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Proc. R. Soc. A (2011) 467, 2149–2167
doi:10.1098/rspa.2010.0576
Published online 2 February 2011
Resonant oscillations of an inhomogeneous gas
between concentric spheres
BY BRIAN R. SEYMOUR1, *, MICHAEL P. MORTELL2
AND
DAVID E. AMUNDSEN3
1 Department
of Mathematics, University of British Columbia,
Vancouver, Canada V6T 1Z2
2 Department of Applied Mathematics, University College Cork,
Cork, Republic of Ireland
3 School of Mathematics and Statistics, Carleton University,
Ottawa, Canada K1S 5B6
We investigate the effects of nonlinearity, geometry and stratification on the resonant
motion of a gas contained between two concentric spheres. The emphasis is on whether
the motion is continuous, and on how the inhomogeneity, geometry or nonlinearity can
move the motion to a shocked state. Linear undamped theory yields a standing wave of
arbitrary amplitude and an eigenvalue equation in which the higher eigenvalues are not
integer multiples of the fundamental; the system is said to be dissonant. Higher modes,
generated by the nonlinearity, are not resonant and consequently shocks may not form.
When the output is shockless, the amplitude is two orders of magnitude greater than that
of the input. When the eigenvalues for a homogeneous gas are not sufficiently dissonant
and shocks form, the introduction of a stratification in the gas can restore dissonance and
allow a continuous output. Similarly, the introduction of an inhomogeneity can change
a continuous motion to a shocked one, as can an increase in the input Mach number,
or an increase in the geometrical parameter. Various limits of the eigenvalue equation
are considered and previous results for simpler geometries are recovered; e.g. a full sphere,
a cone and a straight tube.
Keywords: nonlinear resonant oscillations; spherical shell; stratified gas; shockless motions
1. Introduction
This paper deals with resonant oscillations of a gas between concentric spheres.
The gas may be inhomogeneous, belonging to a class of stratifications that allow
exact solutions of the Webster horn equation. The nonlinearity in the system
is controlled by the Mach number of the input and the geometry is reflected
in the ratio of the radius of the internal sphere to the distance between the
spheres. We are primarily interested in continuous solutions that are, in effect, a
dominant first harmonic, and how the geometry, nonlinearity and inhomogeneity
*Author for correspondence ([email protected]).
Received 4 November 2010
Accepted 6 January 2011
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B. R. Seymour et al.
can render these shockless solutions to be shocked. The approximate solution
is checked against the corresponding solution of the exact equations, giving
satisfactory results.
For more than 70 years, going back at least to the experiments of Lettau (1939)
there has been significant interest in the forced nonlinear, resonant response
of a gas in a container, see Betchov (1958), Gorkov (1963), Chester (1964),
Seymour & Mortell (1973, 1980), Zaripov & Ilgamov (1976), Cox & Mortell
(1983) and Ilgamov et al. (1996). The majority of this work was focused on
plane waves in a closed, straight tube and in understanding shock formation.
However, recently, motivated by the experiments of Lawrenson et al. (1998)
attention has turned to consider the effects of the shape of the container on
resonant oscillations. Of particular interest is the presence or absence of shocks
in the flow, a characteristic of disturbances of a homogeneous gas in straight
tubes. Here, we examine the effects of nonlinearity, geometry and inhomogeneity
on resonant oscillations between concentric spheres.
Chester (1991) and Ellermeier (1994a), respectively, showed that the effect of
a spherical or cylindrical geometry on the resonant motion of a homogeneous gas
may render the oscillations shockless. Then the experiments of Lawrenson et al.
(1998) demonstrated that resonant oscillations in a cone, a horn and a bulb
were shockless and, moreover, were of technological interest. This phenomenon
was termed resonant macrosonic synthesis (RMS). Following a number of
numerical investigations of the governing equations (Ilinskii et al. 1998, 2001;
Bednarik & Cervenka 2000; Chun & Kim 2000), Mortell & Seymour (2004)
provided the first analytical (as distinct from numerical) explanation for
these experiments, reproducing in broad terms the experimental findings.
The time scale for the evolution of the amplitude is t = 32 t, where 33 is
the small dimensionless Mach number. There have been various analyses of
the modulation of resonance amplitudes in spheres, cylinders and concentric
cylinders and spheres (e.g. Ellermeier 1997; Kurihara & Yano 2006) using
the potential formulation as in Chester (1991) and for macrosonic waves
in cones and bulbs by Mortell et al. (2009) using the formulation in
Mortell & Seymour (2004).
Prior to the experiments of Lawrenson et al. (1998) analytical work had been
concerned largely with the effect of small geometric variations from a straight tube
in the presence of shocks in the flow (Chester 1994; Mortell & Seymour 1972;
Keller 1977; Ockendon et al. 1993; Hamilton et al. 2001). An exception is
Ellermeier (1993, 1994b) who outlined a Duffing-like expansion for a strong
inhomogeneity. Also, but to a lesser extent, the effects of rate-dependent
properties of the gas and changes in the impedance at the end of the cylinder
on preventing shocks were examined by Seymour & Mortell (1973), Mortell &
Seymour (1972), Ellermeier (1983) and Sturtevant (1974). These latter introduced
a damping mechanism to prevent the shocks, but the order of magnitude of the
output was not increased.
This paper is concerned with the nonlinear resonant oscillations of an
inhomogeneous (i.e. density stratified) gas contained in a spherical shell. The
analytical approach used here is a generalization of that in Mortell & Seymour
(2004). The essential thrust of the paper is to examine the interactions
between the nonlinearity, geometry and the inhomogeneity, and their effects
on the resonant oscillations. The governing equations are transformed into
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a standard form before a Duffing-like perturbation expansion is used to find
the amplitude–frequency relation for a continuous motion. The basic assumption
is that the solution is dominated by the first harmonic and hence there is
a restriction on the range of validity of the expansion that yields continuous
solutions, e.g. a sufficiently large increase in the Mach number of the input
could invalidate the expansion. At linear theory, O(3), the standard form yields
the Webster horn equation for which exact solutions are available for certain
forms of the inhomogeneity via the function s(R). In this paper, one of these
is exploited to examine the effects of such an inhomogeneity, and both the
geometry and inhomogeneity of the sound field are unified in s(R). The resulting
eigenvalue equation that determines the various resonant modes is central to
the understanding of shock formation. The modes emerge from linear undamped
acoustic theory. When the eigenvalues are sufficiently incommensurate due to
the geometry or inhomogeneity, i.e. the system is sufficiently dissonant, shocks
do not form, provided the Mach number is not too large. When there are no
shocks, and there is no dissipation, the magnitude of the output amplitude is two
orders greater than the input.
The analysis of resonant oscillations of an inhomogeneous gas in a spherical
shell is a nonlinear wave problem, involving reflections, in a domain of finite
extent. Analytical solutions to such inhomogeneous gas problems are rare,
a notable exception being Whitham (1953). Other than Mortell & Seymour
(2007), as far as we are aware solutions are usually confined to geometrical
acoustics theory, where the role of the inhomogeneity is minimal. The solutions
presented here are for a specific class of inhomogeneities, but is not confined
to the ‘slowly varying’ kind. They yield an insight into the effect of a general
inhomogeneity, e.g. a sufficiently strong inhomogeneity can of itself prevent
a shock in a resonant oscillation. In part then, this paper can be viewed as
a theoretical analysis to try to achieve some understanding of how a strong
inhomogeneity can affect nonlinear resonant oscillations. We show how the
homogeneous gas case arises as a limit from the inhomogeneous gas. In both
cases the limit of a full sphere is deduced from that of concentric spheres.
An analysis of the eigenvalue equation for an inhomogeneous gas is central to
the above, and the various limits arise from this single equation. The eigenvalue
equation for a cone, as in the experiments in Lawrenson et al. (1998), also emerges.
Then the assumption that the flow in the cone is quasi-one-dimensional (see
Ilinskii et al. 1998; Mortell & Seymour 2004) is not necessary as it is a radial
flow in a segment of a spherical shell. Thus, the new results from the flow of
an inhomogeneous gas in a spherical shell also provide a synthesis to a range of
previous disparate results.
Since all the experiments of which we know deal with gases, this paper is
written in the language of gas dynamics. Some of the inhomogeneities considered
may not be realistic for a gas, but could certainly be achieved in an elastic
material. Recently, there has been an interest in functionally graded materials
(FGMs) where the specific form of the inhomogeneity is chosen so that closed form
exact solutions are possible, see Collet et al. (2006). To derive the corresponding
results is just a matter of changing notation.
The major part of the paper uses the model of an inviscid polytropic gas since
the Reynolds number Re 1. However, we include the effect of dissipation in §2c
and find the corresponding amplitude–frequency relation.
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B. R. Seymour et al.
2. Formulation
An inhomogeneous gas is contained between two concentric spheres, 0 < r0 ≤
r ≤ r1 . The interior boundary is fixed, while the external boundary oscillates
periodically at or near a resonant frequency, the fundamental frequency of a
linear free vibration. The motion of the gas is assumed to be radially symmetric.
Using Eulerian coordinates, the undamped governing equations in dimensional
variables are
vu
vp
vr v(ur) 2ur
vu
+u
+
= 0,
+
+
= 0,
(2.1)
r
vt
vr
vr
vt
vr
r
with the equation of state for the isentropic flow of a polytropic gas
g
p
r
g(g − 1) 2
e + ··· .
=
= (1 + e)g = 1 + ge +
ps
rs (r)
2
Pressure and density are measured from their values in a reference state
(ps , rs (r)), where e(r, t) = r/rs − 1 is the condensation, ps a constant, and as (r) =
gps /rs (r) the associated sound speed. For an elastic material, with stress
S related to e through the stress–strain law S(e) = E(e + Ke 2 + O(e 3 )), then
S = −p, E = gps is Young’s modulus, and K = (g − 1)/2 is the ratio of secondto first-order elastic constant. It should be noted that the subscripted variables
ps , rs and as refer to the reference state.
The boundary conditions are
u(r0 , t) = 0,
u(r1 , t) = ul sin(ut),
(2.2)
where l is the maximum boundary displacement with frequency
u. Defining
the reference sound speed at the inner boundary, a0 = gps /rs (0) and r0 =
rs (0), velocity, pressure and density are non-dimensionalized with respect to
(a0 , r0 a02 , r0 ) and (u, p, r) are considered as functions of dimensionless length, x,
and time ta0 /(r1 − r0 ). The shell thickness, r1 − r0 , is fixed, and x is defined by
x=
r − r0
r
=
− L,
r1 − r 0 r1 − r 0
where L = r0 /(r1 − r0 ) is a dimensionless measure of the internal radius to
the thickness of the shell. Hence 0 ≤ x ≤ 1. The dimensionless frequency is
u(r1 − r0 )/a0 .
The dimensionless form of equations (2.1) and (2.2) now is
uuq + uux +
ueq +
and
Proc. R. Soc. A (2011)
1
(1 + e)g−2 ex = 0,
rs (x)
1
2(1 + e)u
=0
[rs (x)(1 + e)u]x +
rs (x)
L+x
u(0, q) = 0,
u(1, q) = M sin q,
(2.3)
(2.4)
(2.5)
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Resonant oscillations between spheres
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where q = ut and the Mach number of the applied input velocity is M =
m 3 = ul/a0 1. (The subscripts q and x in equations (2.3) and (2.4) refer
to differentiation w.r.t. q and x). For the small rate limit, see Seymour &
Mortell (1980), this can be interpreted as a restriction on the amplitude of
the velocity input. Since l is the boundary displacement there is a further
restriction, l/(r1 − r0 ) 1; i.e. the boundary displacement is much less than the
distance between the spheres. The solutions sought have the same period as the
boundary forcing,
u(x, q) = u(x, q + 2p),
(2.6)
and are small deviations from the reference state, which is an exact solution of
equations (2.3) and (2.4). It is noted that for L → ∞ equations (2.3) and (2.4)
yield the equations for a straight tube.
A new dependent variable w(x, q) and new length variable R(x) are defined by
w(x, q) = (L + x)2 rs (x)u(x, q)
and
dR
=
dx
rs (x) ≡ c(R),
(2.7)
with R(0) = 0. With
s(R) = (L + x)2 c(R),
(2.8)
equations (2.3)–(2.5) become
uwq + cw
and
w cs
R
+ s(1 + e)g−2 eR = 0,
1
ueq + [(1 + e)w]R = 0
s
w(0, q) = 0, w(R1 , q) = 33 sin q,
(2.9)
(2.10)
(2.11)
for 0 < R < R(1) = R1 , where 33 = M (L + 1)2 rs (1) 1 is a constraint limiting a
combination of the Mach number of the input and the geometric parameter L to
ensure a continuous motion. It is worth noting that the dimensionless parameter
3 combines the Mach number of the input, the geometry of the shell and the value
of the inhomogeneity at the outer sphere. The requirement 33 1 implies that
M must become ever smaller as the L increases in order to ensure a continuous
output. As L → ∞, we are dealing with a straight closed tube and the resonant
output is shocked for any non-zero M when there is no damping, see Chester
(1964), and Seymour & Mortell (1973).
Equations (2.9) and (2.10) are the canonical forms of the governing equations
(2.3) and (2.4) and are the basis of the perturbation scheme for continuous
motions, where 3 is the perturbation parameter.
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B. R. Seymour et al.
(a) Linear theory
To solve the nonlinear problem defined by equations (2.6) and (2.9)–(2.11)
for a continuous output, the following perturbation expansion is proposed (see
Mortell & Seymour 2004)
w(R, q) = 3w1 (R, q) + 32 w2 (R, q) + 33 w3 (R, q) + · · · ,
(2.12)
e(R, q) = 3e1 (R, q) + 32 e2 (R, q) + 33 e3 (R, q) + · · ·
(2.13)
u(3) = l + 32 d + · · · ,
and
(2.14)
where |ei |, |wi | = O(1), i = 1, 2, 3, . . ., l is the fundamental frequency and 32 d is
the detuning. It should be noted that the assumption here is that the output
is O(3) while the input is O(33 ), see equation (2.11). This is the consequence of
resonance. Further, the motion is assumed to be dominated by the first harmonic.
The linear problem is obtained at O(3):
l
ve1
vw1
+s
=0
vq
vR
and
l
ve1 1 vw1
+
=0
vq
s vR
(2.15)
and
w1 (0, q) = 0,
w1 (R1 , q) = 0.
(2.16)
Eliminating e1 from equation (2.15), w1 (R, q) satisfies
2
w1
v
l
−s
2
vq
vR
2v
1 vw1
s vR
= 0.
(2.17)
The purpose of the change of variable (2.7) and (2.8) was to obtain
equation (2.17)—the Webster horn equation—that can be solved exactly for
specific forms of s(R).
On setting
w1 (R, q) = Af(R) sin q
and
e1 (R, q) =
A
f (R) cos q,
ls(R)
(2.18)
where A is an arbitrary amplitude, f(R) satisfies the eigenvalue problem
d
dR
1 df
s(R) dR
+
l2
f = 0,
s(R)
f(0) = f(R1 ) = 0.
(2.19)
R
The eigenfunction f is normalized so that 0 1 s −1 (R)f2 (R) dR = 1. While
A(u) is arbitrary at O(3), it will be determined as a function of u at O(33 ).
Equations (2.19) are a Sturm–Liouville problem where both the spherical
geometry and the stratification of the gas are unified in s(R), see equations (2.7)
and (2.8). The effect of the geometry is contained in the term (L + x)2 , so that
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specifying a form of s(R) defines a particular class of gas stratification through
c(R). This can be done in many ways (see Varley & Seymour 1988), but here one
simple form is selected where all integrations at O(3) can be performed and the
effect of the stratification can be calculated explicitly:
s(R) = L2 (1 + kR)2 .
(2.20)
Then equations (2.7) and (2.8) imply that R(x) and the density rs (x) are
R(x) =
Lx
L + x(1 − kL)
and
rs (x) =
L4
.
[L + x(1 − kL)]4
(2.21)
It should be noted from equation (2.20) that when kL = 1, rs (x) ≡ 1 and s(x) =
L2 (1 + x)2 = (r/(r1 − r0 ))2 . Thus, exact solutions of the Webster horn equation
are available for a homogeneous gas.
With s(R) given by equation (2.20), the solution to equation (2.19) has the form
f(R) = (1 + kR)F (R) − kF (R),
(2.22)
see Mortell & Seymour (2004), where F (R) satisfies
F + l2 F = 0.
(2.23)
This can be confirmed by direct substitution. The boundary conditions f(0) =
f(R1 ) = 0 imply that l is determined by the eigenvalue equation
tan(lR1 ) =
lR1
,
1 + (l/k)2 (1 + kR1 )
(2.24)
where R1 = R(1). Thus, the eigenvalues are, in general, incommensurate, i.e.
ln = nl1 , n = 2, 3, 4, . . .. This is a direct result of the geometry and/or the
inhomogeneity of the gas as contained in s(R).
The incommensurability of the eigenvalues is the critical element for the
validity of the perturbation expansion and the existence of continuous solutions.
Note that from equation (2.21) kL = 1 is a homogeneous gas, rs (x) = 1 and from
equation (2.21) R1 = 1.
In summary, for an ambient gas density given by equation (2.21), the solution
at O(3) (i.e. linear theory) is given by equation (2.18) where f(R) is given
by equations (2.22), (2.23) and (2.24). All that remains is to determine the
amplitude A of the continuous motion and its dependence on the frequency u.
This requires calculating the nonlinear terms up to O(33 ). When the eigenvalues
are incommensurate, the interaction with the nonlinear terms is such that shocks
cannot form and w1 and e1 , given by equation (2.18), capture the dominant single
mode response, where A(u) is the amplitude of the standing wave.
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(b) Calculation of amplitude A(u)
The first correction to linear theory in the perturbation solution to equations
(2.9) and (2.10), i.e. at O(32 ), is given by
ve2
v w1 ve1
vw2
+s
= −cw1
− (g − 2)se1
,
(2.25)
l
vq
vR
vR cs
vR
ve2 1 vw2
1 v(e1 w1 )
l
+
=−
(2.26)
vq
s vR
s vR
(2.27)
and
w2 (0, q) = 0, w2 (R1 , q) = 0.
On eliminating e2 (R, q), and using equation (2.18), it is clear that the form of
w2 (R, q) must be w2 (R, q) = A2 B(R) sin 2q, where B(R) is determined by
d 1 dB
(2l)2
lf
c
s
+
B = 2 (g + 1)f − 2 +
f ,
s
s
s
c
dR s dR
with B(0) = 0, B(1) = 0. Here, f(R) is the eigenfunction given by equations
(2.22)–(2.23) with zero boundary conditions. Now 2l is not an eigenvalue,
since the eigenvalues are assumed incommensurate, then B(R) exists with no
restrictions on the amplitude A. The solution to O(32 ) is
w(R, q) = 3Af(R) sin q + 32 A2 B(R) sin 2q,
and u(R, q) is calculated from equation (2.7).
The equation at O(33 ) to determine w3 (R, q) is of the form
2
v 1 vw3
2 v w3
l
−s
= [−2ldAf + A3 C1 (R)] sin q + A3 C2 (R) sin 3q,
vq2
vR s vR
w3 (0, q) = 0
and w3 (R1 , q) = sin q,
where C1 (R) and C2 (R) depend on s(R), c(R), w1 and w2 .
On assuming a solution of the form
w3 (x, q) = P(R) sin q + Q(R) sin 3q,
the resulting non-homogeneous ordinary differential equation for Q(R) has a
solution with no restriction on A since 3l is not an eigenvalue of the operator on
the left-hand side. However, the problem for P(R) is of the form
d 1 dP
l2
+ P = −2ldAfs −1 + A3 C4 (R)
dR s dR
s
and
P(0) = 0,
P(R1 ) = 1.
Since l is an eigenvalue, the Fredholm alternative puts the following restriction
on A to ensure a solution P(R):
N1 A3 − 2dlA = N2 ,
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(2.28)
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Resonant oscillations between spheres
where
N1 =
R1
[Q1 (R) + Q2 (R)B(R)] dR,
N2 =
0
1
R1
R1
0
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f 2
s c
l R− −
dR (2.29)
s
s
c
and Q1 (R) and Q2 (R) are functions of s, s , c, c , f and f . The cubic equation
(2.28) for A(u) is the required amplitude–frequency relation. The effects of the
variable geometry of the container and the inhomogeneity of the gas are contained
in the integrands in equation (2.29). The solution for w1 (R, q) is now complete
and the velocity u(x, q) is calculated from equation (2.7). The condition for the
validity of the expansion is |3AB| 1, so that the motion is dominated by the
first harmonic. When this condition is violated shocks can be expected.
Equation (2.1) contain no dissipation, measured by the Reynolds number,
Re = n−1 (L + 1)a0 , where n is the bulk viscosity. When Re 1, energy dissipation
is negligible except at a shock front, see Kurihara et al. (2005). In the
following section, we briefly discuss the effect of dissipation in deriving the
amplitude–frequency relation.
(c) Effects of dissipation
Here, we sketch the calculation of the amplitude–frequency relation when there
is a damping term in the momentum equation (2.3). This term facilitates the
transition to the steady state and structures a shock. We work with the equations
for the transformed variable w, see equation (2.7). Equation (2.9) now has a term
ns(w/s)RR on the right-hand side, where we take n = mm 2 , and equation (2.10) is
unchanged. The perturbation expansion is again given by equations (2.12)–(2.14)
where 3 is replaced by m.
Taking s(R) as in equation (2.20), solutions to the eigenfunction equation
(2.19), are given by equation (2.22) and then satisfying the boundary conditions
we find
2
k
f = c − − l(1 + kR) sin(lR) + k 2 R cos(lR) ,
l
where the eigenvalue l satisfies equation (2.24) and the constant c is determined
R
from the normalization condition 0 1 (f2 (R)/s(R)) dR = 1. Then
w1 (R, q) = f(R)[A sin(q) + B cos(q)],
e1 (R, q) =
−f
[−A cos(q) + B sin(q)],
ls
and A and B are arbitrary at this stage.
At the second order no secularities arise and the generalized approximations
w2 (R, q) and e2 (R, q) can be determined in terms of A and B and the various
equation parameters. As with the undamped case, at third order secularities arise.
This leads to a non-trivial solvability condition on the leading order amplitudes
A and B.
f (R1 )
(2.30)
= DA − UB + G(A3 + AB 2 )
prs (1)(L + 1)2
s(R1 )
and
0 = DB + UA + G(B 3 + A2 B).
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(2.31)
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B. R. Seymour et al.
These ensure that w3 and e3 exist and complete the solutions for w1 and e1 . The
constants D, U and G may be expressed in terms of m, d, l and g and the underlying
leading order solution f. We note that when damping is absent, m = 0, then U = 0
which in turn implies B = 0 from equation (2.31). Then equation (2.30) reduces
to a single cubic equation for A as in equation (2.28). A specific instance of this
generalized amplitude–frequency relation where damping is present is given in §3.
(d) Asymptotic forms of the eigenvalue equation
The eigenvalue equation (2.24) contains the effects of the geometry of the shell
through equation (2.20) and the ambient density stratification of the gas, given
by equation (2.21). Firstly, two limits will be investigated in the context of a
homogeneous gas: the result is given in Chester (1991) for a full sphere and the
‘plane-wave’ case for a shell of large internal radius L and fixed thickness. Then,
we examine limiting cases for an inhomogeneous gas.
(i) Homogeneous gas
When kL = 1 the ambient density, rs (x), is constant (= 1), c(x) ≡ 1, R(x) = x
and hence R1 = 1. The eigenvalue equation (2.24) now only contains the effect of
the spherical geometry and becomes
tan l =
l
1+
(lL)2 (1
+ L−1 )
.
(2.32)
This is the case considered by Kurihara & Yano (2006), but the form of the
eigenvalue equation (there called the resonance radius) is very different from
equation (2.32).
Cone. When L = 1/k, equation (2.32) is the eigenvalue equation for the
frustum of a cone with slope k containing a homogeneous gas, see, Mortell &
Seymour (2004), and is the theoretical underpinning for the experimental results
of Lawrenson et al. (1998). The radially symmetric oscillations in a segment of
a spherical shell give the appropriate context for oscillations in the frustum of a
cone, thus obviating the need for the assumption (see Ilinskii et al. 1998; Mortell &
Seymour 2004) of a quasi one-dimensional motion.
Full sphere. The case of a full sphere containing a homogeneous gas is recovered
in the limit L = r0 /(r1 − r0 ) → 0, since L is a measure of the internal radius of
the shell. Then the eigenvalue equation (2.32) reduces to
tan l = l.
(2.33)
This equation is given in Ellermeier (1997), is implicit in Chester (1991), and
yields incommensurate eigenvalues and hence no shocks for sufficiently small 3.
Plane wave. The ‘plane-wave’ case arises from the limit as L → ∞, when the
internal radius of the shell becomes large compared with the fixed shell thickness
and the radial lines are ‘almost’ parallel within the shell. Then the eigenvalue
equation (2.32) becomes
tan l = 0,
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with solutions l = ln = np, n = 1, 2, 3, . . ., the eigenvalues are commensurate and
shocks result. This is, of course, the case of axial resonance in a closed tube, see
Chester (1964).
Chester (1991) raised two open questions in the case of a homogeneous gas: to
calculate the response in a spherical shell, and to examine the limit to the ‘planewave’ case. The first question is answered here using the techniques developed in
Mortell & Seymour (2004) and the second will be dealt with elsewhere.
Now s(R) = (L + x)2 is simply the square of the radial distance from the
origin, and s(R) = L2 (1 + kx)2 , k = 1/L, has the same form as equation (2.20)
and allows a solution f to equation (2.19) of the forms (2.22) and (2.23). Thus,
the reduction to the homogeneous case from equations (2.22) and (2.23) is
possible. This is not always the case, e.g. for a cylinder. If kL = 1, R1 → 0 as
L → 0. This situation is dealt with later by changing the reference state from
r0 to r1 .
(ii) Inhomogeneous gas
Plane wave. For an inhomogeneous gas in the case of ‘plane-waves’, the relevant
limit is L → ∞ in equations (2.21) and (2.24). Then
rs (x) →
1
,
[1 − kx]4
R(x) →
x
1 − kx
(2.35)
and R1 = 1/(1 − k). Consequently, the eigenvalue equation becomes
l
tan
1−k
=
l
.
1 − k + (l/k)2
(2.36)
This is the eigenvalue equation for the axial resonance of the inhomogeneous
gas (2.35) in a closed tube (see Mortell & Seymour 2007). It should be noted that
the eigenvalues given by equation (2.36) are, in general, incommensurate. Thus,
a sufficiently strong ambient stratification, such as that given by equation (2.35),
can prevent resonant shock formation in a closed cylindrical tube. This result
is in sharp contrast to a geometrical acoustics theory, where a slowly varying
inhomogeneity cannot prevent a shock, and the only effect is on the detuning
parameter, see Mortell & Seymour (1972).
Special case k = 0. The eigenvalue equation (2.24) becomes tan(lR1 ) = 0,
where R1 = L/(1 + L) and rs (x) = L4 /(L + x)4 . Thus, the eigenvalues
are now
2
commensurate, resulting in shocks. We note that s(R) = (L + x) rs (x) = L2 ,
i.e. the geometry and inhomogeneity cancel, effectively giving a straight tube
containing a homogeneous gas in the linear approximation. The eigenfunction
equation (2.19) now has constant coefficients.
When the reference state is at r0 , equation (2.21) implies that R1 → 0 as L → 0
unless Lk = 1. Thus for the cases of an inhomogeneous gas in a sphere and a
spherical shell (Lk = 1), it is convenient to have the reference state at r = r1 , the
outer boundary. Then the dimensionless ambient density has rs (1) = 1, and the
non-dimensional radial coordinate is defined as x = r/r1 .
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B. R. Seymour et al.
Full sphere. For the full sphere, 0 ≤ r ≤ r1 , integration of equations (2.7) and
(2.8) gives
1 + kR 2
s(R) =
, k = −1,
(2.37)
1+k
R(x) =
and
(2 + k)x − 1 − k
,
x + (1 − x)k(1 + k)
R(1) = 1, R(0) = −
rs (x) = [x + (1 − x)k(1 + k)]−4 ,
1
k
(2.38)
(2.39)
with
rs (1) = 1,
The eigenvalue equation is
rs (0) = [k(1 + k)]−4 .
(2.40)
1
1
=l 1+
,
tan l 1 +
k
k
(2.41)
yielding incommensurate eigenvalues and shockless motions.
It is noted that the structure of equation (2.41) is the same as that of
equation (2.33) when there is no inhomogeneity. The consequence is that
varying the inhomogeneity given by equation (2.39), by varying k, will not yield
commensurate eigenvalues in equation (2.41) and hence produce shocks.
Spherical shell. For a spherical shell, r0 ≤ r ≤ r1 , with r/(r1 − r0 ) = L + x,
L = r0 /(r1 − r0 ), 0 ≤ x ≤ 1,
R(x) =
(x + L)(1 + L)2 + (1 + k(1 + L))(x − 1)
,
(x + L)(1 + L)2 − k(1 + k(1 + L))(x − 1)
with R(1) = 1 + L. We note that
R(0) =
L(1 + L)2 − (1 + k(1 + L))
1
→−
2
L(1 + L) + k(1 + k(1 + L))
k
as L → 0, in agreement with equation (2.38), the result for a full sphere. Also
(1 + kR)(1 + L) 2
s(R) =
1 + k(1 + L)
and
(1 + L)2
rs (x) =
(1 + L)(x + L) − k(1 + k(1 + L))(x − 1)
with
rs (1) = 1,
rs (0) =
1+L
L+
(1+k(1+L))
1+L
4
,
(2.42)
4
.
(2.43)
Also rs (0) → (1/(k(1 + k)))4 as L → 0, which agrees with equation (2.40), the
result for a full sphere.
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Resonant oscillations between spheres
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The eigenvalue equation is
tan l[R(0) − R(1)] =
lk[(1 + kR(0))/(1 + kR(1)) − 1]
.
l2 (1 + kR(0)) + k 2 /(1 + kR(1))
(2.44)
When L → 0, R(1) → 1, R(0) → −1/k, and equation (2.44) reduces to equation
(2.41); i.e. the eigenvalue for the sphere containing the inhomogeneous gas is the
limit of that for the shell as the inner radius goes to zero.
The condition for the homogeneous gas, rs (x) ≡ 1 is 1 − k − k 2 = 0 in the
case of a sphere, from equation (2.39), and (1 − k 2 )(1 + L) = k in the case of
a spherical shell, from equation (2.42). These conditions coincide as L → 0, and
yield rs (0) = 1/(k(1 + k)) = 1.
The results for a sphere containing a stratified gas can thus be deduced from
the results for a spherical shell containing the gas. This is the analog of obtaining
the result given in Chester (1991) for a sphere from the case of a spherical shell
for a homogeneous gas, see, equations (2.32) and (2.33).
Thus, the model of the spherical shell yields, in the appropriate limits, the
full sphere and ‘plane wave’ results for both a homogeneous and a stratified gas.
It also yields the result for the experimentally important case of the cone, see
Lawrenson et al. (1998) and Mortell & Seymour (2004).
3. Numerical results
In this section, continuous solutions obtained by the perturbation procedure given
in §2a–c are illustrated and compared with those found directly by a numerical
procedure. All numerical examples were performed in Matlab using one of two
schemes. For cases where damping is present and there are no shocks, the PDE
solver by Shampine (2005) is used with a spatial resolution of 200 points. For
cases with no damping or with shocks, a finite volume method based on the
underlying characteristic flux is applied using a spatial resolution of between 100
and 200 points, see Ghidaglia et al. (1996).
We examine the response of the gas at fixed m for various values of the two
parameters L and k, representing changes in geometry and density, restricting the
density profiles to those that decay with radial distance. We also vary m = M 1/3
for fixed L and k.
Figure 1 compares the continuous numerical solution for an inhomogeneous
gas with the leading order perturbation solution for w(R, q), given by equation
(2.15), when L = 4, k = −1, m = 0.03 and n = 0.25m 2 , where n is the bulk viscosity.
A more rigorous test of the perturbation expansion, when L = 3, k = 1/3, m = 0.03
and n = 0.25m 2 , i.e. for a homogeneous gas, is shown in figure 2. The leading order
in w is not influenced by the correction at O(m 2 ), while including the correction
at O(m 3 ) is a significant improvement. In this case, only odd modes contribute
to the expansion for w. This is not the case for the corresponding condensation
e(R, q). Figure 3 shows the effect of increasing the input Mach number when
L = 2, k = 1/2 as the solution curve changes from being continuous for m = 0.03
to being shocked at m = 0.1. This is the increasing effect of nonlinearity that
eventually dominates that of geometry to produce a shock. The reader might note
for the m = 0.1 case that the presence of numerical viscosity gives the appearance
of a continuous solution.
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B. R. Seymour et al.
0.020
0.015
0.010
0.005
ω
0
−0.005
−0.010
−0.015
−0.020
0
1
2
3
4
5
6
θ
Figure 1. Comparison of numerical solution (solid line) to leading order dominant first harmonic
approximation (dashed line) m = 0.03, L = 4, k = −1, n = 0.25m 2 at x = 0.5.
0.20
0.15
0.10
0.05
ω
0
−0.05
−0.10
−0.15
−0.20
0
1
2
3
4
5
6
θ
Figure 2. Comparison of numerical solution (solid line) to leading order (dashed line) and thirdorder approximation (dashed-dotted line) with m = 0.03, L = 3, k = 1/L, n = 0.25m 2 at x = 0.5.
The response curves for a continuous solution of a homogeneous gas L = 2,
k = 1/2, m = 0.03 and n = mm 2 are given in figure 4 for m = 0, 0.25 and 1.0.
The equations for the amplitudes A and B, where w1 (R, q) = f(R)[A sin q +
B cos q], are
−13.37 + 23.67AB 2 − 30.99mB + 23.67A3 − 6.39dA = 0
and
74.35A2 B + 97.37mA + 74.35B 3 − 20.06dB = 0.
We note that the curves bend to the right (i.e. a hardening response) and there
is no hysteresis for sufficiently large n.
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Resonant oscillations between spheres
(a) 0.1
ω
0
−0.1
(b) 0.2
ω
0
−0.2
(c) 0.5
ω
0
−0.5
0
1
2
3
4
θ
5
6
Figure 3. (a–c) Effect of increased forcing amplitude with L = 2, k = 0.5, n = 0 at x = 0.5 and
m = 0.03, 0.05, 0.1.
2.5
2.0
1.5
|ω|
1.0
0.5
0
−15
−10
−5
0
δ
5
10
15
20
Figure 4. Resonant response hysteresis curves for m = 0.03, L = 2, k = 0.5 and n = 0 (dashed line),
n = 0.25m 2 (solid line), n = m 2 (dashed-dotted line).
In figure 5, L = 8, m = 0.03 and m = 0 while k takes on the values k = 0.125, 0,
−1. This shows how the inhomogeneity affects the solutions. In particular, k = −1
gives a continuous output, while k = 0 yields a solution containing a shock since
the eigenvalues are commensurate; see the special case in §2d. The case k = 0.125
corresponds to a homogeneous gas and contains a shock. The shock disappears
for the inhomogeneous gas given by k = −1.
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2164
B. R. Seymour et al.
(a) 0.5
ω
0
−0.5
(b) 0.5
ω
0
−0.5
(c) 0.1
ω
0
−0.1
0
1
2
3
4
5
6
θ
Figure 5. Effect of changing the inhomogeneity, k = 0.125, 0, −1 (a–c), for m = 0.03, L = 8, n = 0 at
x = 0.5. k = 0.125 corresponds to a homogeneous gas.
0.6
0.4
k
0.2
0
−0.2
−0.4
5
10
15
20
L
25
30
35
40
Figure 6. Transitions between shocked and continuous solutions in terms of geometry and
inhomogeneity parameters L and k. Thresholds are shown for output Mach numbers m = 0.03
(solid line) and m = 0.06 (dashed line). Continuous solutions lie outside the bands and shocked
solutions lie inside. The curve L = 1/k (dashed-dotted line) corresponds to a homogeneous gas.
The regions in the L − k space where solutions are continuous or shocked are
indicated in figure 6 for m = 0.03 and m = 0.06. The homogeneous gas case is
given by the curve L = k −1 . This figure summarizes where shocked and shockless
solutions occur for fixed m but different L and k, i.e. different inner radii and gas
inhomogeneities. The regions in the L − k space depend on the value of m.
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Resonant oscillations between spheres
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For a full sphere, L = 0, containing a homogeneous gas, rs (x) ≡ 1, the
transformation (2.7) is w = x 2 u. Then w1 in equation (2.12) is
w1 = Af(x) sin q,
f(x) = xF (x) − F (x),
F (x) = a cos(lx) + b sin(lx). (3.1)
The boundary conditions f(0) = 0 and f(1) = 0, yield the eigenvalue equation
tan l = l, and
3
(3.2)
u(x, q) ∼ 2 w1 (x, q) ∼ 3[x + O(x 2 )] as x → 0.
x
The above agrees with the results in Chester (1991) where a velocity potential is
used. So a full sphere yields a continuous solution, with an amplitude–frequency
relation as in Chester (1991), provided the conditions for the validity of the
expansion are satisfied.
4. Conclusions
The non-dimensional form of the governing equations is cast in a canonical
form. The consequences are that secular terms are obviated in the perturbation
expansion and the eigenvalue equations arise from the Webster horn equation.
This allows us to examine a specific class of gas inhomogeneities, of which a
homogeneous gas is one. Various limiting cases, a cone, a full sphere and a
plane wave, follow from the eigenvalue equation for concentric shells, both for
homogeneous and non-homogeneous cases.
The perturbation expansion to yield continuous solutions is based on the
assumption that the flow is a standing wave dominated by the first harmonic.
This imposes restrictions on the values of the dimensionless parameters, which,
when violated, introduce shocks in the flow.
The effects of, and interactions between, nonlinearity, geometry and
inhomogeneity are examined in the context of resonant oscillations of a gas
contained between two concentric spheres. It is found that when modes
are incommensurate, weakly nonlinear oscillations may be continuous. These
continuous motions may become discontinuous, i.e. contain a shock, by increasing
the Mach number of the input—a nonlinear effect, by changing the inhomogeneity,
or by increasing the radius of the inner sphere while maintaining a fixed distance
between the spheres—a geometric effect. A continuous motion is a standing wave
at O(m) that is maintained by a forcing motion at O(m 3 ). The continuous single
mode approximation is found by a perturbation scheme where the amplitude of
the standing wave is determined at O(m 3 ). The approximations are confirmed by
means of a numerical solution of the full equations.
The research reported in this paper was supported in part by NSERC Discovery Grant A9117
(BRS), and NSERC Discovery Grant 249732 and CFI New Opportunities Grant 7361 (DEA).
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