**Visualization of sound propagation and scattering in
rooms**

**Takatoshi**** Yokota [1*], Shinichi Sakamoto [2^] and Hideki Tachibana [2, **]**

email: taka@iis.u-tokio.ac.jp

2Institute of Industrial Science,

( Received

Abstract: This paper presents visualization of
transient sound propagation in 2-dimensional room sound fields in which the typical shapes of
concert halls are modeled by applying the finite difference
time domain method. As a basic study on room acoustic design, sound propagation
in rooms, scattering effect of acoustic diffusers and refection characteristics
of suspended panel arrays are investigated. Through the investigation, it has been
confirmed that this kind of visualization technique is very effective to get
intuitive comprehension of complex acoustic phenomena which occur in rooms. The
technique can be useful tool for discussion on room and acoustic treatment
between acoustic engineers and architects.

1. INTRODUCTION

Visualization technique of sound propagation in a room
is very effective not only for education on room acoustics but also for
acoustical design work of various kinds of auditoria. For this purpose, such
computer simulation techniques as ray-tracing and image-source methods have been
developed and being widely used. This kind of techniques based on geometrical
acoustics are effective to some extent when obtaining rough estimate of sound reflection
and absorption in a room but it is impossible to exactly deal with such
complicated phenomena as sound refection, diffraction and scattering. To
overcome this problem, the authors have been investigating the application of
‘‘the finite difference time domain (FDTD) method

2. OUTLINE OF CALCULATION BY FDTD
METHOD

In a 2-dimensional sound field, sound wave is expressed
by the following partial differential equations. Equations (1) and (2) are the
momentum equations in x- and y-directions, respectively, and Eq. (3) is the continuity equation.

where p is the sound pressure, ux and uy are the particle velocities in x- and y-directions,
respectively, _ is the density of the air and _ is the volume elastic modulus
of the air.

The spatial and time derivatives of an arbitrary
function f , can be approximated by the central finite
difference forms as

(f(x+Ä÷/2)-f(x-Ä÷/2))/Äx , (f(y+Äy/2- f(y-Äy/2))Äy and (f(t+Ät/2)- f(t-Ät/2))/Ät , respectively. Here the Äx and Äy are the spatial intervals,
and Ät is the discrete time step. When applying the staggered grid system with square-grids (Äx=Äy) shown in Fig. 1, the
following equations are

Fig. 1 Discretization
of the sound field by staggered meshes.

Fig. 2 Sound pressure distribution around the
source position as the initial condition of calculation.

Fig. 3 Discretization
of sound field near boundary.

Zn= ñc 1+√1-án / 1- √ 1- án (11)

In the calculation mentioned in sections 3 and 4 in
this paper, it is assumed that án=0,2 (corresponding to Zn= 7,
357 Ns/m3) for over all frequencies to simplify the boundary condition. (The
authors have been investigating the method to simulate the boundary condition
of arbitrary normal acoustic impedance in the FDTD
method [3].)

Under these initial and boundary conditions, the sound
pressure and particle velocities at each grid point were calculated
successively using Eqs. (4), (5), and (6).

Fig. 4 Sound propagation in a rectangular room
(a), fan-shaped room (b) and elliptic room (c) without diffusing treatment.

3. VISUALIZATION OF SOUND PROPAGATION IN ROOMS

In architectural design of concert halls and theaters, rectangle (so called ‘‘shoe-box style’’),
fan-shape, round shape and ellipse are often chosen. By the difference of such
room shapes, acoustic properties are much varied and therefore the design of
fundamental room shape is an essential problem not only from the architectural
viewpoint but also from the acoustic viewpoint.

In order to examine the acoustic characteristic determined
by such fundamental room shape, sound propagation characteristics in the
2-dimensional rectangular, fan-shaped and elliptic rooms shown in Fig. 4 were calculated
by the FDTD method. In this calculation, it was assumed
that these three 2-dimensional rooms have the same area of 518.4 m2.

Figures 4 shows the typical calculation results
in the form of ‘‘snap shot’’ in the time lapse after the emission of the
impulse source. (When demonstrating the
results by computer animation, the successive propagation of the wave front of
the impulse can be clearly visualized.) In each figure, the black circle
indicates the source position and the white one indicates the receiving
position for the calculation of impulse response mentioned later. Comparison of
these figures reveals that the propagation of the wave front is much different
in each hall. In the case of the rectangular room, it is clearly seen that the number
of wave front increases with the progress of time, whereas in the cases of the
fan-shaped and elliptic rooms, a tendency that the wave front defects and
concentrates is seen. Especially, in the case of the elliptic room, it is clearly
seen the wave front focuses at around the source position and its symmetrical
point alternately. Figure 6(a) shows the impulse responses at the receiving
point in each room. In these results, it is seen that the reflections are dense
and smoothly diminishing in the case of the rectangular room, whereas the reflections
are scattered and uneven in the fan-shaped and the elliptic rooms.

Fig. 5 Sound propagation in a rectangular room
(a), fan-shaped room (b) and elliptic room (c) with diffusing treatment.

Fig. 6 Calculation results of impulse responses at the
receiving points shown in Figs. 4 and 5.

4. THE EFFECT OF SOUND SCATTERING BY DIFFUSERS

In concert halls and theaters,
wall and ceiling are often made irregular to increase sound diffusivity. To
examine the effect of such diffusion treatments, the FDTD
calculation was again performed for the three types of rooms by making their
walls irregular. As the shape of irregular wall, a zigzag shape (Type-2 in Fig.
7) was assumed. The snap shots of the calculation results are shown in Fig. 5.
By comparing the results with those in the case of no diffusion treatment shown
in Fig. 4, it is clearly seen that the distinct wave fronts have been much diminished
and scattered in all of the three rooms.

The impulse responses at the receiving points in the three
rooms were calculated in this case, too. The results are shown in Fig. 6(b) in
comparison with those without diffusion treatment. In these results, it is
obviously seen that the impulse responses have become much denser and smoother
than the case of no diffusion treatment shown in the upper figures. When
hearing these impulse responses through a loudspeakers or headphones, it can be
clearly judged that the reverberation decays have much improved to be natural
and smooth by the diffusion treatment, although the early .uttering sounds
caused by the sound concentration are still slightly remaining in the cases of
the fan-shaped and elliptic rooms. This fact indicates that the general
tendency of sound concentration caused by the fundamental room shape can not be
prevented by this kind of diffusion treatment on the room boundaries.

In order to examine the effect of sound scattering by diffusers
in more detail, a further study was performed on the rectangular room. In this
study, four kinds of zigzag shapes shown in Fig. 7 were assumed. Among them,
Type- 1, Type-2 and Type-3 are similar in shape but the size was varied in
three steps. The ratio of the height of the apex to the width of a triangle was
set 0.15 according to the results of the experimental study made by Ishii [4].
Type-4 is a ‘‘two-way’’ diffuser composed of Type-3 and Type-1. Figure 8 shows
the calculation results. To compare these results with those in the case of no
diffusion treatment shown in Fig. 4, it is clearly seen that the sound is
scattered after the first refection on the diffusive boundaries and the space
is filled with sound pressure fluctuation. In the results of Type-1, Type-2 and
Type-3, it is seen that the scattering effect is dependent on the size of the
diffusers. That is, in the case of Type-1, relatively strong and continuous
wave fronts are still remaining, whereas they are much diminished in the case
of Type-3. In the result of Type-4, the effectiveness of ‘‘two-way’’ diffuser
can be observed.

In the calculation by the FDTD
method, instantaneous sound pressure at every mesh point is obtained. By squaring
the sound pressure, instantaneous potential energy distribution in the room can
be obtained and consequently the time variation of acoustic diffusivity in the
room can be evaluated quantitatively from a viewpoint of the spatial uniformity
of sound energy [5].

Type1=0.4m [per element] ,Type2=1.5m [per element], Type3=3.0 m [per element] and Type4=3.0m [as shown above in <d>]

Fig. 8 Comparison of sound propagation in the
rectangular room with four types of diffusing treatments.

5. REFLECTION CHARACTERISTICS OF SUSPENDED PANEL
ARRAYS

In order to provide early reflections to the stage and
audience areas, suspended panel arrays (so called ‘‘floating clouds’’) are
often equipped in concert halls. To investigate the effect of such suspended
panel arrays, the refection characteristics of typical arrangements of panel
arrays were examined by the 2-dimensional FDTD
calculation. As shown in Fig. 9, three kinds of arrangements were examined in
this study: Type-1 is a straight arrangement with straight panels, Type-2 is a
terraced arrangement with straight panels and Type-3 is a terraced arrangement
with curved panels. The size of each panel is 1.7 m in both cases of the
straight panels and the curved ones. These panels are arranged at an interval
of 2.5 m in horizontal direction in all of the three cases. In the two cases of
the terraced arrangement, the gap between the adjacent panels in vertical
direction is 0.44 m. In this calculation, the suspended panels were assumed to
be perfectly reflective. The snap shots of the calculation results are shown in
Fig. 9, in which sound reflection on the surface of each panel and sound diffraction
and transmission through each gap can be obviously seen. In the case of the
straight arrangement (Type-1), it is seen that the reflections from each panel
forms a continuous wave front, whereas in the cases of the terraced arrangement
(Type-2 and Type-3), each reflections are separated and the wave front is
divided in space. In the case of Type-3, the wave front of each reflection is
almost semi-circular and crossing each other, whereas the strength (sound
pressure) is relatively weak. Regarding this kind of suspended panel arrays, the
frequency characteristic of sound reflection is another essential problem and
the authors are investigating this problem based on the Fresnel-
Kirchhoff diffraction theory [6].

Fig. 9 Comparison of sound reflections by the
differences of shape and arrangement of suspended panel arrays

6. CONCLUSIONS

As a basic study on visualization of room acoustics using
numerical simulation technique, the sound propagation in rooms of different
shapes, the scattering effect of acoustic diffusers and the refection
characteristics of suspended panel arrays have been investigated by applying the
FDTD method. As a result, it has been found that this
kind of visualization technique is very effective to get intuitive comprehension
of acoustic phenomena in rooms. It will be a useful tool for acoustic education
not only for students and acoustic engineers but also for architects who design
concert halls and theaters. Besides the studies on
visualization of acoustic phenomena in room acoustics introduced in this paper, the authors are now
developing the technique to auralize the results of
numerical simulation [7]. By combining these visual and aural simulation
techniques, researches on room acoustics could be further advanced in the
future.

REFERENCES

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[2] S. Sakamoto, Y. Tokita
and H. Tachibana, ‘‘Calculation of impulse responses of rooms by using of the .nite di.erence method’’, Proc. ASA and ASJ 3rd Jt. Meet., p. 1307 (1996).

[3] S. Sakamoto and H. Tachibana, ‘‘Calculation
of impulse responses in 3-D sound field with absorptive boundary by the finite
difference method’’, Proc. Inter-Noise, 97, 1597 (1997).

[4] K. Ishii, ‘‘Acoustical study of auditorium
—Design and execution of works—’’, Rep. Inst. Ind. Sci. Univ. Tokyo, 8, 14 (1958).

[5] T. Yokota, T. Seimiya,
S. Sakamoto and H. Tachibana, ‘‘Difference in acoustic effect of sound diffusers
due to room shapes’’, J. Acoust. Soc. Jpn.
(E), 21, 283 (2000).

[6] T. Yokota, S. Sakamoto and H. Tachibana, ‘‘Sound
reflection characteristics of suspended panel array’’, Proc. 17th Int. Congr. Acoust., Vol. 1 (2001).

[7] S. Sakamoto, T. Seimiya,
T. Yokota and H. Tachibana, ‘‘Multidimensional sound .eld
reproduction based on numerical simulation —A trial of 4 channel system for
2-dimensional sound field—’’, Proc. Autumn Meet. Acoust. Soc. Jpn.,
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