FUNDAMENTALS OF ACOUSTICS(16) 声学基础(英文版教学课件)

2 z
p0,1 A0,1 cos
ye
• Cutoff frequency ------截止频率
The lowest characteristic eigenfrequency other than (0,0)
Geometry figure of high order waves
FUNDAMENTALS OF ACOUSTICS (16)
Sound waves travel in the pipe
p
nx ny nz 0


pnx ,ny ,nz
jt
pnx ,ny ,nz Anx ,ny ,nz cos kx x cos k y y cos kz z e
• Consider high order waves (nx=1, ny=0) • Can it travel in the rectangular cavity with
lx

2
ly
is large in any size ?
For l x

2

2soຫໍສະໝຸດ nx 1, ny 0can travel in the rectangular cavity
Each
pnx ,ny ,nz
c0 nx ny nz 2 lx l y lz
2 2 2
has its own characteristic eigenfrequency
f nx , n y , nz
when
z
2 2
or
nx 0, ny 1
c0 2
f fnx ,ny
In this case , there is only ordinary plane waves travel in the pipe
If
f fnx ,ny
for (0,1)

ly
2 j t c0 ly
k
j (t kx )

c0
Vp c0
(nx , ny ) (0,0)
• Depend on if it can be stimulated by
the source frequency • Example: lx l y 1m, • A rectangular cavity , with source frequency is 100Hz
1 cos 2 2
2
450
p Pnx ,ny Anx ,ny cos kx x cos k y ye
nx 0 ny 0 nx 0 ny 0
pnx ny Anx ny cos kx x cos k y ye
2 2 2 x 2 y
jk z z
e
jt
k z k ( k k ) nx , n y c0
The limiting value of c for which kz remains real ,This defines the cutoff frequency ny 2 c n 2
f nx n y
0

2
(
x
lx
) (
ly
)
Characteristic eigenfrequency
3. The nx, ny mode
• 1. (0,0) order mode • 2. (nx ny) order mode
For (0,0) order, the pressure is
p0.0 A0.0e
1 ( z sin x cos )] c0
cos
c0
)e
z sin ] c0
cos
c0
nx lx
2
nxc0 cos l x
nx c0 2 sin 1 cos 1 ( ) lx c0 n ( )2 ( x )2 c0 lx
Is cutoff frequency
nx lx 2 n lx x 2
(nx,0 ) high order waves can not travel in z direction (nx,o) high order waves can travel in z direction
Example
j (
pnx ,0
nx Anx ,0 cos e lx

c0
)2 (
nx 2 ) z lx
e jt
x


z
nxc0 cos l x
0
cos 1,
nx c0 nx cos 1 lx 2lx so nx c0 fc 2lx
0
Which means there are not high order waves travel in the z direction (except nx 0, ny 0)
From
z
f nx n y
c0 nx 2 n y 2 ( ) ( ) 2 lx ly
The lowest characteristic eigenfrequency is
nx 1, ny 0
f nx n y c0 2 ny 2 nx 2 ( ) ( ) lx ly 1 0 170 Hz 1 1
nx 0, ny 0
pnx ,0
x
nx Anx ,0 cos xe lx
x
j (

c0
)2 (
nx 2 ) z lx
e
jt



(a)
(a)
节线

(1,0)
(b)
节线
z
(2,0)
Physical expression
• High order wave consists of several group
of planes. And these planes are traveled at angle
x


z
pnx ,0 Anx ,0 e 2 Anx ,0 cos(
j [ t
1 ( z sin x cos )] c0 j [ t
Anx ,0 e
j [ t