## Abstract

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.

## 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 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 (1994*a*), 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 *τ*=*ε*^{2}*t*, where *ε*^{3} 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, 1994*b*) 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 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*(*ε*), 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 §2*c* and find the corresponding amplitude–frequency relation.

## 2. Formulation

An inhomogeneous gas is contained between two concentric spheres, 0<*r*_{0}≤*r*≤*r*_{1}. 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
2.1
with the equation of state for the isentropic flow of a polytropic gas
Pressure and density are measured from their values in a reference state (*p*_{s},*ρ*_{s}(*r*)), where *e*(*r*,*t*)=*ρ*/*ρ*_{s}−1 is the condensation, *p*_{s} a constant, and the associated sound speed. For an elastic material, with stress *Σ* related to *e* through the stress–strain law *Σ*(*e*)=*E*(*e*+*Ke*^{2}+*O*(*e*^{3})), then *Σ*=−*p*, *E*=*γp*_{s} is Young’s modulus, and *K*=(*γ*−1)/2 is the ratio of second- to first-order elastic constant. It should be noted that the subscripted variables *p*_{s},*ρ*_{s} and *a*_{s} refer to the reference state.

The boundary conditions are
2.2
where *l* is the maximum boundary displacement with frequency *ω*. Defining the reference sound speed at the inner boundary, and *ρ*_{0}=*ρ*_{s}(0), velocity, pressure and density are non-dimensionalized with respect to ( and (*u*,*p*,*ρ*) are considered as functions of dimensionless length, *x*, and time *ta*_{0}/(*r*_{1}−*r*_{0}). The shell thickness, *r*_{1}−*r*_{0}, is fixed, and *x* is defined by
where *L*=*r*_{0}/(*r*_{1}−*r*_{0}) is a dimensionless measure of the internal radius to the thickness of the shell. Hence 0≤*x*≤1. The dimensionless frequency is *ω*(*r*_{1}−*r*_{0})/*a*_{0}.

The dimensionless form of equations (2.1) and (2.2) now is
2.3
2.4and
2.5
where *θ*=*ωt* and the Mach number of the applied input velocity is *M*=*m*^{3}=*ωl*/*a*_{0}≪1. (The subscripts *θ* and *x* in equations (2.3) and (2.4) refer to differentiation w.r.t. *θ* 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*/(*r*_{1}−*r*_{0})≪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,
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 equations (2.3) and (2.4) yield the equations for a straight tube.

A new dependent variable *w*(*x*,*θ*) and new length variable *R*(*x*) are defined by
2.7
with *R*(0)=0. With
2.8
equations (2.3)–(2.5) become
2.9
2.10and
2.11
for 0<*R*<*R*(1)=*R*_{1}, where *ε*^{3}=*M*(*L*+1)^{2}*ρ*_{s}(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 *ε* combines the Mach number of the input, the geometry of the shell and the value of the inhomogeneity at the outer sphere. The requirement *ε*^{3}≪1 implies that *M* must become ever smaller as the *L* increases in order to ensure a continuous output. As , 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 *ε* is the perturbation parameter.

### (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)
2.12
2.13and
2.14
where |*e*_{i}|, |*w*_{i}|=*O*(1), *i*=1,2,3,…, *λ* is the fundamental frequency and *ε*^{2}*δ* is the detuning. It should be noted that the assumption here is that the output is *O*(*ε*) while the input is *O*(*ε*^{3}), 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*(*ε*):
2.15
and
2.16
Eliminating *e*_{1} from equation (2.15), *w*_{1}(*R*,*θ*) satisfies
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
2.18
where *A* is an arbitrary amplitude, *ϕ*(*R*) satisfies the eigenvalue problem
2.19
The eigenfunction *ϕ* is normalized so that . While *A*(*ω*) is arbitrary at *O*(*ε*), it will be determined as a function of *ω* at *O*(*ε*^{3}).

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 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*(*ε*) can be performed and the effect of the stratification can be calculated explicitly:
2.20
Then equations (2.7) and (2.8) imply that *R*(*x*) and the density *ρ*_{s}(*x*) are
2.21
It should be noted from equation (2.20) that when *kL*=1, *ρ*_{s}(*x*)≡1 and *s*(*x*)=*L*^{2}(1+*x*)^{2}=(*r*/(*r*_{1}−*r*_{0}))^{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
2.22
see Mortell & Seymour (2004), where *F*(*R*) satisfies
2.23
This can be confirmed by direct substitution. The boundary conditions *ϕ*(0)=*ϕ*(*R*_{1})=0 imply that *λ* is determined by the eigenvalue equation
2.24
where *R*_{1}=*R*(1). Thus, the eigenvalues are, in general, incommensurate, i.e. *λ*_{n}≠*nλ*_{1}, *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, *ρ*_{s}(*x*)=1 and from equation (2.21) *R*_{1}=1.

In summary, for an ambient gas density given by equation (2.21), the solution at *O*(*ε*) (i.e. linear theory) is given by equation (2.18) where *ϕ*(*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 *ω*. This requires calculating the nonlinear terms up to *O*(*ε*^{3}). When the eigenvalues are incommensurate, the interaction with the nonlinear terms is such that shocks cannot form and *w*_{1} and *e*_{1}, given by equation (2.18), capture the dominant single mode response, where *A*(*ω*) is the amplitude of the standing wave.

### (b) Calculation of amplitude *A*(*ω*)

The first correction to linear theory in the perturbation solution to equations (2.9) and (2.10), i.e. at *O*(*ε*^{2}), is given by
2.25
2.26and
2.27

On eliminating *e*_{2}(*R*,*θ*), and using equation (2.18), it is clear that the form of *w*_{2}(*R*,*θ*) must be , where *B*(*R*) is determined by
with *B*(0)=0, *B*(1)=0. Here, *ϕ*(*R*) is the eigenfunction given by equations (2.22)–(2.23) with zero boundary conditions. Now 2*λ* 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*(*ε*^{2}) is
and *u*(*R*,*θ*) is calculated from equation (2.7).

The equation at *O*(*ε*^{3}) to determine *w*_{3}(*R*,*θ*) is of the form
where *C*_{1}(*R*) and *C*_{2}(*R*) depend on *s*(*R*), *c*(*R*), *w*_{1} and *w*_{2}.

On assuming a solution of the form
the resulting non-homogeneous ordinary differential equation for *Q*(*R*) has a solution with no restriction on *A* since 3*λ* is not an eigenvalue of the operator on the left-hand side. However, the problem for *P*(*R*) is of the form
and
Since *λ* is an eigenvalue, the Fredholm alternative puts the following restriction on *A* to ensure a solution *P*(*R*):
2.28
where
2.29
and *Q*_{1}(*R*) and *Q*_{2}(*R*) are functions of *s*, *s*^{′}, *c*, *c*^{′}, *ϕ* and *ϕ*^{′}. The cubic equation (2.28) for *A*(*ω*) 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 *w*_{1}(*R*,*θ*) is now complete and the velocity *u*(*x*,*θ*) is calculated from equation (2.7). The condition for the validity of the expansion is |*εAB*|≪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*=*ν*^{−1}(*L*+1)*a*_{0}, where *ν* 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 *νs*(*w*/*s*)_{RR} on the right-hand side, where we take *ν*=*μm*^{2}, and equation (2.10) is unchanged. The perturbation expansion is again given by equations (2.12)–(2.14) where *ε* 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
where the eigenvalue *λ* satisfies equation (2.24) and the constant *c* is determined from the normalization condition . Then
and *A* and *B* are arbitrary at this stage.

At the second order no secularities arise and the generalized approximations *w*_{2}(*R*,*θ*) and *e*_{2}(*R*,*θ*) 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*.
2.30
and
2.31

These ensure that *w*_{3} and *e*_{3} exist and complete the solutions for *w*_{1} and *e*_{1}. The constants **Δ**, *Ω* and *Γ* may be expressed in terms of *μ*,*δ*,*λ* and *γ* and the underlying leading order solution *ϕ*. We note that when damping is absent, *μ*=0, then *Ω*=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, *ρ*_{s}(*x*), is constant (=1), *c*(*x*)≡1, *R*(*x*)=*x* and hence *R*_{1}=1. The eigenvalue equation (2.24) now only contains the effect of the spherical geometry and becomes
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 , since *L* is a measure of the internal radius of the shell. Then the eigenvalue equation (2.32) reduces to
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 *ε*.

*Plane wave*. The ‘plane-wave’ case arises from the limit as , 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
2.34
with solutions *λ*=*λ*_{n}=*nπ*, *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 ‘plane-wave’ 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*)=*L*^{2}(1+*kx*)^{2}, *k*=1/*L*, has the same form as equation (2.20) and allows a solution *ϕ* 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, as . This situation is dealt with later by changing the reference state from *r*_{0} to *r*_{1}.

#### (ii) Inhomogeneous gas

*Plane wave.* For an inhomogeneous gas in the case of ‘plane-waves’, the relevant limit is in equations (2.21) and (2.24). Then
2.35
and *R*_{1}=1/(1−*k*). Consequently, the eigenvalue equation becomes
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 , where *R*_{1}=*L*/(1+*L*) and *ρ*_{s}(*x*)=*L*^{4}/(*L*+*x*)^{4}. Thus, the eigenvalues are now commensurate, resulting in shocks. We note that , 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 *r*_{0}, equation (2.21) implies that as 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*=*r*_{1}, the outer boundary. Then the dimensionless ambient density has *ρ*_{s}(1)=1, and the non-dimensional radial coordinate is defined as *x*=*r*/*r*_{1}.

*Full sphere*. For the full sphere, 0≤*r*≤*r*_{1}, integration of equations (2.7) and (2.8) gives
2.37
2.38and
2.39
with
2.40
The eigenvalue equation is
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, *r*_{0}≤*r*≤*r*_{1}, with *r*/(*r*_{1}−*r*_{0})=*L*+*x*, *L*=*r*_{0}/(*r*_{1}−*r*_{0}), 0≤*x*≤1,
with *R*(1)=1+*L*. We note that
as , in agreement with equation (2.38), the result for a full sphere. Also
and
2.42
with
2.43
Also as , which agrees with equation (2.40), the result for a full sphere.

The eigenvalue equation is 2.44

When , , , 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, *ρ*_{s}(*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 , and yield *ρ*_{s}(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 §2*a*–*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*,*θ*), given by equation (2.15), when *L*=4, *k*=−1, *m*=0.03 and *ν*=0.25*m*^{2}, where *ν* is the bulk viscosity. A more rigorous test of the perturbation expansion, when *L*=3, *k*=1/3, *m*=0.03 and *ν*=0.25*m*^{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*,*θ*). 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.

The response curves for a continuous solution of a homogeneous gas *L*=2, *k*=1/2, *m*=0.03 and *ν*=*μm*^{2} are given in figure 4 for *μ*=0, 0.25 and 1.0. The equations for the amplitudes *A* and *B*, where , are
and
We note that the curves bend to the right (i.e. a hardening response) and there is no hysteresis for sufficiently large *ν*.

In figure 5, *L*=8, *m*=0.03 and *μ*=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 §2*d*. 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.

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*.

For a full sphere, *L*=0, containing a homogeneous gas, *ρ*_{s}(*x*)≡1, the transformation (2.7) is *w*=*x*^{2}*u*. Then *w*_{1} in equation (2.12) is
3.1
The boundary conditions *ϕ*(0)=0 and *ϕ*(1)=0, yield the eigenvalue equation , and
3.2
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.

## Acknowledgements

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).

- Received November 4, 2010.
- Accepted January 6, 2011.

- This journal is © 2011 The Royal Society