Holm et al.

QUANTIFYING DIFFRACTION IN TIME DOMAIN WITH FINITE ELEMENT METHODQUANTIFYING DIFFRACTION IN TIME DOMAIN WITH FINITEELEMENT METHODJUHA HOLM, AKI MÄKIVIRTA
Genelec Oy, Iisalmi, Finlandjuha.holm@genelec.com

Finite element method applied in frequency domain can show diffraction effects, but separation of the excitation fromthe diffracted wave fronts is difficult as the excitation dominates the pressure field. Time domain simulation overcomes
these problems by enabling separation of the pressure fields in time. Time domain separation of the excitation and diffracted wave fronts enable not only time domain analysis but also specific frequency domain analysis of the diffractedsound. This paper demonstrates how time domain finite element analysis can be used to simulate diffraction in loudspeakers. The method is demonstrated in three geometries where the models are excited with one cycle of Gaussian
windowed 5 kHz sine wave. The geometries demonstrate how rounded edges and a waveguide in front of the source canreduce diffraction.
1 INTRODUCTION
Historically the engineering problem of managing andminimizing diffraction effects has been tackled withphysical prototypes, empirical rules of thumb, and various simulation methods. Already in 1957 Olson [1]presented an empirical study of frequency responses forten geometries, showing that rounded and slanted enclosures have less diffraction than rectangular or cylindrical enclosures. In computational modelling, ray tracing
traditionally used in room acoustics [2] has been enhanced also for modelling diffraction effects but theresults do not always agree well with measurements [3].
Also image-source methods have been used by severalauthors [4][5] but the image-source method is only valid
for high frequencies where the wavelength is shortcompared to dimensions of the geometry [4]. Today thefinite element method (FEM) in the frequency domain is
successfully used to predict acoustic pressure and diffraction [6][7].
Frequency domain FEM analysis result can present theeffects of diffraction. Diffracted energy sums with theoriginal sound source causing frequency specific level
deviations in the pressure response. Separation of theexcitation from the diffracted energy is difficult as theexcitation typically dominates the pressure field.
This paper describes how time-domain FEM can beused to simulate diffraction. In time domain the excitation and diffracted wave front can be separated using
time windowing. FEM allows visualization of the soundfield progression in the simulated air space to improveunderstanding of the reasons and sources for diffraction
in complex geometries. This enables optimizing thegeometrical shapes before prototyping.

2 DIFFRACTION MODELIGN METHODS
A finite element method for pressure acoustics physics[8] is used for solving the problem. The precision of thetime domain solution is limited by the mesh modeling.
In this work, triangular mesh elements with maximum
element size of 11 mm are used. Two-dimensional axisymmetric simulation is used for lowering the computational cost of calculation. Post-processing allows geometries to be presented in three dimensions.
The simulated length of time is 2 ms. The simulation
time is long enough to enable the calculation of thepropagation of the exciting wave front, and the first andsecond order diffractions. The time step used in calculations is 0.8 s. A short time step is needed for simulation convergence and to achieve the required precisionof the solved pressure values.
The time domain excitation is a bidirectional pressure
pulse with one cycle of a 5 kHz sine wave, windowedwith a Gaussian time window (Figure 1). The time window is essential for the convergence of calculations.
Gaussian filtering broadens the frequency content of theexcitation.
FEA modeling calculates propagation of the pressure
pulse in the air around the enclosure. Because the bidirectional pulse has short duration, it is possible to isolate
the diffraction pressure in the time domain for typicalloudspeaker enclosure sizes, and to visualize and quantify the diffraction-related effects.
Diffraction is visualized using three types of postprocessing plots. The first is the pressure as function oftime on the acoustical axis of the source. The second is
the pressure magnitude within the simulated air space(simulation domain) at a given time. The third is fre-

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

1

Holm et al.

QUANTIFYING DIFFRACTION IN TIME DOMAIN WITH FINITE ELEMENT METHOD

quency domain analysis of the original impulse and thediffracted pressure waves. It is also possible to animate
the pressure as a function of time in the simulation domain.

The time domain method is demonstrated in three geometrically complex enclosure shapes. First geometry(Case A) is a cylinder with the exciting source at thecentre of the circular plane (Figure 4). Second (Case B)
is the cylinder with rounded edges (Figure 4). The third
geometry (Case C) is the cylinder with rounded edges
and the source located at the throat of a waveguide(Figure 4).
3Figure 1. The stimulus is one cycle of Gaussian windowed 5 kHz sine wave
Table 1. Main dimensions of the three enclosures
CaseFacediameter[cm]

Depth[cm]Rounding
radius[cm]

Waveguide
depth [cm]

A

26

24--

B

25

243-

C

25

243

1.3

Figure 4. Cylindrical enclosure andspherical air space around it (Case A).

RESULTS

3.1 Case A: Cylindrical enclosure
Initial wave front is a hemispherical because of the flat
plane on the cylinder (Case A, 0.2 ms and 0.6 ms, Figure 6). The cylindrical enclosure shows strong diffraction at the edges of the cylinder. Distance from thesound source to the edges is equal. Therefore diffracted
wave fronts sum constructively on the acoustical axis of
the source (Case A, 1.1 ms, Figure 6).
The diffracted wave fronts cause second and higher
order diffractions from the cylinder edges later (Case A,1.5 ms, Figure 6). Polarity of the first order diffraction
effect is inverted as theory suggests, and scaled in level(Case A, Figure 5). Envelope of the diffracted wave
front is similar to the initial impulse suggesting that thespectral content of the diffracted wave front remains
similar to the excitation spectrum.

Figure 4. Cylindrical enclosure withrounded edges (Case B).

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

Figure 4. Cylindrical enclosure
with rounded edges and source atthroat of the waveguide (Case C)

2

Holm et al.

QUANTIFYING DIFFRACTION IN TIME DOMAIN WITH FINITE ELEMENT METHOD

dB SPL

Case C

Case B

Case A

Figure 6. Pressure around enclosure. Three geometries.

Case A

Case B

Case C

Figure 5. Pressure at 0.2 m on-axis of source.
3.2 Case B: Enclosure with rounded edges
Diffracted wave fronts sum on the acoustical axis but
are lower in amplitude (Case B, 1.1 ms, Figure 6). The
second order diffractions can still be seen (Case B, 1.6
ms, Figure 6). Diffracted wave amplitude reduces morethan with Case A (Case B, Figure 5). Time domain
shape of the diffracted pressure wave no longer corresponds with the pulse. This suggests larger changes inthe spectrum of the diffracted pressure wave.
3.3 Case C: Rounded edges and waveguide
Addition of a waveguide affects the initial wave front
shape. The wave front is no longer hemispherical (CaseC, 0.6 ms, Figure 6). The amplitude of the wave front islower at the sides (Case C, 1.1 ms, Figure 6). This suggests the waveguide increases directivity of radiation.
Diffraction wave fronts can be seen but are low in amplitude and spread out in time. Higher order diffractions
cannot be seen with the selected amplitude scale (CaseC, 1.1 ms and 1.6ms, Figure 6).

Rounded corners and the waveguide change the envelope and magnitude of the diffracted waves. Diffractions are not visually distinguishable from the excitation
anymore (Case C, Figure 5). This suggests altered spectrum of the diffracted wave.
3.4 Frequency domain results
The time domain waveforms can be analyzed for spectral content. A simple approach is to use time windowing to separate the diffractions from the excitation. The
window function affects the frequency response in a
known way, and time-frequency resolution trade-off has
to be considered. For frequency domain analyses(Figure 7 and Figure 8) the diffracted wave has been
separated using time windowing. High frequency content in the diffraction energy is reduced by roundedcorners, as expected.

AES 51st International Conference, Helsinki, Finland, 2013 August 22–24

3

Holm et al.

QUANTIFYING DIFFRACTION IN TIME DOMAIN WITH FINITE ELEMENT METHOD
tional cost. Further limitation is posed by the soundsource model. In this work, a 20-mm dome diaphragm
was modeled by a point source. This is adequate as longas the source can be considered acoustically small compared to the wavelength. Modeling the source with arigid piston may cause problems with model convergence. More realistic models of the audio source arepossible with further increases in computational cost.
The demonstrated method can be used in simulating
also other acoustical problems requiring time domain
separation of the reflections from the excitation, such as
acoustical reflection.

Figure 7. Case A. Frequency content in the excitation(blue) and in the first order diffraction (green).

Finally, time domain FEM method presented in thispaper should be compared with finite-difference timedomain modelling (FDTD) which has also been used infinding solutions for acoustical wave propagation problems in time domain, for example by several authors insimulating the seat-dip effect [9][10].
5REFERENCES[1] Harry F. Olson, Acoustical Engineering, 1st ed.: D.
Van Nostrand Company Inc., 1957.
[2] James Porter and Earl Geddes, "Loudspeaker
Cabinet Edge Diffraction," JAES, vol. 37, no. 11,
pp. 908-918, 1989.

Figure 8. Case B. Frequency content in the excitation(blue) and in the first order diffraction (green).
4 CONCLUSIONS AND DISCUSSION
This paper demonstrates how FEM analysis in the time
domain can be used in studying diffraction. Time domain modeling allows diffraction effects to be separated
from the excitation.
Results agree with theory of diffraction. Inversion ofpolarity in the diffracted wave front is demonstrated [3].
Rounded corners of the enclosure can reduce the amplitude of the diffraction. Using a waveguide can increase
the source directivity and further reduce diffraction.
Rounded corners and a waveguide can alter the diffracted spectrum.
Diffraction products can be separated from the excitation waveform also by time windowing or deconvolution because the time-domain excitation waveform
is precisely known in advance. Deconvolution can enable separation of diffraction products when the excitation overlaps with diffraction results in the time domain.
In applying finite element analysis there is a tradeoff
between the geometry size and the highest simulated
frequency. Size of the geometry limits the length of time
domain system response that can be simulated. This
then limits the order of diffraction that can be studied.
Computational costs limit the size of the solvable problem. Three-dimensional models multiply the computa-[3] Finn T. Agerkvist, "A Study of Simple Diffraction
Models," in AES 102nd Convention, Munich,
March 1997.
[4] U.P. Svensson. (2001, Oct) Time domain modelsof edge diffraction. [Online].
http://www.tele.ntnu.no/users/svensson/ED.html[5] Ville Pulkki, Tapio Lokki, and Lauri Savioja,"Implementation and Visualization of Edge
Diffraction with Image-source Method," in 112th
AES Convention, Munich, April April 2002.
[6] Juha Backman, "Computation of Diffraction forLoudspeaker Enclosures," JAES, vol. 37, no. 5, pp.
353-362, May 1989.
[7] Juha Holm, Applying the Finite Element Method
for Modelling Loudspeaker.: Master thesis, AaltoUniversity, Finland , 2010.
[8] COMSOL, Acoustics Module User's guide 4.3.,
2012.
[9] Aki Haapaniemi, Simulation of Acoustic Wall
Reflections.: Master thesis, Aalto University,Finland, 2012.
[10] D. Mardare, and G. Soulodro J. LoVetri,"Modeling of the seat dip effect using the finitedifference time-domain method," J. Acoust. Soc.
Am., pp. 100:2204–2212, 1996.

AES 51st International Conference, Helsinki, Finland, 2013 August 22–244