to measured sound

field. The FEM/BEM method yielded better results than the waveguidemesh method since it had a frequency-dependent complex impedance of

the damping material, while the waveguide mesh technique used a simple

resistive impedance.The vibration behavior of the enclosure plates due to driver excitation

was simulated by the FEM/BEM method in SYSNOISE. Some of the sim-

ulations were encouraging although the results are not accurate enough

and the model only partially included the vibroacoustic couplings within

the system. This problem is important since the wall vibrations at lowfrequencies are of interest to the overall radiation of the loudspeaker. Es-

pecially the lowest modes should be simulated accurately enough.The internal sound field is of less interest in a closed box than in a

vented box structure since it can radiate only through walls or due to driver

cone loading. The simulation of the latter effect requires that the acoustic

impedance matching of the driver and the enclosure is known.

Finally, only the external sound field is of interest to the listener. It canbe solved with the element methods if the driver cone behavior as well as

wall vibrations are modeled accurately enough. In this study we did not try

this since the piston model of the driver diaphragm is not accurate enoughand the wall vibrations are not known precisely enough.

The computational efficiency of the simulations was fairly good: a typ-

ical run-time was about 15 minutes for the uncoupled FEM/BEM problems

and a few minutes for waveguide methods. Although even faster simula-tion would be wished for fast experimentation, such a speed is certainlytolerable.

As a conclusion, the simulations with element-based models in our study

have shown that the vibroacoustic behavior of a relatively simple loud-speaker can be simulated precisely and efficiently enough for the sound

fields. Yet some crucial behavior, especially the mechanical vibration of

walls and tile detailed behavior of the driver cone are not detailed enoughto be useful. Our next step is to develop these crucial parts further, to ex-

tend the modeling to vented box loudspeakers, and to compute the external

sound field based on this knowledge.

25

REFERENCES

[1] Bank G., and W,'ight J., "Loudspeaker Enclosures" in Loudspeaker and

Headphone Handbook, (ed. J. Borwick), Focal Press, Oxford, 1997.

[2] SYSNOISE home page: http://www, lmsintl, com.

[3] I-DEAS (SDRC) home page: http://www, sdrc. corn

[4] Comet/Acoustics home page: http://www, autoa, corn

[5] ABAQUS home page: http://www.hks, corn

[6] ANSYS home page: http://www, ansys, corn

[7] MSC/NASTRAN home page: http ://www.raacsch. corn

[8] Karjalainen M., "DSP Software Integration by Object-Oriented Pro-

gramming, A Case Study of QuickSig," IEEE ASSP Magazine, April1990.

[9] Backman J., "Effect of Panel Damping on Loudspeaker Enclosure Vi-bration,'' AES 101st Convention, Los Angeles, Nov. 8 - 11, 1996, pre-

print n:o 4395 (C-5).

[10] Backman J., "Computing the Mechanical and Acoustical Resonancesin a Loudspeaker Enclosure," AES 102nd Convention, Munich, March

22 - 25, 1997, preprint n:o 4471 (J6).

[11] Blevins R. D., Formulas for Natural Frequency and Mode Shape, VanNostrand Reinhold Company, New York, 1979, pp. 258 & 261.

[12] Fahy F., Sound and Structural Vibration, Academic Press, London1985, pp. 221 - 226.

[13] Richardson E. G., Technical Aspects of Sound, vol. I, Elsevier Publish-

ing Company, Amsterdam, 1953, pp. 3 - 4.

[14] Morse P. M., and Ingard K. U., Theoretical acoustics, Princeton Uni-versity Press, USA, 1965.

[15] Zienkiewicz O. C., and Taylor R_ L., The Finite Element Method Vol.i: Basic formulation and linear problems, 4. ed. McGraw-Hill, London,1989.

26

[16] Zienkiewicz O. C., and Taylor R. L., The Finite Element Method Vol. 2:Solid and fluid mechanics, dynamics and non-linearity, 4. ed. McGraw-Hill, London, 1991.

[17] Schatz A. H., Thom_e V., and Wendland W. L., Mathematical Theory

of Finite and Boundary Element Methods, (DMV Seminar; Bd. 15),Birkh/iuser, Basel, 1990.

[18] Brebbia C. A., Zelles J. C. F., and Wrobel L. C., Boundary ElementTechniques: Theory and Applications in Engineering, Springer Verlag,Berlin, 1984.

[19] Brebbia C. A., and Ciskowsld, R. D. (ed.) Boundary Element Methods

in Acoustics, Computational Mechanics Publications, Southampton,1991.

[20] Botteldooren D., "Finite-difference time-domain simulation of low-frequency room acoustic problems," The Journal of the Acoustical

Society of America, vol. 98, no. 6, pp. 3302-3308, 1995.

[21] Savioja L., Backman J., J/irvinen A., and Takala T., "Waveguide meshmethod for low-frequency simulation of room acoustics," in Proc. 15th

Int. Congr. Acoust., Trondheim, Norway, June 1995, vol. 2, pp. 637-640.

[22] Strikwerda J., Finite Difference Schemes and Partial Differential

Equations, Wadsworth&Brooks, Pacific Grove, CA, 1989.

[23] Smith J. O., "Physical modeling using digital waveguides," ComputerMusic J., vol. 16, no. 4, pp. 74-87, 1992 Winter.

[24] Viilim/iki V., Huopaniemi J., Karjalainen M., and Janosy Z., "Physicalmodeling of plucked string instruments with application to real-time

sound synthesis," Journal of the Audio Engineering Society, vol. 44,no. 5, pp. 331-353, 1996 May.

[25] Jaffe D., and Smith J. O., "Extensions of the Karplus-Strong plucked

string algorithm," Computer Music J., vol. 7, no. 2, pp. 56-69, 1983

Summer, Reprinted in C. Roads (ed.), The Music Machine (MIT Press,Cambridge, MA, USA, 1989), pp. 481-49.

27

[26] Van Duyne S., and Smith J. O., "Physical modeling with the 2-d

digital waveguide mesh," in Proc. 1993 Int. Computer Music Conf.,Tokyo, Japan, Sept. 1993, pp. 40-47.

[27] Huopaniemi J., Savioja L., and Karjalainen M., "Modeling of reflec-

tions and air absorption in acoustical spaces -- a digital filter designapproach," in Proceedings of IEEE 1997 Workshop on Applications of

Signal Processing to Audio and Acoustics_ Mohonk, New Paltz_ NewYork, Oct. 19-22 1997.

[28] Savioja L., Karjalainen M., and Takala T., "DSP formulation of a

finite difference method for room acoustics simulation," in Proc. 1996

IEEE Nordic Signal Processing Symp., Espoo, Finland, Sept. 1996, pp.455-458.

[29] Van Duyne S., and Smith J. O., "The tetrahedral digital waveguide

mesh," in Proc. 1995 IEEE Workshop on Applications of Signal Pro-

cessing to Audio and Acoustics, New Paltz, NY, USA, Oct. 1995.

[30] Savioja L., and V_limSki V., "Improved discrete-time modeling of

multi-dimensional wave propagation using the interpolated digitalwaveguide mesh," in Proceedings of the International Conference on

Acoustics, Speech and Signal Processing, Mfinchen, Germany, April19-24 1997, vol. 1, pp. 459-462.

28

The element is centralized on the front plate.

In the case of rim speaker, the dimensions

are according to next picture. [ _ 120,/ 1_4t _.,]

The material of the plates [ _ --'is 20mm MDF.

I \_ { \

x speaker

5/ _ 110

speaker (rim case)

The inner dimensions of the cabinet are 600mm, 400mm and 250 mm.

Figure 1: Mounting of the loudspeaker (asymmetric position) and the closed

box loudspeaker with microphone array and damping wool inside.

29

Microphone grid:

· o -_ _ _ _,_ distance between .o -_ 1J -" _ :2microphonesis 40mm r0c0

a · · _, · a 7

'4

r14c5 _metal frame single microphone

Figure 2: The structure and dimensions of the electret microphone array.

Electret microphone capsulesr ....... 2 + Bias

_- _ _ _}- _ _-;°_'_ Preamplifier

....: o "_ o-: \ o

__ _ Digitallycontrolledanalog switches

Figure 3: Principle of the electret microphone array, the multiplexer, and

the amplifier circuit.

30

0.05 a) undamped 0.05 b) damped,kl,

0 _ ......................... 0 _'

-0,05 -0.05

0 0.1 0,2 0,3 sec 0 0.1 0.2 0.3 sec

dB I I _l I I I I I I

/_\ A A A C) undamped

20 __a nd_k_ /__/_'__ /__

-20

I I I I

100 200 300 400 500 600 700 800 900 f/Hz

dB i i J i i i i i i

sand d)damped

20-_ e_

0 fr

-20

I I I I I I I I I

100 200 300 400 500 600 700 800 900 f/Hz

Figure 4: Examples of responses inside the enclosure: (a) and (b) impulse

responses from the electric excitation of the driver element to sound pres-

sure in microphone mesh point r14c5 (see Fig. 2), 12 cm from back plane

for (a) empty enclosure in sand and (b) damped enclosure with 10 cm wool,

in sand; (c) magnitude response in the same position for undamped enclos-

ure in sand and free standing; (d) magnitude response in the same position

for damped enclosure in sand and free standing.

31

0 200 400 600 800 1000 1200

I I I I

........................ z .........................

-20 _ i L

0 200 400 600 800 1000 1200

40 , I

-200 200 400 600 800 1000 1200

Figure 5: Results of vibration measurements; top: accelerance (accelera-

tion/force) of an isolated side plate with impact test at a corner of the plate;

middle: accelerance of the side plate of the freely supported undamped en-

closure at point 260 mm from front panel and 297 mm from top plate

(driver side); bottom: side plate veloc!ty at the same point, as a responseto electrical excitation of the driver element.

32

' ' I' ' I '1 I ' , I ,l

! ' ! ' i

II I II I ii ii I i!

,I i!

II

ii

0 1O0 200 300 400 500 600 700 800 900 Hz

Figure 6: Analytic modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh in sand-supported undampedenclosure as measured (solid line)and computed from analytic modeling with rigid walls (dash-dot line).

Figure 7: Lowest vibrational mode corresponding to a clamped (left) anda simply supported (right) boundary condition.

33

H(z_ ! _.r'--"_ _ ,

t t t

q

Figure 8: A two-dimensional waveguide mesh, which consists of one-

dimensional digital waveguides interconnected at their crossings. At bound-

aries there are filters H(z) implementing the reflection characteristics ofeach surface.

34

dB

o 'A .... .......... ,,............. :..:,..̂........,o 'I..._..I,...[A.._........._,.̂....._..,.l...r_/.......

........, _!1.:_ .,/il _1:i..i,,.\-20

'1" "!! l.-30 ............ -- ....

' : i (!_{'' .!! r-40.............. i........... i........ :............. !...... : .......

._o.......................i.......i.....................'ii'_/': ,'6(_(30 200' 360 4()0 500 600 760 800 9()0 HZ

Figure 9: FEM modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in sand-supported undampedenclosure as com-puted from FEM modeling (solid line) and measured (dash-dot line) re-sponse.

dB' _! _ , !

! '_: /_ - !i)_'. i' i : :o ;...._:... _ . . i......... ; ............... ! ....... i.......

.' , _ :: ,-" ,_ - -,I-.; ....... : ,I........ : ......

: . ', :r i I'\'

-20 . _.. . ..,. .......... .i ..

-_o :: !\ ! !-40 : : ......i i j....

.-_o......;............................. i...... i ...I.i ................

_B ' ' ' ' '-6 0 260 3()0 460 500 600 700 800 900 Hz

Figure 10: BEM modeling: magnitude response (diaphragm velocity tosound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in sand-supported undamped enclosure as com-puted from BEM modeling (solid line) and measured (dash-dot line) re-sponse.

35

dB _ _ _ _ _ _

-10 ......... / '...... '.il ..... '......... : ....... i ............... .........: / !i _ ! : i !

.......... / ' ..... i' ...... i........ :........ i........ i .......i ' ii i ! ! i

-20 ,. '.'_'"¢' .... i_ '"!....... i........ ' ........ '........i i" i i i i

..... ! ........ ". ' )......... i........... i........... :.........! ! _.' ' i /.^

-30..... : ..... :....._, : : : . ........ :._.. ..:

/

: ......... : ..... /::: ......... :........-40........................... :...

....... _..... i ....... _...... :..... i ........ i ........ i ...... i ........

i i i i i i i1O0 2[)0 300 400 500 600 700 800 900 Hz

Figure 11: FEM modeling: magnitude response (diaphragm velocity to

sound pressure) in point 120 mm from back plate and r12c2 in the micro-phone mesh as measured in damped enclosure with 50 mm mineral wool as

computed fi'om FEM modeling (solid line) and measured (dash-dot line)response.

_' :. .....':"' i ....i'i ....... . ...................

.... i/ :'i:" ...."'1 I' ' : I :i ' '

-30Ii.- ........... :'_" I

_:'_!....t.!?,'./! _ !¢I

'501 250 3()0 400 5()0 6()0 7()0 8()0 950 HZ

Figure 12: Vibration modeling: magnitude response, diaphragm velocityto side wall velocity, at point 260 mm from front plate and 297 mm fi'om

top plate (driver side) in undamped enclosure: computed (solid line) and

measured (dash-dot line) response.

36

.'!"....... . ....

. '..,

,'.· . .: ·... : .. ,..........; .,': '"....:. .....

"i,,' i : ..........

. .,..' ...,.".." "' ' '..', '.'.'.·.' .''......

;.. ".,....../i·i .' ......._: i :!.... ?:

· ' · '" "'" '·""''".i

! ? _............i i: ..... ......... /_o

1 ' ._' !.'....... ..:'. ." .: .........

oq...i' i.",'_?_ ..' ." .........I ..":...... ,.';;,!_f_,_ :.......... ..' /- 0.2

-l_J,' ..... :,, :7_?_:_!4 ,..' :,'.... ,.,

.'...... ;::': '::....... ,. .." ._'/0.3

.? '..:?'.....,.., ..'" '.

-0,1 ' " '.:"'. ,..../n,5

0,2

0.3

Figure 13: Finite difference time domain simulation of a wavefront insidethe enclosure. A two-dimensional slice at the height of 230 mm from back

plate is visualized. The primary wavefront is approaching the bottom ofthe enclosure and behind that the first reflections from the sidewalls can

be seen.

37

dE ..... ! !

.,o......_....l..X,........f..::_.._:,..:...,...........,:_............l_,..v,.,..,.[.....̂...' / i¥ ' ! \,_l,,: :,' : I/ _: ':I II

._o...._. .-.x:,..........i..._._:::_...,..>,.._^......:,7.......i_...I.,\.

-4o .......:.........i W!..:.........:......!...........i.(...,'--.........:..._.......

-so.........i........;.........:......................................v..:......................: i .

'6hi0''_0 i i i200 300 400 5()0 600 700 O00 O00 HZ

Figure 14: Waveguide mesh modeling: magnitude response (diaphragmvelocity to sound pressure) in sand-supported undamped enclosure in point

120 mm from back plate and r12c2 in the microphone mesh as computed

from modeling (solid line) and measured (dash-dot line) response.

dB !

0 ......... ; ................ - ...............................................

; /% : ; /'1

-lo ..... : / ........ / ....... !....... _..........;..........;.........>.': /! i i !

-2o..... :...... !",' '.'"::.......... i .........::........ ::....i ,: :_'!_:__<",_- _'_

_30

:1 /

-40 ..... !..... !.. ;......................;....................i...........

'5_ O0 i i i200 300 400 500 600 700 800 900 Hz

Figure 15: Waveguide mesh modeling: magnitude response (diaphragm

velocity to sound pressure) in point 120 mm from back plate and r12c2in the microphone mesh in damped enclosure with 50 mm mineral wool as

computed from modeling (solid line) and measured (dash-dot line) response.

38