PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Building FEM low frequency room models
through modal decay time measurements
Roberto MAGALOTTI1; Valentina CARDINALI2
1,2

B&C Speakers S.p.A., Italy

ABSTRACTThe treatment of weakly damped rectangular rooms by Morse and Ingard suggested the idea of using
participation factors to assess the contribution of individual surfaces – or groups of surfaces – to the decay
times of low frequency modes in rooms of generic geometry. This approach requires the numericalcomputation, through Finite Element Method (FEM) models, of specific eigenfrequencies andeigenfunctions in conditions of rigid walls.
By extracting the experimental values of modal decay times from Room Impulse Response (RIR)
measurements of different environments, including a reverberation room, new FEM low frequency models
were built by optimizing the acoustic admittance at the room boundaries. Comparison of the outcome of thesimulation models to the measured behavior shows that the technique has good predicting power and could
be used for the assessment of real boundary conditions.
Keywords: Room acoustics, Finite Element Method, Modal decay time, Acoustic admittance

1. INTRODUCTION
Finite Element Method simulations are increasingly being used in the field of Room acoustics,
thanks to the relative ease of setup and the availability of large amounts of processing power atreasonable costs. Despite that, the literature on the use of FEM models in the modal frequency band ofsmall rooms appears to be scarce.
The main difference in the characterization of close environments above or below the Schroeder
frequency is that, in the second case, one needs to switch from a single reverberation time based on
statistical considerations to a whole set of modal decay times. This set of values, associated with therelevant eigenfrequencies and eigenfunctions, creates a sort of “fingerprint” of the room at low
frequencies.
This work, inspired by Morse and Ingard’s treatment of weakly damped rectangular rooms (1), tries
to bridge the gap between modal behavior measurement and FEM simulations.

2. THEORY
2.1 Weakly damped room
An important issue in the modeling of a weakly damped room in the modal frequency band isfinding relations between the boundary conditions at the walls and the decay times of the modalresonances.
In the case of small wall admittance and generic room shape, the modal damping can be calculated
starting from the corresponding system of eigenfunctions for rigid walls. The rigid wall eigenfunctions
IN , solutions to the Helmholtz equation, are frequency independent and associated to real eigenvalues.
They consist of an orthogonal set with the following normalizationන ߶ே ߶ெ ൌ ܸȦ଴ே ߜேெ(1)௏12

rmagalotti@bcspeakers.com
vcardinali@bcspeakers.com4081

where V is the room volume and the quantityȦ଴ே ൌͳන ߶ ଶܸ ௏ ே(2)is proportional to the energy stored in the single modal component. When a small wall admittanceis turned on, following (1, pg. 562) we can apply a variational procedure to estimate the neweigenvalues from the full rigid wall eigensystem, thus obtaining a modal damping proportional to theimaginary part of the corresponding new complex eigenfrequency (or modal wave vector KN). To afirst order approximation, the result is‫ܭ‬ே؆ఎಿ െ݅ଶ଴ ෍ ߚ௝ න ߶ேʹܸȦ ேௌೕ(3)௝

where {K N, IN} are the rigid wall eigenfrequencies and eigenfunctions, while the j indexes the roomwalls, each with an associated average specific admittance and surface area {E j,Sj }. For the sake ofsimplicity, we suppose the E j to be real and frequency independent, a condition seldom encountered inpractice. Let us write this result in a different fashion‫ܭ‬ே؆ఎಿ െ݅෍ ߳ேೕ ߚ௝ ܵ௝ʹܸ(4)௝where߳ேೕ ൌ ‫߶ۃ‬ே ଶ ΤȦ଴ே ‫ۄ‬ௌೕ(5)and the notation ‫ۄڄۃ‬ௌ indicates surface average. Each participation factor HNj is proportional to thefraction of energy lost (per unit time and per unit area) by the mode N through the surface S j .
2.2 Rectangular room
It is well known (1,2) that in the case of a rectangular room the set of eigenfunctions is߶ே ൌ ߨܰ௬ ‫ݕ‬ߨܰ௫ ‫ݔ‬ߨܰ௭ ‫ݖ‬

‫ܮ‬௬
‫ܮ‬௫
‫ܮ‬௭

(6)

Ȧ଴ே ൌ ͳൗቀ߳ேೣ ߳ே೤ ߳ே೥ ቁ(7)ܰ௫ ǡ ܰ௬ ǡ ܰ௭ ൌ0,1,2,…

(8)

where HNj = 1 if Nj = 0 and HNj = 2 if N j > 0.
It can be easily verified that‫߶ۃ‬ே ଶ ‫ۄ‬ௌೣ ൌͳ߳ே೤ ߳ே೥(9)where S x is the surface of a wall perpendicular to the x direction, with similar formulas for the y andz directions.
It is now straightforward to see that the participation factorsHNj introduced in equation (5) are thegeneralization, in the case of generic room shape, of the normalization factors of the rectangular room.
Indeed, these factors are responsible for the correct evaluation of the total energy stored in each mode.
As it is explained in (1, p. 572) referring to the rectangular room:
The reason for the factors H1 is that, to first order, the mean energy contained inthe mode (0;N y ;Nz), for example, is twice that contained in a mode (N x > 0;N y ;Nz),because the mean square of 1 is 1, whereas the mean square of cos( SN xx/L x) is 1/2.
Since the rate of withdrawal of energy from the x walls is the same in both cases, thefractional rate of withdrawal when N x = 0 is half that when N x > 0.

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2.3 Modal decay times
It is customary to define a modal decay time MT 60 , in analogy to reverberation time, as the timerequired for the sound energy of a single mode to decay by 60 dB(3,4). Starting from equation (4), thisleads to the expression‫଺ܶܯ‬଴ǡே ൌ͸ ሺͳͲሻܸσ௝ ߳ேೕ ߚ௝ ܵ௝ܿ(10)This formula has the same structure as the classical Sabine’s equation for reverberation time, andexpresses the modal decay times based on the geometry of the room (volume and surface areas), thenormalized admittances and the participation factors HNj.
In (5) it was shown that the formula can predict the modal decay times computed by the FEM modelof a rectangular room, once the wall admittances are known. The goal of this work is testing theformula experimentally by building FEM simulations of real rooms and fitting the wall admittances tomeasured modal decay times.

3. EXPERIMENTS
The experiments described in this section were based on measurements of RIR in real environments.
The sound sources were low frequency loudspeaker systems, and the measurement technique was thesinusoidal log sweep.
For the FEM model, we adopted the strategy of dividing the internal surfaces of rooms in group s ofuniform acoustic admittance, in order to minimize the number of free parameters.
3.1

Rectangular rehearsal roomFig. 1 – Music rehearsal room. In the small images, surface groups for FEM modelingThis experiment concerns a small rectangular rehearsal room for musical instruments, made ofgypsum board walls, having an internal volume of 42 m 3 . The room was still in construction when themeasurements were performed, so there was not any kind of furniture inside, and the opening for thedoor was closed with a gypsum board slab during the measurements.
The computation of the FEM model confirmed that it is possible, in this case, to use the theoretical,
rectangular room values of 1 and 2 for the participation factors H, according to the classification ofmodes.

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Figure 2 shows the comparison of the modal decay times from measurement and from FEM
simulation.
2.5

MT60 (s)

21.5

10.5
Measurement
Simulation
060

70

80

90

100

110120Frequency (Hz)Fig. 2 – Measured and simulated modal decay times for the rehearsal room
All simulated values are within 0.5 seconds of the measured ones. More importantly, the method
appears to capture the ups and downs of the modal decay times, even when the numbers are notpredicted exactly; see for example the three MT 60 between 70 and 90 Hz or the two around 110 Hz.
3.2 Quasi-rectangular room
The second experiment concerns a larger room with concrete walls, of nearly rectangular shape,
having an internal volume of 71 m 3, used as a testing chamber at B&C Speakers. In this case, theparticipation factors had to be computed from the simulation of rigid walls eigenfunctions.

Fig. 3 – B&C Speakers testing room and surface groups for simulation

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The choice of surface groups was less obvious than in the purely rectangular case: the small vert ical
wall in the far corner had to be grouped with one of the adjacent walls. Furthermore, we noticed thatmodeling the door as a separate surface with its own admittance improved the results.
5MT60 (s)4321Measurement
Simulation
040

50

60

70

80

90

100

Frequency (Hz)Fig. 4 – Measured and simulated decay times for B&C testing room
As can be seen in Figure 4, this case shows the best fit of all between simulated and measuredvalues. Despite larger absolute values, the difference from prediction to measurement is even smaller
than in the rectangular room, with two exceptions: the decay time at 82 Hz is grossly underestimated,probably because of some feature not included in the model (for example, there are two small windows
in the room); and the axial mode at 56 Hz is not included because the measurement was very noisy.
3.3

Reverberation room

Fig. 5 – Large reverberation room and surface groups

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The last experiment involved a large reverberation room of more than 280 m 3, having a first axial
mode as low as 15 Hz (6). It has a definitely non-rectangular shape, with slanted walls and several
columns and beams inside; but we wanted to test whether the method could produce reasonable resultswith only three surface groups.
181614MT60 (s)121086Measurement

4Simulation
20
10

20

30

40

5060

Frequency (Hz)Fig. 6 – Measured and simulated decay times in the reverberation room
Figure 6 shows that the variance in modal decay times is larger than in previous cases, and themodal density grows much faster before 60 Hz; but despite that, and the assumptions andsimplifications involved, the model can still paint a fair picture of the relation between bound ary
conditions, geometry, and modal decay times.

4. CONCLUSIONS
Careful examination of the assumptions behind low frequency modal analysis in room s led to anextension to generic geometries of the participation factors defined by Morse and Ingard for the case ofrectangular rooms.
This extension has been verified first by comparison with FEM models of rectangular rooms, thenwith models of generic geometries corresponding to real environments. By fitting the boundary
conditions of numerical models, and in particular the acoustic admittance of walls, the model exhibits
good agreement with measured values of modal decay times. This allows for more realistic models ofsmall rooms in low frequency, with modal decay time profiles closely replicating the acoustic“fingerprint” of real rooms, and paves the way for using the method to assess the acoustic impedanceof partitions by measuring modal decays.

ACKNOWLEDGEMENTS
We gratefully acknowledge the support of Donato Masci and the team of Studio Sound Service S.r.l.
for the measurements in the music rehearsal box; of Massimo Garai and Dario D’Orazio from theUniversity of Bologna for the measurements in the reverberation room; of Daniele Ponteggia fromStudio Ponteggia for invaluable help in the development of the soft ware for modal decay timecomputation.

REFERENCES
1. Morse P. M., Ingard K. U., Theoretical Acoustics, Princeton University Press, New Jersey, USA, 19864086

2. Pierce A.D., Acoustics – An Introduction to Its Physical Principles and Applications, Acoustical Societyof America (1991)
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4. Mäkivirta A., Antsalo P., Karjalainen M., Välimäki V., “Low-Frequency Modal Equalization Of
Loudspeaker-Room Responses”, 111th Convention of the Audio Engineering Society, Convention Paper
5480, New York, NY, USA (2001)
5. Magalotti R., Cardinali V., “A simulation test bench for decay times in room acoustics”, 2018 COMSOL
Conference, Lausanne, Switzerland (2018)
6. D'Orazio D., De Cesaris S., Guidorzi P., Barbaresi L., Garai M., Magalotti R., "Room Acoustic
Measurements Using a High SPL Dodecahedron", 140th Convention of the Audio Engineering Society,Convention Paper 9507, Paris, France (2016)

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