WCU 2003, Paris, september 7-10, 2003
NONLINEAR STANDING WAVES IN ACOUSTIC RESONATORS WITH AN ARBITRARY
REFLECTION COEFFICIENTS AT THEIR WALLS
M. Červenka, M. Bednařı́k, P. Konı́ček
Czech Technical University in Prague, Department of Physics,
Technická 2, 166 27 Prague, Czech Republic
E-mail: [email protected]
Model equations
The nonlinear standing waves in a cylindrical resonator, see figure 1, driven by means of external force
may be described by an one-dimensional model equation in the second approximation, see [6], which has for
constant radius form
2
∂2ϕ
∂ ∂ϕ 2
da
2∂ ϕ
− c0 2 +
= −x −
2
∂t
∂x
∂t ∂x
dt
2
∂ϕ γ − 1 ∂ ∂ϕ
−
+ ax +
−a
∂x
2c20 ∂t ∂t
∂ 3/2 ϕ
d1/2 a
∂3ϕ
+ x 1/2 , (1)
+ b 2 − 2d
∂x ∂t
∂t3/2
dt
Abstract
This paper deals with the problems of description
of finite-amplitude standing waves in gas-filled acoustic resonators, where the frontal resonator walls are
of arbitrary reflection coefficients. The concrete wallreflection coefficients form may be connected with the
finite wall stiffness, radiation of acoustic energy from
the resonator cavity or utilization of selective absorbing materials for nonlinear effects suppression. It is
assumed that the standing waves are driven by vibrating piston or by external volume force, with arbitrary
frequency of driving signal. One-dimensional model
equation of the second order is used to derive a set of
two non-homogenous Burgers equations describing two
contrapropagating waves, which are connected at the
resonator walls by the boundary conditions considered.
The equations are solved numerically in frequency domain.
where ϕ is a velocity potential, a(t) is driving acceleration, x is spatial coordinate along the resonator body, t
is time, c0 is a small-signal sound speed, γ is the ratio of
specific heats. Here b is the diffusity coefficient and d is
a coefficient including influence of the boundary layer
1
1
4
1
−
ζ + η +κ
, (2a)
b =
ρ0
3
cV
cp
√ ν0
γ−1
,
(2b)
1+ √
d =
r
Pr
Introduction
Provided that acoustic standing wave is driven into
high amplitude, the nonlinear effects are possible to distort originally harmonic wave and transform thus acoustic energy into higher harmonic components which result in heightened dissipation of acoustic energy. There
are many methods of these nonlinear effects suppression such as appropriate shape of the resonant cavity
and multifrequency driving signal, see [1], [2].
Other possibility is utilization of suitable materials at
the resonator walls which absorb energy of the higher
harmonic components or cause complying with the resonant condition only for the fundamental harmonics.
It is necessary to derive suitable model equation supplemented with the appropriate boundary conditions to
study behaviour of the nonlinear standing waves on
these conditions.
R0
r
a(t)
where ρ0 is ambient density, η and ζ are shear and
bulk viscosities, κ is thermal conductivity coefficient,
cp and cV are constant pressure and volume specific
heats, ν0 = η/ρ0 is kinematic viscosity, r is the resonator radius and P r is the Prandtl number.
The half-order derivative in equation (1) is defined as
1
t
1
∂f (t ) dt
∂2f
√
√
=
.
(3)
1
π −∞ ∂t
t − t
∂t 2
Equation (1) is written in coordinates moving with
the resonator body; acoustic velocity may be obtained
using its solution from definition of the velocity potential as
∂ϕ
,
(4)
v=
∂x
acoustic pressure p = p − p0 as
2
ρ0 ∂ϕ
∂ϕ
− ρ0 a x + 2
+ ax −
p = −ρ0
∂t
2c0 ∂t
2
∂ ϕ
1
da
4
ρ0 ∂ϕ 2
. (5)
+ 2 ζ+ η
+x
−
2 ∂x
3
∂t2
dt
c0
R1
x=l
x=0
Figure 1: Acoustic resonator.
1283
WCU 2003, Paris, september 7-10, 2003
or written in a compact form
This equation in the second approximation is derived
from the Navier-Stokes equation.
If we assume the velocity potential as two counterpropagating waves, see [3],
∂v± γ + 1 ∂v±
−
v±
=
∂x
∂τ±
2c20
=
ϕ(x, t) = ϕ+ (t1 = µt, x1 = µx, τ+ = t − x/c0 )+
±
+ ϕ− (t1 = µt, x1 = µx, τ− = t + x/c0 ), (6)
For simple waves in nondissipative fluids we can
write, see [5],
p±
c0 ± (γ − 1)v± /2 2γ/(γ−1)
=
(14)
p0
c0
where µ is a small dimensionsless parameter, see [3],
[4], and the driving acceleration as
a = a(t) = a+ (x, τ+ ) + a− (x, τ− ) ≈ µ2 ,
a(t) =
∞
(7)
and after neglecting the high-order terms
ak e jkωt =
p± ≈ ±ρ0 c0 v± + ρ0
(8)
k=−∞
∞
=
∞
ak jkωτ− −jkω cx
0,
e
e
2
k=−∞
(9)
X=
x
,
l
T± = ωτ± ,
V± =
P ± =
p±
,
π 2 ρ0 c20
After neglecting terms of the order µ3 and higher we
obtain a set of equations
X ∂A±
γ+1
∂V ±
∂V ±
= π2
ΩV ±
+ π2Ω
±
∂X
2
∂T±
2 ∂T±
∂ 1/2 V ±
Ω ∂2V ±
∓
πΩD
, (17)
± πGT V
1/2
2 ∂T±2
∂T±
b ∂ 2 v−
∂ 1/2 v−
∂a−
+
− 2dc0
. (10b)
2
1/2
∂τ−
c0 ∂τ−
∂τ
−
Here
where
∂ϕ+ ∂ϕ−
∂ϕ
=
+
= v+ + v− .
∂x
∂x
∂x
(11)
Ω=
In case of the steady-state conditions, the terms with
the time derivatives can be neglected and set (10) is reduced to set of equations
=
x ∂a+
+
2c20 ∂τ+
ω
,
ω0
GT V =
bω
,
c20
(12a)
V±
.
π
(18)
The driving is assumed to be periodic, so acoustic
quantities can be expanded into Fourier series
d ∂ 1/2 v+
,
c0 ∂τ 1/2
+
∂v− γ + 1 ∂v−
−
v−
=
∂x
∂τ−
2c20
d
D=√ .
ω
For non-dimensional acoustic pressure we then obtain
P ± = ±
−
a±
, (16)
lω02
πc0
l
is the fundamental resonant frequency for rigid-walled
resonator cavity and l is the resonator cavity length.
Non-dimensional form of equation (13) is
+
b ∂ 2 v+
2c30 ∂τ+2
A± =
ω0 =
∂v−
∂v−
∂v−
− 2µc20
+ (γ + 1)v−
=
2µc0
∂t1
∂x1
∂τ−
∂v+ γ + 1 ∂v+
−
v+
=
∂x
∂τ+
2c20
v±
,
πc0
where
∂v+
∂v+
∂v+
− 2µc20
+ (γ + 1)v+
=
∂t1
∂x1
∂τ+
b ∂ 2 v+
∂ 1/2 v+
∂a+
−
+ 2dc0
, (10a)
= −x
2
1/2
∂τ+
c0 ∂τ+
∂τ
−2µc0
v=
(15)
Numerical analysis
For purposes of numerical analysis it is convenient to
introduce non-dimensional variables
a− x,τ−
= −x
γ+1 2
v + ....
4 ±
Acoustic pressure may be computed from solution of
equation (13) using the first term of the equation (15).
ak jkωτ+ jkω cx
0 +
e
e
2
k=−∞
a+ (x,τ+ )
+
b ∂ 2 v±
d ∂ 1/2 v±
x ∂a±
±
∓
. (13)
c0 ∂τ 1/2
2c20 ∂τ±
2c30 ∂τ±2
V
(12b)
±
≈
N
k=−N
b ∂ 2 v−
d ∂ 1/2 v−
x ∂a−
− 3
+
,
= 2
c0 ∂τ 1/2
2c0 ∂τ−
2c0 ∂τ−2
A± ≈
N
Ak jkT± ±jkπΩX
e
e
.
2
k=−N
−
1284
Vk± e jkT± ,
(19a)
(19b)
WCU 2003, Paris, september 7-10, 2003
Distribution of acoustic pressure spectra along the resonator cavity
After substituting the series (19) into set (17), we obtain
5000
4500
N
dVk±
±
2γ + 1
(k − m)Vm± Vk−m
+
= jπ
Ω
dX
2
4000
3500
m=k−N
3000
p [Pa]
jkπ 2 ΩXAk ±jkπΩX πk2 GT V Ω ±
∓
+
e
Vk ∓
4 2
|k|
∓π
ΩD[1 + j · sign(k)]Vk± . (20)
2
2000
1500
1000
If we assume the acoustic pressure reflection coefficients, see [5], at X = 0 and X = 1 to be R0 = R0 (ω)
and R1 = R1 (ω), with respect to (11), we can write
boundary conditions for acoustic velocity in form
X =0:
500
0
0
=0 ⇒
Vk+
+
R1 Vk− e 2jkπΩX
0.8
1
4
= 0.
Set of ODEs (20) is solved numerically by means of
the Runge-Kutta method of the 4th order, the two-point
boundary value problem (21) is solved using the shooting method.
Non-dimensional acoustic velocity spectra components may be expressed as
Vk = Vk+ e −jkπΩX + Vk− e jkπΩX
(22)
and acoustic velocity time-domain dependence then
2
x 10
1.5
p [Pa]
+ R1 V
0.6
(21b)
−
1
0.5
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
X [-]
40
30
v [m/s]
V
0.4
Figure 2: Comparison of numerical results. Red lines
belong to solution of Eqs. (13)+(15), the blue lines to
Eqs. (1)+(5).
R0 V + + V − = 0 ⇒ R0 Vk+ + Vk− = 0,
+
0.2
X [-]
(21a)
X =1:
2500
20
10
v(X, T ) = 2πc0
+
(Vk+
N (Vk+ + Vk− ) cos kπΩX+
0
0
0.2
0.4
X [-]
k=1
− Vk− ) sin kπΩX cos kT −
Figure 3: Distribution of acoustic pressure and
velocity spectra for R0 = 1, R1 = −1.
− (Vk+ + Vk− ) cos kπΩX−
− (Vk+ − Vk− ) sin kπΩX sin kT . (23)
The figure 3 shows the acoustic pressure and velocity spectra distribution in case of R0 = 1, R1 = −1
and Ω = 1.5 (resonance condition for the fundamental
harmonics). It can be seen that higher harmonics are
suppressed and the first harmonic component is of high
amplitude in comparison with the case of the figure 2.
This can be explained by effect of the “time-reversal reflection”, see [7]. Wave traveling towards the wall with
the reflection coefficient R1 = −1 is distorted due to
acoustic nonlinearities and it goes back to non-distorted
form after the reflection.
The figure 4 shows the model case where all the
energy of the higher harmonics is absorbed by the resonator boundaries. Here Ω = 1, R0 = R1 = 1 for
the fundamental harmonics and R0 = R1 = 0 for the
higher ones.
Numerical results
The following figures present some numerical results. For all cases the resonator cavity is filled with
room-conditions air, the resonator length l = 15 cm,
non-dimensional acceleration A = 5 × 10−4 , GT V =
10−2 , D = 0.
The figure 2 shows the comparison of numerical results obtained from Eqs. (13)+(15), and (1)+(5). Here
Ω = 1 and R0 = R1 = 1. The agreement of the numerical results obtained is evident, the algorithm for Eq.
(13) numerical solution is faster, more stable and it is
possible to process more harmonic components. It is
also unnecessary to use so high additional-viscosity coefficient GT V , see [1].
1285
WCU 2003, Paris, september 7-10, 2003
10000
p [Pa]
8000
6000
4000
2000
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
X [-]
25
v [m/s]
20
15
10
5
0
0
0.2
0.4
X [-]
Figure 4: Distribution of acoustic pressure and
velocity spectra for R0 = R1 = 1 for the fundamental
harmonics and R0 = R1 = 0 for the higher ones.
Acknowledgements
This research has been supported by CTU internal
grant No. 0313813, GAČR grant No. 202/01/1372 and
research project No. J04/98:212300016.
References
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Doren, E. A. Zabolotskaya,“Nonlinear standing
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Am. 104, pp. 2664-2674, 1998.
[2] P. T. Huang, J. G. Brisson, “Active control of finite amplitude acoustic waves in a confined geometry”, J. Acoust. Soc. Am. 102(6), pp. 3256-3268,
1997.
[3] V. E. Gusev, “Buildup of forced oscillations in
acoustic resonators”, Sov. Phys. Acoust. 30(2), pp.
121-125, 1984.
[4] O. V. Rudenko, S. I. Soluyan, “Theoretical foundations of nonlinear acoustics”, Consultants Bureau, New York, 1977.
[5] D. T. Blackstock, “Fundamentals of physical
acoustics”, A Willey-Interscience Publication,
John Willey & Sons, Inc., New York, 2000.
[6] M. Bednařı́k, M. Červenka, “Nonlinear Waves in
Resonators”, Nonlinear Acoustics at the Turn of
the Millenium”, Institute of Physics, pp. 165-168,
2000.
[7] M. Tanter, J.-L. Thomas, F. Coulouvrat, M. Fink,
“Breaking of time reversal invariance in nonlinear
acoustics ”, Physical Review E, vol. 64, no. 1, pp.
016602/1-7, 2001.
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