Frequency response of nonlinear oscillations of air column
in a tube with an array of Helmholtz resonators
N. Sugimoto, M. Masuda, and T. Hashiguchi
Department of Mechanical Science, Graduate School of Engineering Science, University of Osaka,
Toyonaka, Osaka 560-8531, Japan
共Received 10 January 2003; revised 20 June 2003; accepted 23 June 2003兲
Nonlinear cubic theory is developed to obtain a frequency response of shock-free, forced
oscillations of an air column in a closed tube with an array of Helmholtz resonators connected
axially. The column is assumed to be driven by a plane piston sinusoidally at a frequency close or
equal to the lowest resonance frequency with its maximum displacement fixed. By applying the
method of multiple scales, the equation for temporal modulation of a complex pressure amplitude
of the lowest mode is derived in a case that a typical acoustic Mach number is comparable with the
one-third power of the piston Mach number, while the relative detuning of a frequency is
comparable with the quadratic order of the acoustic Mach number. The steady-state solution gives
the asymmetric frequency response curve with bending 共skew兲 due to nonlinear frequency upshift in
addition to the linear downshift. Validity of the theory is checked against the frequency response
obtained experimentally. For high amplitude of oscillations, an effect of jet loss at the throat of the
resonator is taken into account, which introduces the quadratic loss to suppress the peak amplitude.
It is revealed that as far as the present check is concerned, the weakly nonlinear theory can give
quantitatively adequate description up to the pressure amplitude of about 3% to the equilibrium
pressure. © 2003 Acoustical Society of America. 关DOI: 10.1121/1.1600719兴
PACS numbers: 43.25.Gf, 43.25.Vt, 43.25.Cb 关MFH兴
I. INTRODUCTION
Novel methods for generation of shock-free, highamplitude oscillations of gas in a tube 共or a container兲 have
recently attracted much attention in view of applications to
thermoacoustic devices. They commonly exploit resonance
in one form or another by exciting an acoustic system at a
frequency close to or equal to one of its resonance frequencies 共eigenfrequencies兲. As the amplitude of oscillations of
gas becomes high, however, there emerges a shock, i.e., discontinuity in pressure, etc., so that increase in pressure amplitude of oscillations tends to be suppressed as the excitation is increased.
Whether or not the shock emerges is crucially determined by the relation between resonance frequencies and
frequencies of higher harmonics of the excitation. If the resonance frequencies are ordered as multiples of the fundamental one, just as in the case of a closed tube of uniform cross
section, then the tube is called being consonant and otherwise dissonant.1,2 As the amplitude of excitation is increased
in a consonant tube, each frequency of higher harmonics
generated by nonlinearity hits the respective resonance frequencies so that higher modes are gradually excited and energy in the fundamental mode is pumped up into higher
modes and dissipated there. This cascade process of energy
flow is the mechanisms behind emergence of the shock and
the resulting suppression of increase in pressure amplitude of
the fundamental mode.
In order to annihilate the shock, the cascade process
should be blocked by any means. At present, there are two
methods confirmed experimentally at high pressure level.
One is the method devised by Lawrenson et al.2 and Ilinskii
et al.3 at MacroSonix. The essential point lies in making the
1772
axial cross section of a container nonuniform so as to render
it dissonant. By vibrating the whole container on a shaker,
shock-free, high-amplitude oscillations have been achieved
experimentally and theoretical analysis has also been made.
The other method is to exploit wave dispersion, as devised by the present authors,4 although it was proposed originally in a different problem.5 Sound speed in gas is usually
constant independent of frequency; therefore, no dispersion
occurs in propagation. But, by connecting external agents to
the tube, the phase speed can be made dependent on a frequency. In fact, this is achieved by connecting an array of
Helmholtz resonators to a tube of uniform cross section axially. By sinusoidally driving the bellows mounted at one end
of this tube, shock-free, high-amplitude oscillations of about
10% of the equilibrium pressure have been generated at the
other closed end. Because the phase speed now depends on
the frequency, resonance frequencies are no longer ordered
as multiples of the fundamental one, and the tube becomes
dissonant automatically without any change of cross section
of the tube.
In the present context, it may be worthwhile to mention
the dispersion. For propagation in a tube of nonuniform cross
section, the phase speed cannot be defined strictly even in the
case of an exponential horn because the amplitude decays or
grows in the direction of propagation. The propagation speed
is then determined by the characteristics and is still given by
the sound speed. In this sense, the system still remains hyperbolic. In the tube with the array, on the contrary, a plane
sinusoidal wave can be propagated so that the phase speed is
clearly defined, which is now different from the sound speed
and dependent on the frequency. The connection of the array
changes the hyperbolic system originally to be dispersive.
J. Acoust. Soc. Am. 114 (4), Pt. 1, October 20030001-4966/2003/114(4)/1772/13/$19.00
© 2003 Acoustical Society of America
Besides the two methods, Rudenko et al.6 have already
proposed ideas to block the cascade process by introducing
special absorbers of the second harmonics through the
boundary condition. Andreev et al.7 have confirmed the effects experimentally, though at even lower pressure level.
For suppression or reduction of the second harmonics, Gusev
et al.8 have abandoned a usual monochromatic excitation to
control the driver actively by adding to the fundamental sinusoidal excitation the second harmonic one with a phase difference. While these methods focus on increase in so-called
quality factor of a resonator, it is questionable whether or not
shock-free oscillations are ultimately achieved.
The purpose of this paper is to formulate nonlinear
forced oscillations of an air column in the tube with the array
of Helmholtz resonators and to derive theoretically a frequency response corresponding to the one obtained
experimentally.4 Supposing each resonator is small in effect,
one-dimensional motion of air is assumed over the cross section of the tube except for a boundary layer on the wall and
vicinity of resonator’s orifices open to the tube. Supposing
also that the axial spacing between neighboring resonators is
small in comparison with a wavelength, the continuum approximation for the resonators is made so that the effects of
the discrete distribution in array may be smeared out per
axial length.
Because the pressure level observed in the experiment is
still small relative to the equilibrium pressure, weakly nonlinear theory is developed by using the asymptotic method of
multiple 共two兲 scales.9 The bellows used in the experiment
are modeled as a plane piston reciprocating sinusoidally by
taking account of the correspondence between the displacement of the bellows and the one of the piston. The boundary
condition for the piston is usually of three types, either one
of the maximum displacement, maximum speed, or maximum acceleration being held constant, the first of which is
used in the present theory.
As the ratio of the maximum displacement of the piston
to the tube length is much smaller than unity, so is the piston
Mach number defined by the ratio of the maximum piston
speed to the sound speed, which is comparable in order with
the former. Denoting the piston Mach number by ␧ p , and a
typical acoustic Mach number in the tube by ␧, respectively,
the situation in the experiment corresponds to a case where ␧
is of order ␧ 1/3
p and the relative detuning of a frequency from
the resonance one is of order ␧ 2 (⫽␧ 2/3
p ). It is emphasized
that the rigorous cubic nonlinear theory starting from the
formulation based on the above assumptions is necessary to
obtain a frequency response correctly up to the third order in
the pressure amplitude.
The problem is formulated in Sec. II and the lossless
linear theory is first described in Sec. III. Section IV is devoted to the nonlinear theory by using the method of multiple scales to derive the equation for slow modulation of a
complex pressure amplitude of the fundamental mode. From
the steady-state solution to the equation, the frequency response is obtained and compared with the experiment. In
Sec. V, an effect of the jet loss at the throat of the resonator
is considered in order to compensate the discrepancy beJ. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
FIG. 1. Illustration of a tube with an array of Helmholtz resonators.
tween the theory and experiment as the amplitude becomes
high.
II. FORMULATION OF THE PROBLEM
A. Basic equations
We start by formulating the problem. Suppose a straight,
rigid tube of radius R and of length l, to which an array of
Helmholtz resonators is connected 共see Fig. 1兲. One end of
the tube is allowed to be displaced by a plane piston while
the other end is closed by a flat plate. The tube is closed
hermetically with the resonators inclusive. Each resonator is
assumed small in the sense that the cavity’s volume V is
much smaller than the tube’s volume per spacing Ad, A
(⫽ ␲ R 2 ) being the cross-sectional area of the tube and d the
axial spacing. This ratio is denoted by ␬ (⬅V/AdⰆ1), and
called a size parameter of the array. Taking the axial spacing
to be much smaller than a wavelength of oscillation, the
continuum approximation is made for discrete distribution of
the resonators to average its effect per unit axial length.
The Reynolds number is usually sufficiently high that
effects of viscosity and heat conduction are limited only
within the boundary layer developing on the tube wall. In the
outside of the boundary layer called a region of an acoustic
main flow, these lossy effects are negligible. The boundary
layer is thin and the array is small ( ␬ Ⰶ1), so the acoustic
main flow may be regarded as being almost one-dimensional.
Under these assumptions, the basic equations for the main
flow have already been presented in Ref. 10.
The equations of continuity and of motions are combined into the following equations:
冋
册冋
册
2
a
⳵
⳵
u⫾
⫹ 共 u⫾a 兲
共 a⫺a 0 兲 ⫽⫾
⳵t
⳵x
␥ ⫺1
A
冖
v n ds,
共1兲
with the signs ordered vertically where x and t denote, respectively, the axial coordinate along the tube and the time,
while u and a denote, respectively, the axial velocity of the
air and the local sound speed. The latter is defined by
a 2⫽
冉 冊
p
dp
⫽a 20
d␳
p0
( ␥ ⫺1)/ ␥
,
Sugimoto et al.: Nonlinear frequency response of air column
共2兲
1773
with a 0 ⫽ 冑␥ p 0 / ␳ 0 where p and ␳ denote, respectively, the
pressure and density of the air and the subscript 0 attached to
p and ␳ implies the respective values in equilibrium, ␥ being
the ratio of the specific heats, and a 0 is the linear sound
speed. Since the lossy effects are neglected in the acoustic
main flow, the adiabatic relation p/ p 0 ⫽( ␳ / ␳ 0 ) ␥ is assumed
to hold.
Through the right-hand side of 共1兲 are included the effects of the boundary layer and of the array of Helmholtz
resonators. Here, v n represents the velocity directed inward
normal to the boundary of the axial cross section of the
acoustic main-flow region, ds being the line element along
the boundary. Where the tube wall exists, v n is the velocity at
the edge of the boundary layer v b , given by
冉
v b ⫽ 1⫹
␥ ⫺1
冑Pr
冊
冑␯
冉 冊
⳵ ⫺ 1/2 ⳵ u
,
⳵ t ⫺1/2 ⳵ x
共3兲
q⫽
冕
t
⫺⬁
f 共 x, ␶ 兲
冑t⫺ ␶
d␶ .
共4兲
⳵␳ c
⫽Bq,
⳵t
共5兲
where ␳ c denotes the mean density of the air in the cavity,
and B denotes the cross-sectional area of the throat.
Thus, the integral in 共1兲 consists of two contributions as
follows:
1
A
冖
冉
v n ds⫽2 1⫹
␥ ⫺1
冑Pr
冊
冉 冊
冑␯ ⳵ ⫺ 1/2 ⳵ u
R * ⳵ t ⫺1/2
␬ ⳵␳ c
⫺
,
⳵x
␳ ⳵t
共6兲
where R * is the reduced radius of the tube defined by
R/(1⫺BR/2Ad), and q is set equal to ␳ w at the orifice on
the tube side. The derivation of the right-hand side of 共6兲
may be facilitated by multiplying A and ds by d, respectively, noting that d(ds) corresponds to the area element on
the surface bounding the region of the acoustic main flow.
Here, q in 共5兲 is expressed in terms of the excess pressure p c⬘
(⫽p c ⫺p 0 ) in place of the density. Assuming the adiabatic
relation for the air in the cavity, ␳ c is expanded into the
Taylor series with respect to p c⬘ to yield q in the following
form:
1774
冋
册
⳵
共 ␥ ⫺1 兲 2 共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 3
p ⬘c ⫺
p⬘ ⫹
p c⬘ ⫹¯ .
⳵t
2␥p0 c
6 ␥ 2 p 20
共7兲
L
⳵q
2L 冑␯ ⳵ 1/2
⫽⫺p c ⫹ p⫺
共 ␳ w 兲,
⳵t
r ⳵ t 1/2 0
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
共8兲
where the last term represents the friction at the throat wall.10
Using the lowest relation of 共7兲 to evaluate ␳ 0 w, substitution
of 共7兲 for q into 共8兲 leads to the following equation:
⳵ 2 p c⬘ 2 冑␯ ⳵ 3/2p c⬘
⫹
⫹ ␻ 20 p ⬘c
⳵t2
r * ⳵ t 3/2
⫽ ␻ 20 p ⬘ ⫹
␥ ⫺1 ⳵ 2 p ⬘c 2 共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 ⳵ 2 p c⬘ 3
⫺
2␥p0 ⳵t2
⳵t2
6 ␥ 2 p 20
⫹¯ ,
On the other hand, where the tube opens to the resonator, v n is the velocity of the air flowing out of the throat into
the tube, ⫺w. Because the throat is much shorter than a
typical wavelength, the mass flux density q averaged over
the cross section of the throat may be regarded as being
uniform along the throat. While no account of motions of the
air is taken in the cavity, the rate of increase in the mass
therein must balance with the mass flux flown into it. This
requires that
V
Ba 20
The behavior of the resonator is governed by 共5兲 supplemented by the equation of motion for the air in the throat.
The momentum balance of the air in the throat of length L
requires that
where ␯ denotes the kinematic viscosity taken constant, Pr
being the Prandtl number, and the derivative of minus halforder of a function f (x,t) is defined as
⳵ ⫺ 1/2 f
1
⫺1/2 ⬅
⳵t
冑␲
V
共9兲
where p ⬘ (⫽p⫺ p 0 ) is the excess pressure in the tube, r *
(⫽r/c L ), c L being L ⬘ /L e with L ⬘ ⫽L⫹2r and L e ⫽L⫹2
⫻0.82r, is the reduced radius of the throat by taking account
of the end corrections on both ends, and ␻ 0 (⫽ 冑Ba 20 /L e V)
is the natural angular frequency of the resonator.4 The derivative of three-half order is defined by differentiating the derivative of minus half-order twice with respect to t.
B. Reduction by the velocity potential
Addition and subtraction of 共1兲 with the upper and lower
signs lead, respectively, to
⳵u
⳵u
2a ⳵ a
⫹u ⫹
⫽0,
⳵t
⳵ x ␥ ⫺1 ⳵ x
共10兲
and
冉
冊
⳵a
⳵a
⳵u a
2
⫹a ⫽
⫹u
␥ ⫺1 ⳵ t
⳵x
⳵x A
冖
v n ds.
共11兲
In order to eliminate a from these equation, we introduce a
velocity potential ␾ defined by
u⬅
⳵␾
.
⳵x
共12兲
Substituting 共12兲 into 共10兲 and integrating this with respect to
x, we have
a 2 ⫽a 20 ⫺ 共 ␥ ⫺1 兲
冋
冉 冊册
⳵␾ 1 ⳵␾
⫹
⳵t 2 ⳵x
2
.
共13兲
This is simply the Bernoulli’s theorem. Multiplying 共11兲 by a
and using 共13兲 to eliminate a 2 , we derive
Sugimoto et al.: Nonlinear frequency response of air column
冉 冊
冉 冊
⳵ 2␾
⳵ 2␾
⳵ ⳵␾
2
⫺a
2
2 ⫽⫺
0
⳵t
⳵x
⳵t ⳵x
2
⫺ 共 ␥ ⫺1 兲
␥ ⫹1 ⳵ ␾
⫺
2
⳵x
2
tube length. Using this angular frequency ␲ a 0 /l, ␻ is made
dimensionless by introducing ␴ as
⳵ ␾ ⳵ 2␾
⳵t ⳵x2
冖
⳵ 2␾ a 2
⫺
⳵x2
A
␻⫽
v n ds.
p ⬘ ⫽⫺ ␳ 0
⳵␾
␳0
⫹
⳵ t 2a 20
冋
冋冉 冊 冉 冊 册
冉 冊 冉 冊册
⳵␾
⳵t
␳ 0 共 ␥ ⫺2 兲 ⳵ ␾
⫹ 2
⳵t
2a 0 3a 20
2
⫺a 20
3
⳵␾
⳵x
⳵␾ ⳵␾
⫹
⳵t ⳵x
¯
⳵ 2␾
⳵ t̄ 2
⫺
⫽
2
2
⫹¯ .
¯
⳵ 2␾
⳵ x̄ 2
␬ ā 2 ⳵␳
¯c
2
冋冉 冊
冉 冊
⳵
¯
⳵␾
⳵ t̄
⳵ x̄
⳵ x̄
2
ā 2
¯␳
⫽1⫺␧ 共 ␥ ⫺2 兲
共16兲
共17兲
关 x, t, ␾ , a, ␳ , ␳ c , p ⬘ , p c⬘ 兴
¯ , a 0 ā, ␳ 0¯␳ , ␳ 0¯␳ c ,
⫽ 关 lx̄, 共 l/a 0 兲 t̄ , lu 0 ␾
共18兲
where the quantities with the overbar imply the dimensionless ones, and u 0 is a typical speed of the air, say the maximum speed in the acoustic main flow. Here, two Mach numbers are defined: one is the acoustic Mach number ␧
associated with the air speed and the other the piston Mach
number ␧ p . Both Mach numbers are assumed to be much
smaller than unity, as
共19兲
In a lossless case of a tube without the array of resonators, it
is known that the air column will resonate in the lowest
mode when a half-wavelength ␲ a 0 / ␻ coincides with the
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
册
冉 冊
¯
⳵ ⫺ 1/2 ⳵ 2 ␾
2
⳵ t̄ ⫺1/2 ⳵ x̄
,
共21兲
¯
⳵␾
⳵ t̄
⫺␧
2
共 ␥ ⫺2 兲
冋冉 冊 冉 冊 册
2
¯
⳵␾
⫹
⳵ t̄
2
¯
⳵␾
2
⳵ x̄
共22兲
¯␳ c ⫽1⫹␧ p̄ c⬘ ⫺␧ 2
共 ␥ ⫺1 兲 2
共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 3
¯p c⬘ ⫹␧ 3
¯p c⬘
2
6
共23兲
On the other hand, 共9兲 is normalized as
⳵ 2 p̄ ⬘c
⳵ t̄
2
⫹␦r
⳵ 3/2p̄ c⬘
⳵ t̄
3/2
⫹ 共 ␲␴ 0 兲 2 p̄ c⬘
⫽ 共 ␲␴ 0 兲 2 p̄ ⬘ ⫹␧
We next normalize the equations and the boundary conditions by making the following replacement:
␻Xp
u0
Ⰶ1 and ␧ p ⬅
Ⰶ1.
a0
a0
⳵ x̄ 2
⫺ ␦ ā 2
2
⳵ t̄ ⳵ x̄
⫹¯ .
C. Normalization
␧⬅
¯
⳵ 2␾
¯ ⳵ 2␾
¯
⳵␾
and
where x p denotes the position of the piston surface; the displacement amplitude, X p , and the angular frequency of excitation, ␻, are taken real and positive, c.c. implying the
complex conjugate to all preceding terms, if any.
␳ 0 a 0 u 0 p̄ ⬘ , ␳ 0 a 0 u 0 p̄ c⬘ 兴 ,
⫹ 共 ␥ ⫺1 兲
⫹¯ ,
and
⳵␾
⫽0 at x⫽l,
⳵x
2
¯
共 ␥ ⫹1 兲 ⳵ ␾
2
with ¯␳ ⫽(1⫹␧ ␥ p̄ ⬘ ) 1/␥ , ¯␳ c ⫽(1⫹␧ ␥ p̄ c⬘ ) 1/␥ , and ā 2 ⫽1
¯ / ⳵ t̄ ⫹␧( ⳵ ␾
¯ / ⳵ x̄) 2 /2兴 , where ā 2 /¯␳ and ¯␳ c are
⫺␧( ␥ ⫺1) 关 ⳵ ␾
expanded, respectively, as
⳵ ␾ dx p i
⫽
⫽ ␻ X p ei ␻ t ⫹c.c.
⳵x
dt
2
1
at x⫽x p ⫽ X p ei ␻ t ⫹c.c.⫽X p cos共 ␻ t 兲 ,
2
⫺␧
␧¯␳ ⳵ t̄
⫺␧
共15兲
The boundary conditions are imposed at the piston surface and the closed end so that the velocity of the air may be
equal to the one of the respective surfaces. Taking the origin
of the coordinate x at the mean position of the piston surface
over one period and the closed end at x⫽l, the respective
conditions are expressed as follows:
共20兲
By the replacement 共18兲, 共14兲 with 共6兲 is normalized as
共14兲
In passing, substituting 共13兲 into 共2兲, p ⬘ is expressed inversely in terms of ␾. In fact, by expanding p/ p 0
⫽(a 2 /a 20 ) ␥ /( ␥ ⫺1) around a⫽a 0 and truncating the series at
the cubic terms, it follows that
␲a0
␴.
l
⫻
⳵ 2 p̄ c⬘ 3
⳵ t̄ 2
2
共 ␥ ⫺1 兲 ⳵ 2 p̄ c⬘
⳵ t̄ 2
2
⫺␧ 2
共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲
6
⫹¯ ,
共24兲
with ␴ 0 ⫽l ␻ 0 / ␲ a 0 , where ␦ and ␦ r are the parameters representing the boundary-layer effects due to the tube wall and
the throat wall, respectively, and are given by
冉
␦ ⫽2 1⫹
␥ ⫺1
冑Pr
冊
冑␯ l/a 0
and ␦ r ⫽2
R*
冑␯ l/a 0
r*
.
共25兲
In 共24兲, p̄ ⬘ is given, after normalization of 共18兲, by
p̄ ⬘ ⫽⫺
⫹
冋冉 冊 冉 冊 册
冋 冉 冊 冉 冊册
¯
⳵␾
⳵ t̄
⫹
␧
¯
⳵␾
2
⳵ t̄
2
¯
␧ 2 共 ␥ ⫺2 兲 ⳵ ␾
2
3
⳵ t̄
⫺
¯
⳵␾
2
⳵ x̄
3
⫹
¯ ⳵␾
¯
⳵␾
⳵ t̄
⳵ x̄
2
⫹¯ .
共26兲
On the other hand, the boundary conditions are normalized as
Sugimoto et al.: Nonlinear frequency response of air column
1775
¯
⳵␾
␲␴ c i ␲␴¯t
c
¯
e
⫽i
⫹c.c. at x̄⫽ ei ␲␴ t ⫹c.c.,
⳵ x̄
2␧
2
共27兲
¯
⳵␾
⫽0 at x̄⫽1,
⳵ x̄
共28兲
and
where c represents the dimensionless amplitude X p /l (Ⰶ1)
and ␲␴ c corresponds to ␧ p .
III. LOSSLESS LINEAR THEORY
At first, it is instructive to discuss the lossless linear
case. Taking the limit as c, ␦, and ␦ r →0, with c/␧ fixed, the
linearized problem is governed by the following equations:
⳵ p c⬘
⳵ 2␾ ⳵ 2␾
,
2 ⫺
2 ⫽␬
⳵t
⳵x
⳵t
共29兲
⳵ 2 p ⬘c
⫹ 共 ␲␴ 0 兲 2 p c⬘ ⫽ 共 ␲␴ 0 兲 2 p ⬘ ,
⳵t2
共30兲
p ⬘ ⫽⫺
⳵␾
,
⳵t
共31兲
with the boundary conditions given by
⳵␾
␲␴ c i ␲␴ t
⳵␾
⫽i
e
⫽0 at x⫽1.
⫹c.c. at x⫽0 and
⳵x
2␧
⳵x
共32兲
Here and hereafter, the overbar is omitted.
Setting
冋册冋 册
␾
⌽共 x 兲
p ⬘ ⫽ F 共 x 兲 ei ␲␴ t ⫹c.c.,
p c⬘
G共 x 兲
共33兲
and substituting this into 共29兲–共31兲 to eliminate F and G, we
derive
d2 ⌽
⫹k 2 ⌽⫽0,
dx 2
where k is given by
冉
k 2 ⫽ 共 ␲␴ 兲 2 1⫹
共34兲
冊
␬ ␴ 20
.
␴ 20 ⫺ ␴ 2
共35兲
Imposing the boundary conditions 共32兲, the solutions are easily obtained as
冋册冋 册
␾
i/ ␲␴
␲ 2 ␴ 2 cos关 k 共 x⫺1 兲兴 c i ␲␴ t
p⬘ ⫽ 1
e
⫹c.c.,
k sin k
2␧
p c⬘
s
This is a resonance supported by the array of resonators
where the air column oscillates in unison. It seems to be
worth examining this new resonance, but we are concerned
here with the resonance when sin k vanishes, i.e., k⫽m ␲
(⬎0) (m⫽1,2,3, . . . ).
Unlike in the case of the tube without the array, there are
⫾
(⬎0) for a given value of m.
two resonance frequencies ␴ m
They are determined from the quadratic equation ␴ 4 ⫺ 关 m 2
⫹(1⫹ ␬ ) ␴ 20 兴 ␴ 2 ⫹m 2 ␴ 20 ⫽0 in ␴ 2 as
⫾ 2
兲 ⫽ 21 兵 m 2 ⫹ 共 1⫹ ␬ 兲 ␴ 20
共␴m
⫾ 冑关 m 2 ⫹ 共 1⫹ ␬ 兲 ␴ 20 兴 2 ⫺4m 2 ␴ 20 其 ,
共36兲
with s⫽ ␴ 20 /( ␴ 20 ⫺ ␴ 2 ).
It is noted in 共36兲 that k becomes purely imaginary for
␴ 0 ⬍ ␴ ⬍ 冑1⫹ ␬ ␴ 0 . Then, cos关k(x⫺1)兴 and k sin k take, respectively, cosh关兩k兩(x⫺1)兴 and ⫺ 兩 k 兩 sinh兩k兩. As ␴ approaches
␴ 0 , k tends to diverge and the evanescence occurs in the
limit. This is due to the side-branch resonance where each
resonator reflects back the incident wave totally. As ␴ approaches 冑1⫹ ␬ ␴ 0 , on the other hand, k tends to vanish.
1776
FIG. 2. Graphs of the resonance frequencies ␴ m⫾ (m⫽1, 2, and 3兲: 共a兲
displays the dependence on ␴ 0 with the value of ␬ fixed at 0.2, and 共b兲
displays the dependence on ␬ with the value of ␴ 0 fixed at 2.5.
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
共37兲
⫺
with the signs ordered vertically. It is easily found that ␴ m
⫹
⬍ ␴ 0 ⬍ 冑1⫹ ␬ ␴ 0 ⬍ ␴ m . Figure 2 shows the dependence of
␴ m⫾ (m⫽1, 2, and 3兲 on ␴ 0 or ␬ for either one fixed. In the
tube without the array, the resonance frequencies are given
by ␴ ⫽m simply. In Fig. 2共a兲, where ␬ is fixed to be 0.2, it is
⫹
⫺
tends to m as ␴ 0 vanishes, whereas ␴ m
tends to
seen that ␴ m
⫹
⫺
and ␴ m
vanish. On the other hand, as ␴ 0 becomes large, ␴ m
tend to 冑1⫹ ␬ ␴ 0 and m/ 冑1⫹ ␬ , respectively, for mⰆ ␴ 0 .
⫺
’s that correspond to the
For a large value of ␴ 0 , it is ␴ m
resonance frequencies in the tube without the array. But,
Sugimoto et al.: Nonlinear frequency response of air column
since the tube then becomes consonant, the merit of connection of the array is lost. For a fixed value of ␴ 0 ⫽2.5, on the
⫾
on ␬ is shown in Fig. 2共b兲.
other hand, the dependence of ␴ m
⫺
decrease,
As ␬ increases, the resonance frequencies ␴ m
⫹
⫾
are exwhereas ␴ m increase. For a small value of ␬ , ␴ m
pressed asymptotically as
冋
册
␬ ␴ 20
␴ m⫹ ⫽ ␴ 0 1⫹
⫹¯ ,
2 共 ␴ 20 ⫺m 2 兲
and
冋
␴ m⫺ ⫽m 1⫺
␬ ␴ 20
2 共 ␴ 20 ⫺m 2 兲
册
共39兲
␴ m⫾
We now develop a nonlinear theory by taking account of
small but finite magnitude of ␧ and also lossy effects. While
the magnitudes of c and of ␧ p are defined clearly, the one of
␧ is left ambiguous and should now be related to c. When
the driving frequency ␴ is off the resonance, the linear solution 共36兲 suggests that ␧ is of the same order as c or ␧ p . This
is because the quantities on the left-hand side of 共36兲 are
regarded as being of order unity. But, as ␴ is set closer to the
resonance frequency, ␧ becomes much larger than c, as the
term ‘‘resonance’’ implies. Then, ␧ is determined not only by
c but also by a detuning ⌬␴, indicating how far the driving
frequency is set from the resonance one. Since the experiment exploits the resonance in the lowest mode, we consider
a case where ␴ is near the lowest resonance frequency ␴ ⫺
1
(⬅ ␴ 1 ). Setting ⌬ ␴ ⫽ ␴ ⫺ ␴ 1 , k sin k in the denominator of
共36兲 may be approximated as ⫺ ␲ (dk/d␴ ) 兩 ␴ ⫽ ␴ 1 ⌬ ␴ , so ␾,
p ⬘ , and p c⬘ are estimated to be of order c/␧ 兩 ⌬ ␴ 兩 . For this to
be of order unity, ␧ is determined as
c
.
兩⌬␴兩
⳵
⳵
⳵
⫹␧ 2 ⫹¯ .
⫽
⳵t ⳵t0
⳵t2
A. Perturbation procedures
In light of the ordering mentioned above, the exponential factor exp(i␲␴t) in 共27兲 may be written as exp关i␲(␴1
⫹␧2␴⬘)t兴 by setting ⌬ ␴ ⫽␧ 2 ␴ ⬘ , ␴ ⬘ being a quantity of order
unity. Regarding ␧ 2 t as a slow variable compared with t, we
employ the method of multiple 共two兲 scales by introducing
two variables t 0 (⬅t) and t 2 (⬅␧ 2 t). Then, the boundary
condition 共27兲 may be given, without the overbar, as
The lowest-order problem of ␧ takes the same form as 共29兲 to
共31兲 with ␾, p ⬘ , and p ⬘c replaced by ␾ (0) , f (0) , and g (0) ,
respectively. But, they are now subjected to the homogeneous boundary conditions: ⳵ ␾ (0) / ⳵ x⫽0 at x⫽0 and x⫽1,
since ⳵ ␾ / ⳵ x in 共41兲 is expanded around x⫽0 and E is assumed to be of order ␧ 2 . Thus, the lowest-order solutions are
given by the eigensolutions of the system. Taking, in 共36兲,
the formal limit as c/␧→0, k→ ␲ , and ␴ → ␴ 1 , and setting
( ␲ 2 ␴ 2 /k sin k)c/2␧ to be ␣, the solutions are obtained as
冋 册冋 册
␾ (0)
i/ ␲␴ 1
f (0) ⫽
1
cos关 ␲ 共 x⫺1 兲兴 ␣ ei ␲␴ 1 t 0 ⫹c.c.,
(0)
s1
g
共44兲
with s 1 ⫽ ␴ 20 /( ␴ 20 ⫺ ␴ 21 ) ( ␴ 1 ⫽ ␴ 0 ), where ␲ 2 ␴ 21 (1⫹ ␬ s 1 )
⫽ ␲ 2 and the complex amplitude ␣ including the detuning is
assumed to depend on t 2 but unspecified at this order.
Here, we make the following remark. The constant s 1
becomes positive or negative depending on the values of ␴ 0
and ␴ 1 . Therefore, the pressure in the cavity differs from the
one in the tube in magnitude and phase. If ␴ 0 is chosen
greater than ␴ 1 , as is usually the case, the pressure amplitude in the cavity becomes higher than the one in the tube.
1. First-order problem
Let us now proceed to the first-order problem. Then, the
equations take the following form:
⳵ 2 ␾ (1) ⳵ 2 ␾ (1)
⳵ g (1)
2 ⫺
2 ⫺␬
⳵x
⳵t0
⳵t0
冋
⫽⫺ ␬ 共 ␥ ⫺1 兲 g (0) ⫹ 共 ␥ ⫺2 兲
⫺
共41兲
冉 冊
⳵ ⳵ ␾ (0)
⳵t0 ⳵x
2
⫺ 共 ␥ ⫺1 兲
册
⳵ ␾ (0) ⳵ g (0)
⳵t0 ⳵t0
⳵ ␾ (0) ⳵ 2 ␾ (0)
,
⳵t0 ⳵x2
共45兲
and
⳵␾
E
⫽i ei ␲␴ 1 t 0 ⫹c.c.⫹O 共 ␧ 2 E 兲
⳵x
2
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
共43兲
共40兲
Since we assume the lossless case here, ␧ diverges, of
course, as ⌬␴ vanishes. The experimental result indicates
that c is of order 10⫺3 , 兩 ⌬ ␴ 兩 is of 10⫺2 , and ␧ is of 10⫺1 .
This suggests that c is of ␧ 3 and ⌬␴ is of ␧ 2 . Also, ␦ and ␦ r
for the lossy effects are estimated to be of ␧ 2 .
c
at x⫽ ei ␲␴ 1 t 0 ⫹c.c.⫹O 共 ␧ 2 c 兲 ,
2
共42兲
By introduction of t 2 , the differential operator is expanded
as
are exchanged for
IV. NONLINEAR THEORY
␧⬇
冋册冋 册 冋 册 冋 册
␾
␾ (0)
␾ (1)
␾ (2)
(0)
(1)
(2)
2
p⬘ ⫽ f
⫹␧ f
⫹␧ f
⫹¯ .
(0)
(1)
p c⬘
g
g
g (2)
共38兲
⫹¯ ,
for ␴ 0 ⬎m, while the expressions of
␴ 0 ⬍m.
where we note that E 关 ⫽( ␲␴ 1 c/␧)exp(i␲␴⬘t2)兴 is dependent
of t 2 and is of order ␧ 2 .
We seek the solutions to 共21兲 and 共24兲 with 共26兲 under
the boundary conditions 共41兲 and 共28兲 in the asymptotic expansion of ␧ as
⳵ 2 g (1)
共 ␥ ⫺1 兲 ⳵ 2 g (0)2
2 (1)
2 (1)
⫹
␲␴
g
⫺
␲␴
f
⫽
,
兲
兲
共
共
0
0
2
⳵ t 20
⳵ t 20
共46兲
with
Sugimoto et al.: Nonlinear frequency response of air column
1777
f (1) ⫽⫺
冉 冊 冉 冊
⳵ ␾ (1) 1 ⳵ ␾ (0)
⫹
⳵t0
2 ⳵t0
2
⫺
1 ⳵ ␾ (0)
2 ⳵x
2
,
共47兲
while the boundary conditions at this order take simply
⳵ ␾ (1)
⫽0 at x⫽0 and x⫽1.
⳵x
冋 册冋 册
␾
f (1) ⫽
g (1)
⌽ (1)
2
F (1)
2
G (1)
2
e2i ␲␴ 1 t 0 ⫹c.c.⫹
冋 册
⌽ (1)
0
F (1)
0
G (1)
0
共49兲
,
(1)
(1)
( j⫽0,2) are functions of x and
where ⌽ (1)
j , F j , and G j
␣ to be determined, and the subscript j implies the coefficient of the jth harmonics, ei j ␲␴ 1 t 0 . Substituting 共49兲 into
共45兲–共47兲, we consider the respective harmonics separately.
and G (1)
to
For the second harmonics, we eliminate F (1)
2
2
(1)
derive the equation for ⌽ 2 as follows:
d2 ⌽ (1)
2
dx 2
⫹k 22 ⌽ (1)
2 ⫽
(1)
F (1)
0 ⫽G 0 ⫽
共48兲
Introducing 共44兲 into the right-hand sides of 共45兲–共47兲,
we find the first-order solutions should be in the following
form:
(1)
For the zeroth harmonics, on the other hand, the righthand side of 共45兲 contains no terms independent of t 0 .
Therefore, we take ⌽ (1)
0 ⫽0. From 共46兲 and 共47兲, we have
冋
2. Second-order problem
Upon completion of the first-order problem, we proceed
to the second-order problem. It is given by
⳵ 2 ␾ (2) ⳵ 2 ␾ (2)
⳵ g (2)
⫺
⫺
␬
⫽K (2) ⫹L (2) ⫹M (2) ,
⳵x2
⳵t0
⳵ t 20
共50兲
with
K (2) ⫽ ␬
␲2
A0 ⫽ 2 ␥ ⫺3⫹ ␬ 共 s 1 ⫺s 2 兲 ␴ 21 ⫺s 2 ⫹ 共 ␥ ⫺1 兲
k2
⫻ 共 s 1 ⫺1 兲 s 1 ␴ 21 ⫹4 共 ␥ ⫺1 兲 s 21 s 2
and
A2 ⫽
再
␴ 21
⍀
冋
册冎
,
⫻共 s 1 ⫺1 兲 s 1 ␴ 21 ⫹4 共 ␥ ⫺1 兲 s 21 s 2
⍀
册冎
,
共51兲
共52兲
i
兵 A ⫹A2 cos关 2 ␲ 共 x⫺1 兲兴 其 ␣ 2 ,
2 ␲␴ 1 0
冉 冊 冋 冉 冊册
冎
册 冋
再冋 冉 冊
册
冉 冊
冎
再
F (1)
2 ⫽ A0 ⫹
共54兲
⳵ ␾ (0)
⳵x
共 ␥ ⫺1 兲共 ␥ ⫺2 兲 ⳵ ␾ (0) ⳵ g (0)2
2
⳵t0 ⳵t0
⫹
共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 ⳵ g (0)3
,
6
⳵t0
2
⫹
⳵ g (0)
⳵t0
2
冎
冉 冊
共58兲
⳵ 2 ␾ (0) ␦ ⳵ ⫺ 1/2 ⳵ 2 ␾ (0)
⫺
,
⳵ t 2 ⳵ t 0 ␧ 2 ⳵ t ⫺1/2
⳵x2
0
冉
冊
共59兲
冉
⳵
⳵ ␾ (0) ⳵ ␾ (1)
⳵ ␾ (0) ⳵ 2 ␾ (1)
2
⫺ 共 ␥ ⫺1 兲
⳵t0
⳵x
⳵x
⳵t0 ⳵x2
冊
冉 冊
⳵ ␾ (1) ⳵ 2 ␾ (0)
共 ␥ ⫹1 兲 ⳵ ␾ (0)
⫺
2
⳵t0 ⳵x
2
⳵x
2
⳵ 2 ␾ (0)
,
⳵x2
⳵ 2 g (2)
⫹ 共 ␲␴ 0 兲 2 g (2) ⫺ 共 ␲␴ 0 兲 2 f (2)
⳵ t 20
⫽⫺2
s 21
1
1
1⫺ 2 ⫺ 共 ␥ ⫺1 兲
cos关 2 ␲ 共 x⫺1 兲兴 ␣ 2 .
4
⍀
␴1
共55兲
1778
⫹
⫹
⳵ ␾ (0)
⳵t0
and
s 21
1
1
s 2 A0 ⫹
1⫹ 2 ⫺ 共 ␥ ⫺1 兲
⫹s 2 A2
4
⍀
␴1
⫹
共 ␥ ⫺2 兲
2
共60兲
1
1
1
1
1⫹ 2 ⫹ A2 ⫹
1⫺ 2
4
4
␴1
␴1
⫻cos关 2 ␲ 共 x⫺1 兲兴 ␣ 2 ,
G (1)
2 ⫽
共53兲
冉
冊
冋冉 冊 冉 冊 册
⳵ ␾ (0) ⳵ g (1) ⳵ ␾ (1) ⳵ g (0)
⫹
⳵t0 ⳵t0
⳵t0 ⳵t0
⫺
M (2) ⫽⫺
共57兲
⳵ g (0)
⳵
⫺ 共 ␥ ⫺1 兲
共 g (0) g (1) 兲
⳵t2
⳵t0
L (2) ⫽⫺2
with ⍀⫽ ␴ 20 / ␴ 21 . Note that k 22 ⫽4 ␲ 2 but k 2 is assumed not to
vanish, i.e., ␴ 1 ⫽ 冑1⫹ ␬ ␴ 0 /2. The second-harmonic solutions are easily obtainable as
⌽ (1)
2 ⫽
再
⫺ 共 ␥ ⫺2 兲
␲2
␥ ⫹1⫹ ␬ 共 s 1 ⫺s 2 兲 ␴ 21 ⫹s 2 ⫹ 共 ␥ ⫺1 兲
共 k 22 ⫺4 ␲ 2 兲
␴ 21
共56兲
These solutions show that while no steady streaming occurs
at this order, the steady but nonuniform pressure distribution
appears in the tube as well as in the cavities. The maximum
兩 ␣ 兩 2 occurs at both ends, while the minimum ⫺ 兩 ␣ 兩 2 / ␴ 21 occurs in the middle. Note that the effect of the array, i.e., ␬,
appears only in the minimum through ␴ 1 .
with k 22 ⫽(2 ␲␴ 1 ) 2 (1⫹ ␬ s 2 ) and s 2 ⫽ ␴ 20 /( ␴ 20 ⫺4 ␴ 21 ) ( ␴ 1
⫽ ␴ 0 /2), where A0 and A2 are given by
再
1
1
1
1
1⫺ 2 ⫹
1⫹ 2
2
2
␴1
␴1
⫻cos关 2 ␲ 共 x⫺1 兲兴 兩 ␣ 兩 2 .
i
兵 k 2 A ⫹ 共 k 22 ⫺4 ␲ 2 兲 A2
2 ␲␴ 1 2 0
⫻cos关 2 ␲ 共 x⫺1 兲兴 其 ␣ 2 ,
再冉 冊 冉 冊
冎
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
⫺
⳵ 2 g (0) ␦ r ⳵ 3/2g (0)
⳵ 2 (0) (1)
⫺ 2
共g g 兲
3/2 ⫹ 共 ␥ ⫺1 兲
⳵ t 2⳵ t 0 ␧ ⳵ t 0
⳵ t 20
共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 ⳵ 2 g (0)3
,
6
⳵ t 20
共61兲
with
Sugimoto et al.: Nonlinear frequency response of air column
f (2) ⫽⫺
⫹
⳵ ␾ (2) ⳵ ␾ (0) ⳵ ␾ (0) ⳵ ␾ (1) ⳵ ␾ (0) ⳵ ␾ (1)
⫺
⫹
⫺
⳵t0
⳵t2
⳵t0 ⳵t0
⳵x
⳵x
冉 冊
共 ␥ ⫺2 兲 ⳵ ␾ (0)
6
⳵t0
3
⫹
冉 冊
1 ⳵ ␾ (0) ⳵ ␾ (0)
2 ⳵t0
⳵x
Q 3 ⫽⫺
2
.
共62兲
The boundary conditions now take
共63兲
⳵␾
⫽0 at x⫽1.
⳵x
(2)
共64兲
At this order, we do not necessarily have to seek the full
solutions. We have only to derive the condition for the complex amplitude ␣ from the boundary conditions. Setting the
solutions in the following form:
冋 册冋 册 冋 册
⌽ (2)
⌽ (2)
␾ (2)
1
3
(2)
i ␲␴ 1 t 0
f (2) ⫽ F (2)
F
e
⫹
e3i ␲␴ 1 t 0 ⫹c.c.,
1
3
(2)
(2)
(2)
g
G1
G3
共65兲
and substituting this into 共57兲–共62兲, we end up with
⫹k 21 ⌽ (2)
1 ⫽B cos关 ␲ 共 x⫺1 兲兴
⫹terms in cos关 3 ␲ 共 x⫺1 兲兴 ,
共66兲
S re ⫽⫺S im ⫽⫺
Q 1 ⫽⫺
再
冑
Q 2⫽
冏
冉
冋
,
共68兲
␴⫽␴1
冊
␬ ␦ r s 21
␲␴ 1 ␦
⫹
,
2 ␴ 21
⍀
共69兲
␲
␥ ⫺1⫹ ␬ ␴ 21 ⫺ 共 ␥ ⫺3 兲 s 1 ⫹2 共 ␥ ⫺2 兲 s 2
␴1
⫺ 共 ␥ ⫺1 兲 s 1 s 2 ⫺ 共 ␥ ⫺1 兲
再
s 21 s 2
⍀
冋冉
册冎
A0 ,
共70兲
冊
␲
1
␥ ⫹1⫺ ␬ ␴ 21 ⫺ ␥ ⫺3⫹ 2 s 1 ⫹2 共 ␥ ⫺2 兲 s 2
2␴1
␴1
⫺ 共 ␥ ⫺1 兲 s 1 s 2 ⫺ 共 ␥ ⫺1 兲
s 21 s 2
⍀
␴ 21
s2 ⫹
共 ␥ ⫺3 兲
␴ 21
s1
␲ ␬ ␴ 1 共 ␥ ⫺1 兲
6 共 ␥ ⫺2 兲 s 21
8
1
1
s s ⫺3 共 2 ␥ ⫺1 兲 s 31 ⫹ 2 3⫺ 2
␴ 21 1 2
␴1
⫺12共 ␥ ⫺1 兲 1⫹
1
␴ 21
s 2 ⫺3 共 2 ␥ ⫺1 兲 s 21
s 21
s1
s 1s 2
.
⍀
⍀
共72兲
Here, note that the fractional derivative of minus half-order
of the exponential function is reduced to the Fresnel
integral11 and given simply as
⳵ ⫺ 1/2 i ␲␴ t
共 1⫺i 兲 i ␲␴ t
1 0⫽
e 1 0 ⫽ 共 i ␲␴ 1 兲 ⫺1/2ei ␲␴ 1 t 0 .
⫺1/2 e
⳵t0
冑2 ␲␴ 1
共73兲
The derivative of three-half order is given by ⫺(1⫺i)
⫻( ␲␴ 1 ) 2 / 冑2 ␲␴ 1 exp(i␲␴1t0) 关 ⫽(i ␲␴ 1 ) 3/2exp(i␲␴1t0)兴. It is
found from 共69兲 that since ␬ is small, the friction loss due to
the throat wall becomes comparably small with the one due
to the tube wall. In addition, when ⍀ is chosen large, it is
made even smaller.
The solution ⌽ (2)
1 is obtained as
⌽ (2)
1 ⫽
B
共 x⫺1 兲 sin关 ␲ 共 x⫺1 兲兴 ⫹C1 cos关 ␲ 共 x⫺1 兲兴
2␲
共74兲
共67兲
where ␮, S (⫽S re ⫹iS im ), and Q 0 (⫽Q 1 ⫹Q 2 ⫹Q 3 ) are defined as
s1
2 dk
⫽
⍀
␲␴ 1 d␴
1
6 共 2 ␥ ⫺3 兲 ⫹
⫹C2 sin关 ␲ 共 x⫺1 兲兴 ⫹terms in cos关 3 ␲ 共 x⫺1 兲兴 ,
S
⳵␣
⫹i 2 ␣ ⫹iQ 0 兩 ␣ 兩 2 ␣ ,
B⫽⫺ ␮
⳵t2
␧
冉 冊
␲␬␴1
8
⫹24共 ␥ ⫺2 兲 s 2 ⫹ 3⫹
with k 21 ⫽( ␲␴ 1 ) 2 (1⫹ ␬ s 1 )⫽ ␲ 2 and
␮ ⫽2⫹2 ␬ s 1 1⫹
⫹
⫺2 共 ␥ ⫺2 兲 3⫹
and
dx 2
8 ␴ 31
⫹ 3⫹
⳵ ␾ (2)
E
⫽i 2 ei ␲␴ 1 t 0 ⫹c.c. at x⫽0,
⳵x
2␧
d2 ⌽ (2)
1
册
再冋
冉 冊冎
再
冉 冊
冋冉 冊
冉 冊
冉 冊 册 冎
3 ␲ 共 ␥ ⫹1 兲
册冎
A2 ,
and
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
共71兲
where C1 and C2 are integration constants to be determined.
By imposing the boundary conditions 共63兲 and 共64兲, it follows that while C1 , i.e., the coefficient of the homogeneous
solution, is left undetermined within the present framework
of the theory,12 C2 must vanish, and B⫽iE/␧ 2 . The latter
gives the equation governing the behavior of ␣ as
i␮
⳵␣
S
E
⫹ ␣ ⫹Q 0 兩 ␣ 兩 2 ␣ ⫽ 2 .
⳵t2 ␧2
␧
共75兲
Here, we examine the value of Q 0 for a large value of ⍀.
Noting that s 1 ⫽1/(1⫺1/⍀)⬇1⫹1/⍀⫹¯ and s 2 ⫽1/(1
⫺4/⍀)⬇1⫹4/⍀⫹¯ so that s 2 ⫺s 1 Ⰶ1, and k 22 ⫽4 ␲ 2 关 1
⫹ ␬ (s 2 ⫺s 1 ) ␴ 21 兴 , it is found that A2 is much larger than the
other terms and therefore Q 2 is dominant in Q 0 . Approximating A2 to be
A 2⬇
␲ 2 共 ␥ ⫹1⫹ ␬ 兲
,
共 k 22 ⫺4 ␲ 2 兲
共76兲
Q 0 may be evaluated as
Q 0 ⬇Q 2 ⬇
␲ 3 共 ␥ ⫹1⫹ ␬ 兲 2
⬎0.
2 ␴ 1 共 k 22 ⫺4 ␲ 2 兲
共77兲
Thus, it is found that Q 0 is positive for ⍀Ⰷ1.
Sugimoto et al.: Nonlinear frequency response of air column
1779
We proceed to complete the second-order problem by
taking account of all terms including those unspecified so far.
Although the calculations are straightforward, the full expressions for ␾ (2) , f (2) , g (2) are too complicated to be reproduced here. By making use of the same approximation leading to 共76兲, we present here only the leading expressions for
the respective modes
⌽ (2)
1 ⬇⫺
i 共 ␥ ⫺3⫺ ␬ 兲
A2 cos关 3 ␲ 共 x⫺1 兲兴 兩 ␣ 兩 2 ␣ ,
16␲␴ 1
3i ␲ 共 ␥ ⫺3⫺ ␬ 兲
⌽ (2)
3 ⬇
2 ␴ 1 共 k 23 ⫺ ␲ 2 兲
⫹
2 ␴ 1 共 k 23 ⫺9 ␲ 2 兲
冉 冊册
F (2)
3 ⬇
冋
A2 cos关 3 ␲ 共 x⫺1 兲兴 ␣ 3 ,
2 共 k 23 ⫺ ␲ 2 兲
⫺1 兲兴 ␣ 3 ⫹
冋
冉 冊册
1
1
1⫹ 2
2
␴1
9 ␲ 2 共 ␥ ⫹1⫹ ␬ 兲
2 共 k 23 ⫺9 ␲ 2 兲
⳵␺
⫺ ␲ ␮ ⌬ ␴ 兩 P 兩 ⫹S re 兩 P 兩 ⫹Q 兩 P 兩 3 ⫽⌫ cos ␺ ,
⳵t
冉
1
S ⫹Q 兩 P 兩 2 ⫾
⌬␴⫽
␲ ␮ re
共80兲
A2 cos关 ␲ 共 x
A2 cos关 3 ␲ 共 x⫺1 兲兴 ␣ 3 ,
共81兲
(2)
G (2)
1 ⬇s 1 F 1 ,
共82兲
(2)
G (2)
3 ⬇s 3 F 3 ,
共83兲
with k 23 ⫽(3 ␲␴ 1 ) 2 (1⫹ ␬ s 3 )⫽9 ␲ 2 and s 3 ⫽ ␴ 20 /( ␴ 20 ⫺9 ␴ 21 ),
where the approximation 共76兲 is used, and s 1 and s 3 in G (2)
1
2
and G (2)
3 are left unapproximated. Here, we assume that k 3
2
⫽ ␲ and ␴ 1 ⫽ ␴ 0 /3. Details will be discussed later.
B. Steady-state solution
Let us examine the steady-state solution to 共75兲. Before
doing this, we transform the equation into a form suitable for
comparison with the results of measurements. So far ␧ has
been used conveniently to make a note of ordering in developing the asymptotic analysis. From the viewpoint of the
experiment, however, it is not a quantity to be measured
easily, while use of ␣ is not also suitable for comparison.
Because an available quantity is the excess pressure in the
tube, we rewrite 共75兲 in terms of the dimensionless excess
pressure relative to p 0 . In view of 共18兲, 共42兲, and 共44兲, we
set
␳ 0 a 0 u 0 p̄ ⬘
1
⫽␧ ␥ p̄ ⬘ ⫽ cos关 ␲ 共 x⫺1 兲兴 Pei ␲␴ 1 t ⫹c.c.⫹¯ ,
p0
2
共84兲
where P⫽2␧ ␥␣ (Ⰶ1) to the lowest order. Noting that
⳵ ␣ / ⳵ t⫽␧ 2 ⳵ ␣ / ⳵ t 2 ⫹¯ , and using P instead of ␣, 共75兲 is
rewritten as
1780
共86兲
共87兲
with ␺ ⫽ ␲ ⌬ ␴ t⫺ ␪ . For the steady-state solution, we drop
⳵ / ⳵ t to have S im 兩 P 兩 ⫽⌫ sin ␺, ␪ ⫽ ␲ ⌬ ␴ t⫹constant, and
共 ␥ ⫺3⫺ ␬ 兲
16
2
⫹
⳵兩 P兩
⫹S im 兩 P 兩 ⫽⌫ sin ␺ ,
⳵t
共79兲
冉 冊
A2 cos关 3 ␲ 共 x⫺1 兲兴 兩 ␣ 兩 ␣ ,
9 ␲ 2 共 ␥ ⫺3⫺ ␬ 兲
␮
␮兩 P兩
⫻cos关 ␲ 共 x⫺1 兲兴 兩 ␣ 兩 2 ␣ ⫺
共85兲
and
i ⳵␣
1
1
cos关 ␲ 共 x⫺1 兲兴 ⫹
1⫺ 2 A2
F (2)
1 ⬇⫺
␲␴ 1 ⳵ t 2
2
␴1
1
1
⫺
1⫹ 2
2
␴1
⳵P
⫹S P⫹Q 兩 P 兩 2 P⫽⌫ei ␲ ⌬ ␴ t ,
⳵t
with Q⫽Q 0 /4␥ 2 , ⌫⫽2 ␲ ␥ ␴ 1 c, and ⌬ ␴ ⫽␧ 2 ␴ ⬘ .
Setting P⫽ 兩 P 兩 exp(i␪) and separating the real and imaginary parts, it follows that
共78兲
A2 cos关 ␲ 共 x⫺1 兲兴 ␣ 3
3i ␲ 共 ␥ ⫹1⫹ ␬ 兲
i␮
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
冊
冑
⌫2
2
⫺S im
.
兩 P兩2
共88兲
It is found that 兩 P 兩 is bounded at ⌫/S im . This suggests that
the peak amplitude is proportional to the magnitude of excitation ⌫, i.e., c. It is also found that the peak frequency is
shifted downward by the dispersion due to the wall friction
while shifted upward by nonlinearity, because S re ⬍0 and
Q⬎0 for ⍀Ⰷ1. If ⌫Ⰶ1 so that 兩 P 兩 Ⰶ1, we can neglect
Q 兩 P 兩 2 P in 共85兲 to have immediately the steady-state solution
as
P⫽
⌫ei ␲ ⌬ ␴ t
.
S⫺ ␲ ␮ ⌬ ␴
共89兲
Thus, we obtain the linear resonance curve given by
兩 P兩⫽
⌫
2
冑共 ␲ ␮ ⌬ ␴ ⫺S re 兲 2 ⫹S im
,
共90兲
which is consistent with 共88兲 in the limit as P→0.
Incidentally, 共90兲 can also be derived from the linear
theory shown previously.4 When the lossy effects are included, the form of p ⬘ in 共36兲 is unaltered but k 2 is modified
as
k 2⫽ ␲ 2␴ 2
再
1⫹ ␬ ␴ 20 / 关 ␴ 20 ⫺ ␴ 2 ⫺ 共 1⫺i 兲 ␦ r ␴ 2 / 冑2 ␲␴ 兴
1⫺ 共 1⫺i 兲 ␦ / 冑2 ␲␴
冎
.
共91兲
See the expression for k below 共4兲 in Ref. 4. But, note the
difference in the sign of argument of the exponential function
for the piston displacement. This difference yields ⫺i instead of ⫹i. Expanding the right-hand side of 共91兲 around
␴ ⫽ ␴ 0 by setting ␴ ⫽ ␴ 1 ⫹⌬ ␴ , we retain the lossless terms
up to the order of ⌬␴ but truncate the lossy terms to the
lowest order. Thus, k is obtained as ␲ ⫹( ␲ ␮ ⌬ ␴ ⫺S) ␴ 1 /2.
Next, noting that ␣ in 共44兲 corresponds to
( ␲ 2 ␴ 2 /k sin k)c/2␧, we approximate k sin k as ⫺ ␲ (k⫺ ␲ )
and take the lowest terms for ␣ to recover 共90兲 after multiplying it by 2␧␥ in the relation between ␣ and P.
Sugimoto et al.: Nonlinear frequency response of air column
V. DISCUSSION OF THE RESULTS
A. Nonlinear frequency response
The nonlinear frequency response in the steady state is
described by 共88兲 theoretically. The peak amplitude P peak
attained in the response curve is determined to be ⌫/S im by
the linear loss, while the peak frequency at P⫽ P peak is lowered by ⌬ ␴ ⫽S re / ␲ ␮ due to the dispersive effects of the
boundary layer. This result is merely an extension of the
linear case. Although the linear downshift is constant, the
shift dependent on the amplitude occurs through Q 兩 P 兩 2 P,
which is responsible for bending 共skew兲 of the response
curve toward the higher or lower frequency side. We examine variations of Q as ␴ 0 is changed with ␬ fixed. This corresponds physically to changing cavity volume and axial
spacing, with the tube length, its cross-sectional area, and the
throat length held constant.
From 共70兲–共72兲 with 共51兲 and 共52兲, it is expected that Q
may diverge at 共i兲 兩 s 1 兩 ⫽⬁; 共ii兲 兩 s 2 兩 ⫽⬁; 共iii兲 k 22 ⫽0; or 共iv兲
k 22 ⫽4 ␲ 2 . The first case 共i兲 where ␴ 1 ⫽ ␴ 0 does not occur
unless ␬ ⫽0 or ␴ 0 ⫽0, as is seen from Fig. 2共a兲 or 共37兲. The
second case 共ii兲 with 2 ␴ 1 ⫽ ␴ 0 may occur if the frequency of
the second harmonics hits ␴ 0 , whereby ␴ 0 and ␬ must satisfy the relation ␴ 0 ⫽ 冑12/(3⫹4 ␬ ). But, because of evanescence at ␴ 0 , no divergence of Q occurs there. In fact, it is
confirmed that all terms proportional to s 2 in Q cancel out
altogether. The third case 共iii兲 corresponds to the one with
2 ␴ 1 ⫽ 冑1⫹ ␬ ␴ 0 for resonance, and indeed occurs when ␴ 0
and ␬ satisfy the following relation:
␴ 0⫽
冑
4 共 3⫺ ␬ 兲
⬅ ␴ cr ⬍2.
3 共 1⫹ ␬ 兲 2
共92兲
The fourth case 共iv兲 corresponds to the second harmonic
resonance when 2 ␴ 1 hits 2 ␴ ⫾ . But this case does not occur
because ␴ 1 ⫽0 nor ␴ 1 ⫽1. Besides the third case, Q diverges in the limit as ␴ 0 →0. This divergence is very rapid as
⫺ ␲ ( ␥ ⫺1)(2 ␥ ⫹1)/32␥ 2 ␬ 3 ␴ 70 . In addition, we have assumed for the second-order solutions that k 23 ⫽ ␲ 2 or ␴ 1
⫽ ␴ 0 /3. The equality k 23 ⫽ ␲ 2 holds for ␴ 1 ⫽ 冑␴ 0 /3, where
␴ 0 ⫽(5⫾ 冑16⫺9 ␬ )/3(1⫹ ␬ ). The equality ␴ 1 ⫽ ␴ 0 /3 holds
for ␴ 0 ⫽ 冑72/(8⫹9 ␬ ). Although these cases have no direct
bearing on 共85兲, they should be avoided in the sense of seeking uniformly valid, asymptotic solutions.
Taking account of such exceptional cases, Q is drawn
versus ␴ 0 in Fig. 3共a兲 for three fixed values of ␬. The broken,
solid, and dotted lines represent the values of Q for ␬
⫽0.1, 0.2, and 0.3, respectively. As is seen, Q takes both
positive and negative values. This means that the response
curve may be bent rightward or leftward, depending on the
choice of ␴ 0 . While Q diverges at ␴ 0 ⫽ ␴ cr and at ␴ 0 ⫽0, it
grows as ␴ 0 increases. In this limit, the asymptotic expression 共77兲 for Q 0 is already available so that Q is approximated as
Q⫽
␲ 3 共 ␥ ⫹1⫹ ␬ 兲 2
Q0
.
2 ⬇Q a ⫽
4␥
8 ␥ 2 ␴ 1 共 k 22 ⫺4 ␲ 2 兲
共93兲
Figure 3共b兲 shows the relative error 兩 (Q⫺Q a )/Q 兩 for three
values of ␬ ⫽0.1, 0.2, and 0.3 by using the same types of
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
FIG. 3. Graphs of the coefficient Q versus ␴ 0 with ␬ fixed at 0.1, 0.2 and
0.3: 共a兲 displays the values of Q where the respective values diverge at ␴ 0
⫽ ␴ cr ⬇1.79, 1.61, and 1.46 and at ␴ 0 ⫽0, and 共b兲 displays the relative
errors 兩 (Q⫺Q a )/Q 兩 of the asymptotic values Q a to Q for the three values of
␬.
lines as in Fig. 3共a兲. It is found that Q a agrees with Q very
well for ␴ ⬎2 and the relative errors are almost less than 1%.
Even below ␴ cr , there is a region in which Q a still gives a
good approximation to Q.
The other parameters ␮ and S are relatively simple. Figure 4 displays the graphs of ␮ and S im versus ␴ 0 for ␬
⫽0.1, 0.2, and 0.3, where the meaning of D will be given
later. Three families of curves labeled ␮, S im , and D are
drawn. In each family, the broken, solid, and dotted lines
represent the values of labeled quantity for ␬ ⫽0.1, 0.2, and
0.3, respectively. In order to draw the graphs of S im , the
values of ␦ and ␦ r are set to be 0.0282 and 0.223 in view of
the experiment to be decribed in the next subsection. The
value of ␦ r is a little larger than the estimation. It is found
that all quantities decrease as ␴ 0 increases, but they are almost constant for ␴ ⬎2, and they decrease as ␬ becomes
smaller.
B. Comparison with the experimental result
We now compare the frequency response with the one
measured by using the tube of length l (⫽3256 mm) and of
diameter 2R (⫽80 mm) with the Helmholtz resonator having a cavity of volume V (⫽49.7⫻10⫺6 m3 ), and a throat of
length L (⫽35.6 mm) and of diameter 2r (⫽7.11 mm) conSugimoto et al.: Nonlinear frequency response of air column
1781
FIG. 4. Graphs of the coefficients ␮, S im , and D vs ␴ 0 for ␬ ⫽0.1, 0.2, and
0.3.
nected with the axial spacing d (⫽50 mm). The size parameter ␬ takes the value 0.198. Figure 5 is the reproduction of
the frequency response reported in Ref. 4 共see Fig. 4兲, where
the half of the peak-to-peak value of the excess pressure
measured on the flat plate at the closed end, ␦ p, relative to
the atmospheric pressure p 0 is drawn versus the frequency of
excitation ␻/2␲. The blank triangle, solid triangle, blank
circle, and solid circle indicate the data measured at the displacement amplitude of the bellows X b ⫽0.5, 1.5, 2.5, and
3.5 mm, respectively. The equilibrium pressure and the mean
temperature in the tube near the closed end are 1.007
⫻105 Pa and 25.6 °C, respectively, so a 0 ⫽347.1 m/s, ␯
⫽1.567⫻10⫺5 m2 /s, Pr⫽0.7089, and ␥ ⫽1.402. At this
temperature, the natural frequency of the resonator ␻ 0 /2␲ is
calculated to be 242.6 Hz with the end corrections. Thus, we
find that ␴ 0 ⫽4.55 and ␴ 1 ⫽0.911.
Since ␬ ⫽0.198 and ␴ 0 ⫽4.55 for the experiment, we
have ␮ ⫽2.43, Q⫽15.2, ⫺S re ⫽S im ⫽0.0430. In Fig. 5, the
FIG. 5. Comparison of the nonlinear frequency response obtained by the
theory and experiment for the displacement amplitude of the bellows X b
⭓0.5 mm, where the broken and solid lines represent, respectively, the
curves without taking account of the jet loss and with it, while the measured
data are indicated by the blank triangle, solid triangle, blank circle, and solid
circle for X b ⫽0.5, 1.5, 2.5, and 3.5 mm, respectively, X b being related to the
one of the piston by X p ⫽1.422X b .
1782
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
FIG. 6. Comparison of the nonlinear frequency response obtained by the
theory and experiment for the displacement amplitude of the bellows X b
⬍0.5 mm, where the broken and solid lines represent, respectively, the
curves without taking account of the jet loss and with it, while the measured
data are indicated by the blank square, solid square, blank triangle, and solid
triangle for X b ⫽0.1, 0.2, 0.3, and 0.4 mm, respectively, X b being related to
the one of the piston by X p ⫽1.422X b .
broken lines represent 兩 P 兩 versus the dimensional frequency
( ␴ 1 ⫹⌬ ␴ )a 0 /2l by using 共88兲. The displacement of the bellows is converted into the one of the piston by multiplying it
with a factor 1.422,4 so that ⌫ is given by 2 ␲ ␥ ␴ 1 X p /l with
X p ⫽1.422X b . The theory overestimates not only the peak
amplitudes but also the peak frequencies considerably, even
for X b ⫽0.5 mm. When higher-order corrections to p ⬘ in 兩 P 兩
on the right-hand side of 共84兲 are taken into account, it is
found that the deviation becomes worse. This is understood
from the nature of asymptotic expansions that inclusion of
higher-order terms does not necessarily improve the accuracy. Rather, the smallness of 兩 P 兩 is required. Hence, it turns
out that the theory based on 共88兲 fails to describe the experimental results adequately.
So, the experimental data for X b smaller than 0.5 mm
are presented, though unpublished so far. Figure 6 shows the
frequency response where the blank square, solid square,
blank triangle, and solid triangle indicate the data measured
at X b ⫽0.1, 0.2, 0.3, and 0.4 mm, respectively, and the equilibrium pressure and the mean temperature are 1.010
⫻105 Pa and 23.1 °C, respectively. As in Fig. 5, the broken
lines represent 兩 P 兩 versus the dimensional frequency calculated by 共88兲 where a 0 ⫽345.6 m/s, ␯ ⫽1.539⫻10⫺5 m2 /s,
Pr⫽0.7095, and ␥ ⫽1.402. While ␻ 0 /2␲ is now shifted to
be 241.3 Hz, note that both ␴ 0 and ␴ 1 are independent of a 0
and therefore of the temperature. It is seen in Fig. 6 that as
X b is decreased, the broken lines tend to approach the data
measured. But, the theory overestimates the peak amplitude
slightly even for the smallest excitation with X b ⫽0.1 mm.
C. Effect of jet loss
The discrepancy may be attributed to some effects not
taken into account in the analysis. One is an effect of jet loss
which occurs at the throat of the resonator. As the amplitude
of oscillatory flows in the throat becomes large, the flow
pattern 共except for direction of flow兲 just entering the throat
Sugimoto et al.: Nonlinear frequency response of air column
differs from the one leaving the throat 共see Fig. 1兲. When the
air is sucked into the throat, it flows toward the orifice omnidirectionally, whereas when it is ejected, it flows in the
form of jet unidirectionally. This asymmetry of flow pattern
on the suction and ejection sides gives rise to the jet loss.
The detailed analysis is given in the Appendix of Ref. 10 and
is not reproduced here. The jet loss introduces the additional
term
⫺
V
␳ 0 a 20 BL e
冏 冏
⳵ p ⬘c ⳵ p ⬘c
,
⳵t ⳵t
共94兲
to the right-hand side of 共9兲 for the behavior of the resonator.
Then, the normalized equation 共24兲 is modified into
⳵ 2 p̄ ⬘c
⳵ t̄ 2
⫹␦r
⳵ 3/2p̄ c⬘
⳵ t̄ 3/2
⫺
⫹ 共 ␲␴ 0 兲 2 p̄ ⬘c
2
共 ␥ ⫺1 兲 ⳵ 2 p̄ c⬘
⳵ t̄
2
3
共 ␥ ⫺1 兲共 2 ␥ ⫺1 兲 ⳵ 2 p̄ ⬘c
2
⳵ t̄ 2
6
i␮
2
冏 冏
⫺␧ ␦ J
⳵ p̄ ⬘c ⳵ p̄ ⬘c
⳵ t̄
⳵ t̄
,
共95兲
where ␦ J ⫽V/BL e ⫽(l/ ␲␴ 0 L e ) 2 . Since the new term is accompanied by ␧, it yields additional terms in the first-order
solution. In fact, it follows that
冏 冏
⳵ g (0) ⳵ g (0)
⫽⫺4 共 ␲␴ 1 s 1 兲 2 共 兩 sin ␰ 兩 sin ␰ 兲 兵 兩 cos关 ␲ 共 x⫺1 兲兴 兩
⳵t0 ⳵t0
⫻cos关 ␲ 共 x⫺1 兲兴 其 兩 ␣ 兩 2 ,
共96兲
with ␣ ⫽ 兩 ␣ 兩 e i ␪ and ␰ ⫽ ␲␴ 1 t 0 ⫹ ␪ . Using the following Fourier expansions:
⬁
2i
关共 ⫺1 兲 n ⫺1 兴 in ␰
兩 sin ␰ 兩 sin ␰ ⫽⫺
e
␲ n⫽⫺⬁ n 共 n 2 ⫺4 兲
兺
⫽⫺
4i i ␰
4i 3i ␰
e ⫹
e ⫹¯⫹c.c.,
3␲
15␲
共97兲
9 ␲ ␧ ␴ 20
兩␣兩␣,
共99兲
兩 cos关 ␲ 共 x⫺1 兲兴 兩 cos关 ␲ 共 x⫺1 兲兴
4
sin共 n ␲ /2兲 in ␩
⫽⫺
e
␲ n⫽⫺⬁ n 共 n 2 ⫺4 兲
兺
共98兲
with ␲ x⫽ ␩ and the sums taken except for n⫽0 and n
⫽⫾2, there arises in 共46兲 the term proportional to cos关␲(x
⫺1)兴exp(i␲␴1t0)兩␣兩␣. Note that although cos关␲(x⫺1)兴 is defined only in the interval 0⭐x⭐1 originally, it is extended
periodically to the outside of it with period 2 and so is the
case with 兩 cos关␲(x⫺1)兴兩. Thus, the solution 共49兲 must include the component ⌽ (1)
1 exp(i␲␴1t0). As is inferred from
the analysis for the second-order problem, this term gives
rise to the equation for ⌽ (1)
1 corresponding to 共66兲 with B
proportional to 兩 ␣ 兩 ␣ . Applying the boundary conditions 共48兲,
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
共100兲
with D 0 ⫽128␬ ␦ J ␴ 31 s 31 /9␲␴ 20 . In terms of P, finally, 共85兲 is
modified as
i␮
⳵P
⫹S P⫹iD 兩 P 兩 P⫹Q 兩 P 兩 2 P⫽⌫ei ␲ ⌬ ␴ t ,
⳵t
共101兲
with
D⫽
64␬ ␦ J ␴ 1 s 31
9␲␥⍀
共102兲
.
In Fig. 4, the graph of D vs ␴ 0 is displayed for ␬ ⫽0.1, 0.2,
and 0.3.
Separating the real and imaginary parts of 共101兲 by setting P⫽ 兩 P 兩 exp(i␪), the only correction to 共86兲 and 共87兲 is to
replace S im with S im ⫹D 兩 P 兩 . Then, the frequency response is
given by
⌬␴⫽
冋
1
S ⫹Q 兩 P 兩 2 ⫾
␲ ␮ re
P peak⫽
⬁
8
8
cos关 ␲ 共 x⫺1 兲兴 ⫹
cos 3 关 ␲ 共 x⫺1 兲兴 ⫹¯ ,
3␲
15␲
⳵␣
E
S
D0
⫹ 2 ␣ ⫹i
兩 ␣ 兩 ␣ ⫹Q 0 兩 ␣ 兩 2 ␣ ⫽ ␴ 1 2 ,
⳵t2 ␧
␧
␧
冑
册
⌫2
⫺ 共 S im ⫹D 兩 P 兩 兲 2 .
兩 P兩2
共103兲
It is found that the jet loss adds D 兩 P 兩 to the linear one S im ,
and then the peak amplitude is determined as
and
⫽
128␬ ␦ J ␴ 31 s 31
and 共75兲 is replaced by
⫽ 共 ␲␴ 0 兲 2 p̄ ⬘ ⫹␧
⫺␧
it follows that ␣ must vanish. To remedy this, it may be
considered to introduce another time variable t 1 (⬅␧t). But,
as ⌬ ␴ and E are assumed to be of order ␧ 2 , the introduction
of t 1 is inconsistent.
To resolve the dilemma, we notice the following point.
The parameter ␬ takes a small value such as 0.1 or 0.2, but it
has been regarded as a quantity of order unity and independent of ␧. As the magnitude of ␧ becomes comparable with
the value of ␬, however, the smallness of the latter may be
taken into account. If ␬ is taken as small as ␧, the contribution from the jet loss had rather be incorporated in the
second-order problem. Then, B in 共67兲 includes the additional term given by
1
2
⫹4D⌫ 兲 .
共 ⫺S im ⫹ 冑S im
2D
共104兲
2
For ⌫ⰆS im
/4D, P peak is given by ⌫/S im , whereas for ⌫
2
ⰇS im /4D, P peak is given by 冑⌫/D. The latter suggests that
the peak amplitude does not increase in proportion to ⌫ as in
the linear regime but to 冑⌫. The changeover may be defined
2
/4D⬅⌫ ch , which gives ⌫ ch ⫽0.0012 for
to occur at ⌫⫽S im
D⫽0.398. This corresponds to 0.34 mm for X b .
The response curves taking account of the jet loss are
drawn in the solid lines in Fig. 5. For the amplitude 0.5 mm,
good agreement is observed for the peak amplitude but the
peak frequency is slightly overestimated. The latter is determined by the accuracy of measurement of the temperature. If
the temperature is higher by 1 degree, then the peak frequency is shifted by 0.09 Hz to the right. For the amplitude
smaller than 0.5 mm, the solid lines in Fig. 6 shows the
Sugimoto et al.: Nonlinear frequency response of air column
1783
response curves with the jet loss. In this case, the peak frequencies are well predicted. Although the peak amplitudes
are overestimated slightly, it is found that account of the jet
loss iproves the results considerably.
As the amplitude increases beyond 0.5 mm, however,
there arises significant discrepancy. It may be attributed to
the following two points. Although the frequency of the sec⫺
ond harmonics never hits ␴ ⫺
2 , the ratio ␴ 2 / ␴ 1 is 1.98 and
very close to 2 numerically. Even if no shock then occurs,
this situation is not preferable to the theory. Secondly, the
frequency of the fifth harmonics, 5 ␴ 1 , hits ␴ 0 for evanescence in the present case. Then the energy absorption at the
fifth harmonics is enhanced so that the theory might be limited. Furthermore if causes of the discrepancy are sought
beyond the present framework, it is open whether or not the
weakly nonlinear theory is still applicable. In addition, another effect of transverse motions of air in the cross-section
might come into play, associated with emergence of acoustic
streaming. To make a firm conclusion, however, it is necessary to check validity of the theory against new experiments
by using tubes in geometrically different configuration taking
account of some restrictions.
VI. CONCLUSIONS
The weakly nonlinear theory has been developed to obtain the frequency response of forced oscillations of the air
column in the closed tube with the array of Helmholtz resonators connected axially. Since the array introduces the wave
dispersion, shock-free, high-amplitude oscillations are obtainable, and for their analysis, the usual asymptotic method
of multiple is successfully applicable. The analysis is based
on the assumptions of one-dimensional motions in the acoustic main flow outside of the boundary layer on the tube wall,
and of the continuum approximation of the discrete distribution of the resonators. Thus, no acoustic streaming involving
transverse motions of air in the cross section is covered, but
the steady but axially nonuniform pressure distribution of
quadratic order is revealed. By completing the second-order
problem, the cubic nonlinear equation describing temporal
slow modulation of the complex pressure amplitude of the
lowest mode has been derived.
In the linear regime, as is well known, the small dissipation and dispersion caused by the boundary layer on the
tube and throat wall limit the peak amplitude and lower the
peak frequency. As the amplitude of excitation is increased,
the nonlinear frequency shift occurs, depending on the amplitude of oscillations and on the natural frequency of the
resonator. The shift is crucially determined by the coefficient
Q in the equation. It takes positive or negative values so that
the response curve may be bent toward the high- or lowfrequency side. But, it is usual that Q is positive and the
response curve is bent toward the high-frequency side, because the natural frequency of resonator is chosen higher
1784
J. Acoust. Soc. Am., Vol. 114, No. 4, Pt. 1, October 2003
than the critical frequency ␴ cr . When the theoretical curve is
compared with the experiment, it is revealed that both results
do not agree with each other quantitatively, although qualitatively the tendency of the bent may be explained. In particular, the theory overestimates the peak amplitude considerably.
The discrepancy has been resolved to some extent by
taking account of the jet loss at the throat. It introduces the
additional nonlinear dissipation to the linear one so that the
peak amplitude is suppressed as the amplitude becomes
large. It is found that the peak amplitude then becomes proportional to the square root of the amplitude of excitation. As
far as the present experiment is concerned, the linear and
weakly nonlinear theory with the jet loss can give a quantitative description up to the pressure amplitude 3% to the
equilibrium pressure. In order to design a tube for generation
of even higher peak amplitude, the results in the present
paper provide useful guidelines.
ACKNOWLEDGMENTS
The authors acknowledge the support by the Grants-inAid from the Japan Society of Promotion of Science and also
from The Mitsubishi Foundations, Tokyo, Japan.
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12
Although C1 should be determined, in nature, by solving an initial-value
problem from start-up, it may be set equal to zero if the combination ␣
⫺␧ 2 i ␲␴ 1 C1 on substitution of the solutions obtained into 共42兲 is replaced
by ␣ newly. Then, 共75兲 关and 共100兲兴 are rewritten in terms of this new ␣ but
their form is subjected to no alterations within the order concerned.
Sugimoto et al.: Nonlinear frequency response of air column