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VISUALIZATION OF SOUND REFLECTION AND DIFFRACTION BY THE FINITE
DIFFERENCE TIME DOMAIN METHOD
Shinichi SAKAMOTO
Institute of Industrial Science The University of Tokyo
4-6-1 Komaba, Meguro-ku, Tokyo, 153-8505 Japan E-mail:
[email protected]
Hideki TACHIBANA
Department of Computer Science Chiba Institute of Technology
E-mail: [email protected]
ABSTRACT Visualization of sound field helps us to intuitively
understand mechanisms of various acoustic phenomena. This report
presents some examples of sound field visualization based on
numerical analysis. In order to analyze such complicated acoustic
phenomena as reflection, diffraction and interference, the finite
difference time domain (FDTD) method is applied to the calculation.
In the FDTD algorithm, all of sound field is computed in discrete
time steps and therefore transient acoustic phenomena can be easily
animated. As an application of the numerical technique to
environmental noise control, sound reflection and diffraction under
various geometrical conditions such as undulations of ground,
several types of noise barriers, embankments,
depressed/semi-underground road structures, etc. are introduced in
this report. It should be added that the physical quantities such
as sound pressure exposure level and sound intensity can be deduced
from the calculation results and therefore the method is useful
from engineering view point. KEYWORDS: Visualization, Sound
reflection and diffraction, FDTD method
INTRODUCTION
To examine and understand such complicated acoustic phenomena as
reflection, diffraction and interference, visualization of sound
fields is very effective not only from an educational viewpoint but
also from a practical engineering viewpoint in building acoustics
and noise control engineering. Recently, such numerical analysis
techniques as finite element method
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(FEM) and boundary element method (BEM) have become popular in
acoustics and they are often applied to the visualization of sound
fields. Among these numerical methods, the authors have been
investigating the application of the finite difference time domain
(FDTD) method [1,2,3] to various problems in architectural
acoustics and noise control engineering. In the FDTD algorithm, the
sound pressure and particle velocity in a sound field are directly
computed in discrete time steps and therefore transient acoustic
phenomena can be easily animated. As an application of this
numerical technique, some examples of visualization of sound
reflection and diffraction around undulations of ground, several
types of noise barriers and depressed/ semi-underground road
structures are introduced in this report.
THEORY The sound wave in a 2-dimensional sound field is
described by the following differential equations. Equations (1)
and (2) are the momentum equations in x- and y- directions,
respectively, and Eq. (3) is the continuity equation.
( ) ( ) 0,,,, =∂
∂+
∂∂
ttyxu
xtyxp xρ (1)
( ) ( ) 0,,,, =∂
∂+
∂∂
ttyxu
ytyxp yρ (2)
( ) ( ) ( ) 0,,,,,, =⎟⎟⎠
⎞⎜⎜⎝
⎛∂
∂+
∂∂
+∂
∂y
tyxux
tyxut
tyxp yxκ (3)
where, p is the sound pressure, ux and uy are the particle
velocities in x- and y- directions, respectively, ρ is the density
of the air and κ is the volume elastic modulus of the air.
The spatial and time derivatives of an arbitrary function f, xf
∂∂ , yf ∂∂ and tf ∂∂ , can be approximated by the central finite
difference forms as ( ) xxxfxxf ∆∆−−∆+ )2()2( , ( ) yyyfyyf ∆∆−−∆+
)2()2( and ( ) tttfttf ∆∆−−∆+ )2()2( , respectively. Here, ∆t is
the discrete time step, and ∆x and ∆y are the spatial intervals. By
adopting the staggered grid system with square-grids (∆x=∆y) as
shown in Fig.1, the following equations are obtained for a discrete
system.
( ) ( ) ( ) ( ){ }jipjiphtjiujiu nnnx
nx ,,1,2/1,2/1
21211 +++ −+∆∆
−+=+ρ
(4)
( ) ( ) ( ) ( ){ }jipjiphtjiujiu nnny
ny ,1,2/1,2/1,
21211 +++ −+∆∆
−+=+ρ
(5)
( ) ( ) ( ) ( ) ( ) ( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
∆
−−++
∆−−+
∆−= −+h
jiujiuh
jiujiutjipjip
ny
ny
nx
nxnn 21,21,,21,21,, 2121 κ (6)
where, ∆h is the size of the square-grid and indices n, n+1/2,
n-1/2 and n+1 denote time steps. As the initial condition, a
continuous distribution of sound pressure was set (see Fig.2).
Several types of profiles of the initial sound pressure
distribution are possible. Among them, following two types of
profiles expressed by the following equations were adopted in this
study.
-
( ) ( )( )⎪⎩
⎪⎨⎧
∆>
∆
-
Under the configuration of the impulse source and the barrier
shown in Fig. 4, instantaneous sound pressure distribution in each
time step was calculated and the sound propagation process was
visualized. The snap shots of the sound pressure distribution for
Types 1, 2 and 3 at every 16 ms are shown in Figures 5(a), 5(b) and
5(c), respectively. In these figures, the amplitude of sound
pressure is shown by monochrome shade. In Fig.5(a) for Type 1
barrier, a simple pattern of sound diffraction at the top of the
straight barrier is clearly seen. The diffraction sound makes a
circular wave front of which amplitude varies according to the
diffraction angle. On the other hand, in the cases of Type 2 and
Type 3 barriers, the sound propagation is much more complicated. In
these cases, multiple reflections occur inside the area between the
overhang of the barrier and the road surface. These multiple
reflections make secondary diffractions as shown in the figures of
48 ms and 64 ms.
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
S S S16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
16 ms
32 ms
48 ms
S S S
(a) Type 1 (b) Type 2 (c) Type 3
Figure 5 Visualization of sound diffraction over noise barriers.
Sound diffraction over embankment[7] The second example is sound
diffraction over embankments. In assessment of noise attenuation by
embankment, which is often used as
(a) Type 1(Straight wall)
H=5
m
H=5
m
W=3 m
(b) Type 2(Arc wall)
H=5
m
W=3 m
(c) Type 3(Inverse-L wall)
(a) Type 1(Straight wall)
H=5
m
(a) Type 1(Straight wall)
H=5
mH
=5 m
H=5
mH
=5 m
H=5
m
W=3 m
(b) Type 2(Arc wall)
H=5
m
W=3 m
H=5
m
W=3 m
(b) Type 2(Arc wall)
H=5
m
W=3 m
(c) Type 3(Inverse-L wall)
H=5
m
W=3 m
H=5
m
W=3 m
(c) Type 3(Inverse-L wall)
Rigid ground
Sound absorbing layer
50 m
25m
2.4 m
2.4 m
Barrier
Sound source (Receiving point)10 m (15 m)10 m (15 m)
Figure 3 Sectional shapes of barriers under investigation.
Figure 4 Calculation domain for the FDTD method.
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countermeasure against aircraft noise, the embankment is
substituted by an equivalent straight barrier and diffraction chart
derived by Maekawa is extendedly applied. The applicability of such
usage of the chart, however, is not fully examined and therefore
the authors made numerical investigation using the FDTD method. In
this report, sound propagation around embankment is focused on and
required outcome will be introduced.
Figure 6 shows cross sectional shapes of barriers and an
embankment under investigation- . Type 1 : thin wall, Type 2 :
thick barrier and Type 3 : trapezoid embankment. Sound source was
assumed to be at 10 m in front of the center of the barriers on a
rigid ground. As an example of the calculation results, snap shots
of transient sound pressure distribution at every 15 ms are shown
in figure 7. In the case of thin wall (Type 1), sound diffraction
at the top edge of the barrier is clearly seen. Circular wave front
is generated at the edge as if secondary sound source were located
at the point. In the case of thick wall, two diffracted waves are
seen and amplitude of sound wave in the shadow area decrease. On
the other hand in the case of embankment, sound wave proceeds along
the slopes of the embankment and amplitude of the diffracted sound
in the shadow area do not so much decrease.
3 m3 m3 m 3 m
2 m
3 m
2 m
3 m
2 m
30°
2 m
30°
2 m
30°30°
2 m2 m
Type 1 Type 2 Type 3
Figure 6 Sectional shape of embankments under investigation.
Sound propagation from depressed/semi-underground road[8] In
order to prevent noise propagation from highways, depressed or
semi-underground structure is used sometimes. However, noise
prediction method for this kind of road structures has not yet been
established
Figure 7 Sound propagation around barrier and embankment.
Type 1 Type 2 Type 3
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and therefore the authors are making a numerical investigation
to develop a practical calculation method applicable to these road
structures. From the results, some examples of sound field
visualization are introduced here.
The cross sections of the road structures under investigation
and the sound source position are shown in Fig.8. These are (a)
depressed road without sound absorption treatment, (b) the same one
with absorption treatment on the sidewalls, (c) semi-underground
road without sound absorption treatment and (d) the same one with
absorption treatment on the sidewalls. As the condition of sound
absorption treatment, 80% sound absorption coefficient was assumed
over all frequencies. The instantaneous sound pressure
distributions at every 50 ms after the emission of the impulse
source are shown in each series of Figures 9 (a) to (d). As for the
depressed structure, in the case where the sidewalls are assumed
reflective shown in the series of Fig.9 (a), remarkable multiple
reflections are seen between the sidewalls and strong sound waves
caused by the multiple reflections propagate intermittently outside
the road structure. On the other hand, in the case where the
sidewalls are absorptive shown in the series of Fig. 9 (b), the
multiple reflection is much reduced and consequently the subsequent
reflections after the direct sound is much diminished compared to
the former case. In the case of semi-underground structure without
absorption treatment shown in the series of Fig.9 (c), very
complicated multiple reflections inside the structure and sound
propagation through the opening are seen. In the case where the
sidewalls are assumed absorptive as shown in the series of Fig.9
(d), the multiple reflections and sound propagation outside are
much diminished in the same manner as in the former case. In the
series of Fig.9 (d), it should be noted that multiple reflection
still remains between the ceiling and the road surface.
(a) Depressed structurewithout absorption
(b) Depressed structurewith absorption
5 m
6 m
20 m
S
5 m
6 m
20 m
SAbsorption
(c) Semi-underground structure
without absorption(d) Semi-underground structure
with absorption
5 m20 m
S7.5 m
5 m6
m20 m
Absorption7.5 m
S
Figure 8 Variation of depressed or semi-underground road
structure under investigation
50 ms
100 ms
150 ms
50 ms
100 ms
150 ms
50 ms
100 ms
150 ms
50 ms
100 ms
150 ms
S S S S
(a) (b) (c) (d)
Figure 9 Sound reflection and diffraction around depressed or
semi-underground road structure.
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Sound propagation over undulations of ground When assessing
environmental noise, sound propagation on uneven ground with gentle
undulations often becomes problem. Especially in assessment of
noise radiated from construction work, the difficulty of the noise
prediction is serious and calculation model for such geography is
being discussed in Japan. In this section, sound propagation over
the undulations of ground is introduced. In this study, five types
of ground (Type 0: flat surface, Type 1 and 2 simple convex and
concave surfaces, Type 3 and 4: two cases of combination of convex
and concave surfaces) which extend over 800 m were set. Radius of
curvature was made 1,000 m for types 1, 2, 3 and 4. In this
calculation, spatial grid size became relatively large and
therefore sound pressure response has mainly low frequency
components. As an example of the calculation result, snap shot of
transient sound pressure distribution of pulse propagation is shown
in Fig. 10.
SS
SS
SS
SS
SS
400ｍ400ｍ
(Height = 1.5 m)
Radius of curvature: 1,000 m
Radius of curvature: 1,000 m
Type 0
Type 1
Type 2
Type 3
Type 4
Figure 10 Instantaneous sound pressure distribution at 1.0 s
after a impulsive source is generated
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When the ground has a concave profile as Type 1, amplitude of
propagating sound near the ground is emphasized. On the contrary,
the amplitude of sound decreases when the ground has convex profile
as Type 2. This tendency is clearly seen in Fig. 10 by the change
of the shade of the wave fronts. In the case where the ground has
successive smooth unevenness as Type 3 and 4, it is seen that the
amplitude of sound changes by position of the sound source; in Type
4 where the source is located on a bottom of concave, amplitude of
sound is weak near the ground, whereas in the Type 3 where it is on
the top of the convex, sound wave has large amplitude.
CONCLUSIONS
In this report, some examples of sound field visualization based
on the numerical calculation using the FDTD method have been
presented. These results can be demonstrated effectively by
computer-animation. Visualization technique may contribute to
intuitive understanding of acoustic phenomena –reflection,
diffraction and interference- in arbitrary sound fields and
therefore it will be very useful not only for acoustic education
but also for practical acoustic engineering. It should be added
that the detailed quantitative evaluation is also possible by
squaring and integrating the impulse responses obtained by the FDTD
method in each frequency band. Hence, the FDTD method is
effectively utilized for engineering purposes.
REFERENCES
[1] Takatoshi Yokota, Shinichi Sakamoto, Hideki Tachibana,
”Sound field simulation method by combining finite difference time
domain calcuration and multi-channel reproduction technique,”
Acoust. Sci. & Tech., 25, (1) 5-23 (2004) [2] Takatoshi Yokota,
Shinichi Sakamoto, Hideki Tachibana, “Visualization of sound
propagation and scattering in rooms,” Acoust. Sci. & Tech., 23,
(1) 40-46 (2002) [3] Shinichi Sakamoto, Takuma Seimiya, Hideki
Tachibana, “Visualization of sound reflection and diffraction using
finite difference time domain method,” Acoust. Sci. & Tech.,
23, (1) 34-39 (2002) [4] J. P. Berenger, “A perfectly matched layer
for the absorpton of electro magnetic waves,” J. Comput. Phys., 114
(1) 185-200 (1994) [5] Shinichi Sakamoto, Takuma Seimiya, Hideki
Tachibana, “Study on sound attenuation performance of noise barrier
with various sectional shapes,” Proceedings of the 2000 meeting of
the Institute of Noise Control Engineering of Japan, 141-144
(2000.9) [6] Shinichi Sakamoto, Hideki Tachibana, “FDTD analysis of
diffraction over various types of noise barriers,” Proceedings of
ICA 2004, I, 525-526 (2004.4) [7] Shinichi Sakamoto, Hedeki
Tachibana, “Numerical analysis on noise attenuation performance of
embankment,” Proceedings of inter-noise2003, in CD-ROM paper No.
N918 (2003.8) [8] Shinichi Sakamoto, Hedeki Tachibana, “Study on
simplified calculation model for prediction of noise radiation from
semi-underground roads,” Proceedings of inter-noise 2002, in CD-ROM
paper No. N634 (2002.8)
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