带家具房间内的声驻波仿真
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2 | EIGENMODES OF A ROOM
Introduction
Resonance can at times be a problem in everyday life. The low bass notes from the music
system or home theater in the living room can shake the windows and make the floor
vibrate. This happens only for certain frequencies — the eigenfrequencies of the room.
It is only in the low-frequency range that the eigenfrequencies are well separated. In the
mid- and high-frequency ranges, the eigenfrequencies are packed so closely, with less than
a halftone between them, that the individual resonances are insignificant for music and
other natural sounds. Nevertheless, the music experience is affected by the acoustics of the
room.
When designing a concert hall, it is extremely important to take the resonances into
account. For a clear and neutral sound, the eigenfrequencies should be evenly spaced. For
the home theater or music system owner, who cannot change the shape of the living room,
another question is more relevant: Where should the speakers be located for the best
sound?
Model Definition
For example, take a room with the dimensions 5 by 4 by 2.6 meters equipped with a flat-
screen TV, a sideboard, two speakers, and a couch. To illustrate the effects on the music,
compute a few resonance frequencies in the vicinity of 90 Hz together with the
corresponding eigenmodes. The eigenmode shows the sound intensity pattern for its
associated eigenfrequency. From the characteristics of the eigenmodes, you can draw some
conclusions as to where the speakers should be placed.
DOMAIN EQUATIONS
Sound propagating in free air is described by the wave equation:
where p is the pressure, and c is the speed of sound. If the air is brought into motion by a
harmonically oscillating source, for example, a loudspeaker, only one frequency f exists in
the room. For that reason it makes sense to look for a time-harmonic solution on the form
The wave equation then simplifies to the Helmholtz equation for p, the amplitude of the
acoustic disturbances:
pΔ–
1
c
2
-----
t
2
2
∂
∂ p
+ 0=
pp
ˆ
e
iωt
=
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