## Abstract

This paper examines marginal conditions for the onset of thermoacoustic oscillations in resonators of a Sondhauss tube and a gas-filled, dumbbell-shaped tube. An analysis is performed using a linear acoustic theory based on the first-order boundary-layer approximation. When a parabolic temperature distribution is assumed along the tube’s neck, a frequency equation is available analytically, whose complex solutions are examined numerically. In the case of the dumbbell-shaped tube, two modes of oscillations exist, one being an antisymmetric mode and the other a symmetric one for pressure variations in the neck, while in the case of the Sondhauss tube, only a mode corresponding to the antisymmetric one exists due to the boundary condition at the open end. Marginal conditions are sought numerically, not only for the lowest branches of both modes, but also for second higher branches, but they are available only for the lowest branch of the antisymmetric mode. For the Sondhauss tube, marginal conditions are obtained by taking account of radiation into free space. Some discussions are provided in comparison with experiments.

## 1. Introduction

A Sondhauss tube is one of the typical examples in which spontaneous occurrence of thermoacoustic oscillations is perceived by sound radiated into ambient air. It is a flask or a bottle-shaped vessel with a long neck, which may be called a Helmholtz resonator if the neck is short, subjected to a temperature gradient along the neck generated by heating a bulb (figure 1). Although generation of sound had long been noticed among glass blowers, it was Sondhauss (1850) who investigated the phenomenon in detail for various shapes of tubes other than the one shown in figure 1. For the literature on Sondhauss thermoacoustic phenomena, see the review paper by Feldman (1968).

However, physical mechanisms of generation of sound are difficult to explain. In fact, Sondhauss gave no adequate explanation. Later, Rayleigh (1945) described a qualitative criterion of generation of sound. In modern language, this criterion is expressed quantitatively in such a way that a temporal mean of the product of an excess pressure *p*′ and a heat flux into a gas *q*′ divided by *ρ**c*_{p}*T* over a cycle should be positive to sustain acoustic oscillations, with *ρ*, *c*_{p} and *T* being the density of gas, its specific heat at constant pressure and temperature (see Howe 1998; Huelsz & Ramos 1999). For the mean to be positive, an appropriate phase difference between *p*′ and *q*′ is required to be established. When the gas is in contact with a solid wall subjected to a temperature gradient, a preferable phase difference is often brought about naturally by thermoviscous effects due to viscosity and heat conduction of gas.

Unfortunately, however, such a criterion alone cannot qualitatively predict a marginal condition for the onset of instability. To answer it, a stability analysis is required. Rott & Zouzoulas (1976) attempted to derive the condition for a tube consisting of two circular, cylindrical tubes connected through a constriction. They applied the theory originally developed by Rott (1969, 1973) to derive marginal conditions for Taconis oscillations in a helium-filled, quarter-wavelength tube in cryogenics (Taconis *et al.* 1949). They also performed experiments to confirm the theoretical results to some extent. Although the marginal conditions are obtained analytically by assuming a step temperature distribution, it is obvious that a smooth distribution would be appropriate in natural environments.

Connecting a cavity to a tube has a great advantage in reducing the natural frequency of the tube. This makes a thermoviscous layer thick, and significantly lowers the temperature ratio of a hot part to a cold one for the onset of spontaneous oscillations. In fact, the ratio is reduced to 2.5–3 compared with a ratio of around 8 in the case of the quarter-wavelength tube. Thus, the oscillations may occur in air, even if the cold end is kept at room temperature. In recent years, application of thermoacoustic phenomena to heat engines has attracted much attention. In this context, a Sondhauss tube has successfully been applied to a thermoacoustic refrigerator (Wheatley *et al.* 1983; Swift 1988). But since an acoustic theory of the Sondhauss tube is not well established, it is a fundamental and interesting problem in its own right.

This paper attempts to derive analytically a marginal condition for the Sondhauss tube, as illustrated in figure 1, using an acoustic theory based on a boundary-layer approximation. This theory was originally developed to derive marginal conditions of the Taconis oscillations by Sugimoto & Yoshida (2007). If a temperature distribution is assumed to be smooth and parabolic along the tube, the conditions are available analytically to provide marginal curves similar to the ones observed (Yazaki *et al.* 1980*a*). Also for other smooth distributions, marginal conditions are numerically obtained. The theory can be extended even more to simulate emergence of self-excited oscillation in nonlinear regimes (Sugimoto & Shimizu 2008).

Nevertheless, the theory is simple in comparison with Rott’s theory and others. This facilitates the understanding of physical mechanisms. As shown schematically in figure 2, thermoviscous effects on an acoustic field appear only through a velocity *v*_{b} at the edge of the boundary layer directed normal to the tube wall and into the gas. In spite of such a simplification, *v*_{b} can yield not only a phase shift to harmonic oscillations, but also hereditary effects for non-harmonic oscillations. In this theory, *v*_{b} plays a key role rather than *q*′. For the explicit forms of *v*_{b} and *q*′, see eqns (7) and (72) in Sugimoto & Yoshida (2007), where *Q*_{n} in eqn (72) is identified as *q*′ in the present context.

Instability is considered to be brought about by an active action of the boundary layer on the gas in an acoustic main-flow region outside of it. If a temporal mean of the product *p*′*v*_{b} is positive over a cycle, the boundary layer does work on the gas in the region to input power locally. Furthermore, if the total power integrated over the entire surface at the edge of the boundary layer is positive, then the energy is piled up in the gas so that oscillations will occur. This criterion is the same, in essence, as Rayleigh’s because *q*′/*ρ**c*_{p}*T* has a dimension of velocity, although it slightly differs from *v*_{b} in numerical coefficients (Shimizu & Sugimoto 2009).

The boundary-layer approximation assumes that a typical thermoviscous layer is *very* thin compared with a tube radius. But it may be applicable even to such an extreme case that the layer becomes comparable in thickness to the radius. As was demonstrated by Sugimoto & Shimizu (2008), it is expected that some aspects of classical thermoacoustic oscillations in geometrically simple configurations would be explained by the boundary-layer approximation. Thus, it is the purpose of this paper to apply the theory to a Sondhauss tube to examine its validity.

In the following, a pressure equation in linear acoustic theory based on a boundary-layer approximation is first presented in §2. Treatment of the gas in the cavity is simplified as a lumped mass. In §3, a stability problem is posed as a boundary-value problem for the pressure equation in the neck. To treat the Sondhauss tube, it is instructive to consider, in parallel, a dumbbell-shaped tube shown in figure 3, in which two identical tubes of figure 1 are connected symmetrically at the open end, with their axes common. Frequency equations are derived for a parabolic temperature distribution using an idea of renormalization developed by Sugimoto & Yoshida (2007). In §4, a stability analysis is performed on the basis of complex solutions to the frequency equations. Finally, discussions and conclusions are given.

## 2. Formulation of the problem

As illustrated in figure 1, it is assumed that the Sondhauss tube consists of a straight, circular tube of uniform cross section as a neck, and of a cavity in the form of a bulb. One end of the neck is open to free space and the other end is connected to a cavity that is closed otherwise. Thus, ambient gas fills the inside of the tube. A dumbbell-shaped tube illustrated in figure 3 is a gas-filled tube with two identical Sondhauss tubes connected symmetrically at both open ends with a common axis.

Letting the neck of the Sondhauss tube be of length *L* and of radius *R* with cross-sectional area *A*, the *x*-axis is taken along the neck, with its origin at the open end and the junction with the cavity at *x*=*L*. In the case of the dumbbell-shaped tube, let the junctions with the cavity be located at *x*=*L* and *x*=−*L*, respectively. Suppose that the tube wall, which is rigid and has enough heat capacity, is subjected to a temperature gradient axially. In the case of the Sondhauss tube, the wall temperature *T*_{e}(*x*) increases from the open end towards the cavity, whose wall is kept at a uniform temperature everywhere equal to *T*_{L} [≡*T*_{e}(*L*)]. In the case of the dumbbell-shaped tube, the temperature distribution is symmetric with respect to *x*=0, i.e. *T*_{e}(*x*)=*T*_{e}(−*x*).

In mechanical equilibrium for the gas to be quiescent, a uniform pressure *p*_{0} prevails in the tube where no gravity is assumed. The temperature of the gas is regarded as being uniform over the cross section and equal to *T*_{e}(*x*). Consistent with this axially non-uniform distribution, the density of gas *ρ*_{e}(*x*) varies according to the equation of state *ρ*_{e}*T*_{e}=*ρ*_{0}*T*_{0}, where the law of an ideal gas is assumed and the subscript 0 implies a value at *x*=0.

When spontaneous oscillations of the gas occur in the tube, an acoustic Reynolds number is usually very high so that thermoviscous effects are confined in a boundary layer on the tube wall. In addition, since a typical axial wavelength, i.e. the tube’s length, is long enough compared with its radius, the pressure may be regarded as being uniform over each cross section. Using the linearized boundary-layer approximation, the excess pressure *p*′(*x*,*t*) over *p*_{0} in the neck, with *t* being the time, is governed generally by the following wave equation (Sugimoto & Yoshida 2007):
2.1
with the minus half-order derivative defined by
2.2
where and *ν*_{e}(*x*) (=*ν*_{0}*T*_{e}/*T*_{0}) denote, respectively, the local adiabatic sound speed and the kinematic viscosity of gas, with the temperature dependence of the shear viscosity being neglected for simplicity. *C* and *C*_{T} are constants given, respectively, by
2.3
with *γ* and *P**r* being the ratio of specific heats and the Prandtl number, respectively.

The gas in the cavity is treated as a lumped mass. Its motions are neglected and only the mass balance is of concern. Letting the mean density of gas in the cavity be *ρ*_{c}, the rate of increase of the mass *ρ*_{c}*V* is set equal to the mass flux flowing into it. Denoting by *u* an axial velocity in the neck, the rate of increase of the mass is given within the linearized approximation as
2.4
where implies the mean value of *u* over the cross section of the tube and the right-hand side is evaluated at the junction with the cavity, with *A* being *π**R*^{2}.

To evaluate , the distribution of the axial velocity over the cross section must be specified (figure 2). In the acoustic main-flow region, *u* is almost uniform and given by *u*′(*x*,*t*), where the prime is attached to designate a disturbance. In the boundary layer, *u* is subjected to a defect ŭ from *u*′ as *u*=*u*′+ŭ. The integral of ŭ over the boundary layer is given by
2.5
where *n* implies a boundary-layer coordinate normal to the tube wall and directed into the gas with its origin at the wall, and infinity corresponds to the edge of the boundary layer. For details, see (A 2), (A 3) and (A 9) in appendix A of Sugimoto & Tsujimoto (2002). Thus, it follows that is given by
2.6
It should be mentioned, however, that relation (2.6) may be questionable near the junction and the open end, especially when the gas flows into the neck, because the boundary-layer theory takes no account of the neck’s ends.

But putting this problem aside, equation (2.6) is used in equation (2.4). This is rewritten, if the adiabatic change is assumed to be in the cavity, as
2.7
where *a*_{c} and *p*′_{c} denote the sound speed in the cavity and the excess pressure averaged therein, respectively, which are assumed to be equal to *a*_{e} and *p*′ at *x*=*L*. Of course *u*′ takes the value at *x*=*L*. This relation is used as a boundary condition to equation (2.1) at the junction with the cavity.

## 3. Frequency equation

### (a) Case of the Sondhauss tube

We now derive a frequency equation by using equation (2.1) to examine the stability. Setting in equation (2.1), where *P*(*x*) is a complex amplitude and *ω* an angular frequency, and noting that the minus half-order derivative of is given by , equation (2.1) is written in the following form:
3.1
with
3.2
in the interval 0<*x*<*L*, where |*δ*_{e}| (≪1) measures a ratio of a local thickness of a boundary layer to the tube radius. A complex amplitude *U*(*x*) of the axial velocity *u*′ is given in terms of *P* as
3.3
where . Then, a complex amplitude of is (1−2*δ*_{e})*U* by relation (2.6).

When the temperature distribution is assumed to be parabolic in the form of
3.4
where *λ* is a positive constant, it was shown by Sugimoto & Yoshida (2007) that equation (3.1) is reduced to an analytically solvable equation to the first order of *δ*_{e} as
3.5
where *Ω* takes a constant value,
3.6
Here and hereafter, all terms higher than the first order of *δ*_{e} are neglected.

To transform equation (3.1) into equation (3.5), *P* is replaced by a new variable *F* through
3.7
with
3.8
while *x* is transformed into a new *complex* coordinate *ξ* through
3.9
with
3.10
Noting that , with *δ*_{0} being *δ*_{e}(0), *ξ* is related to *x* as
3.11
where *η*=1+*λ**x*/*L* and *b*=*C**δ*_{0}. Thus, it follows that the open end at *x*=0 corresponds to *ξ*=0, while the junction at *x*=*L* corresponds to *ξ*=*ξ*_{L} given by
3.12

By using *F* and *ξ*, equation (3.5) is rewritten as
3.13
where *σ* (=*ω**L*/*a*_{0}) denotes a dimensionless angular frequency. Since |*b*|≪1, this equation is solvable by a successive approximation in terms of *b*, i.e. by setting *F*=*F*^{(0)}+*F*^{(1)}*b*+⋯ to seek *F*^{(0)} and *F*^{(1)} successively, as
3.14
where *α* and *β* are arbitrary constants and *k* and *l* are defined as
3.15
with *ψ*=(*σ*^{2}−*λ*^{2}/4)^{1/2}. Note that solution (3.14) is not a straightforward expansion in *b* since *ξ* includes *b* implicitly.

For the Sondhauss tube, the boundary condition at the open end must take account of radiation into free space outside of the tube. Denoting an admittance at the open end by *A*_{0}/*ρ*_{0}*a*_{0}, with *A*_{0} being dimensionless, the condition at *x*=0 is imposed as
3.16
When defining the admittance, use is made of the mean axial velocity , whose complex amplitude is given by (1−2*b*/*C*)*U*. On the other hand, the boundary condition at *x*=*L* is stipulated by equation (2.7) and imposed as
3.17
Here, *κ* designates the ratio of the cavity’s volume to the neck’s volume as
3.18

These conditions are rewritten in terms of *F* as
3.19
where *J* takes *J*_{0} at *x*=0 and *J*_{L} at *x*=*L*, respectively. In calculating the left-hand side, it is convenient to note the following relation:
3.20

Thus, the conditions (3.16) and (3.17) are rewritten as
3.21
at *ξ*=0, and
3.22
at *ξ*=*ξ*_{L}. Substituting solution (3.14) into condition (3.21), this is satisfied if *α* and *β* are chosen as
3.23
and
3.24
where *B* is an arbitrary constant.

Substituting these into condition (3.22), and setting , a frequency equation is obtained from the non-trivial condition for *B* as
3.25
where *W*_{1} and *W*_{2} are given, respectively, by
3.26
with *C*_{b}=2/*C*−1, and
3.27
with
3.28
3.29
3.30
and
3.31

### (b) Case of the dumbbell-shaped tube

A frequency equation in this case is also available in a similar way to the one in the preceding case. A positive interval 0<*x*<*L* and a negative one −*L*<*x*<0 are treated separately, which are connected by imposing matching conditions at *x*=0. The temperature distribution is assumed to be parabolic in each interval as
3.32
Here, the use of the sign ‘±’ or ‘∓’ is noted. When it appears hereafter, the upper sign applies to the positive interval and the lower sign to the negative one. When the signs appear in a single equation more than once, they are understood to be ordered vertically.

The transformation (3.7) is extended to the negative interval as
3.33
Taking the origin of *x* at *ξ*=0,
3.34
with *η*_{±}=1±*λ**x*/*L*, so that the junctions at *x*=±*L* correspond to ±*ξ*_{L}.

Equations for *F*_{±} are given by
3.35
The solutions are available in a similar way to that in the preceding case as
3.36
where *α*_{±} and *β*_{±} are constants.

The matching conditions at *x*=0 require continuity of the pressure and the axial velocity. They are given as
3.37
and
3.38
at *ξ*=0. The boundary conditions at *x*=±*L* are imposed as
3.39
at *ξ*=±*ξ*_{L}, respectively, where *d**P*/d*x*=−*J*_{L}*P* at *x*=−*L*.

Substituting solutions (3.36) into the matching conditions and exploiting (3.20) for *F*=*F*_{±} with the sign of *k* and *l* reversed for *F*_{−}, it follows that
3.40
and
3.41
respectively. Here *A* and *B* with the subscripts s and d denote the sum and difference of *α*_{±} and *β*_{±}, respectively, as
3.42

The boundary conditions at *ξ*=±*ξ*_{L} lead to
3.43
and
3.44
with
3.45
and
3.46
These conditions are rewritten as
3.47
and
3.48

Although there are four homogeneous equations for four unknowns, introduction of the sum and difference decouples them into two sets of homogeneous equations for the sum and for the difference, respectively. Hence, two frequency equations are available. Firstly it follows from the non-trivial condition for *A*_{d} and *B*_{d} in equations (3.40) and (3.48) that
3.49
This is recast into the frequency equation of the following form:
3.50
with
3.51
and
3.52
Here, and are equal to equations (3.26) and (3.27), respectively, without the terms proportional to 1/*A*_{0} for radiation into free space. Since *A*_{s}=*B*_{s}=0, i.e. *α*_{+}=−*α*_{−} and *β*_{+}=−*β*_{−}, this case represents the antisymmetric mode of oscillations in pressure with respect to the origin as a node. This is why the superscript (a) is attached.

On the other hand, it follows from the non-trivial condition for *A*_{s} and *B*_{s} in equations (3.41) and (3.47) that
3.53
This is the other frequency equation given by
3.54
with
3.55
and
3.56
Since *A*_{d}=*B*_{d}=0, i.e. *α*_{+}=*α*_{−} and *β*_{+}=*β*_{−}, this case represents the symmetric mode of oscillations in pressure with respect to the origin as a loop, which is reflected in the superscript (s).

## 4. Stability analysis

The frequency equation for the Sondhauss tube agrees with the one for the antisymmetric mode in the dumbbell-shaped tube, unless radiation into free space is taken into account. Thus, we start with the case for the dumbbell-shaped tube and then consider the one for the Sondhauss tube. We first examine neutral oscillations in a lossless case by neglecting terms with *b*.

### (a) Frequencies of neutral oscillations

In the lossless case with *b*=0, the frequency equations (3.50) and (3.54) for the antisymmetric and symmetric modes are reduced, respectively, to the following equations:
4.1
and
4.2
These determine a frequency of neutral oscillations of each mode, i.e. a natural frequency. It is to be noted that the frequency depends on *κ* and *λ*, but not on material properties of the gas concerned.

Solving equations (4.1) and (4.2) numerically, figure 4 shows the lowest (smallest) frequencies against the temperature ratios *T*_{L}/*T*_{0} [=(1+*λ*)^{2}] for various values of *κ* (=0,0.1,0.2,0.5,1,2,4 and 10). The thick and thin curves represent, respectively, the antisymmetric and symmetric modes. For reference, the broken curve represents the relation *σ*=*λ*/2 on which *ψ* vanishes. As a special case with *κ*=0 corresponds to a straight tube with both ends closed, i.e. a half-wavelength tube, the frequencies of the antisymmetric and symmetric modes start at *T*_{L}/*T*_{0}=1 with *σ*=*π*/2 and *π*, respectively, for the tube length 2*L* to be a half wavelength and one wavelength.

As the sound speed increases with temperature, the frequency increases, in general, with temperature ratio. It should be noted that the cavity lowers the frequency, and the larger its volume becomes, the more the frequency is lowered. When the temperature gradient is absent, i.e. *λ*→0, equations (4.1) and (4.2) are further reduced to
4.3
and
4.4
For a large value of *κ* (≫1), equations (4.3) and (4.4) are solved asymptotically to yield, respectively, *σ* ≈ and *σ*≈*π*/2 as a smallest solution. Thus, it is found that the frequencies of both the modes are lowered significantly. The former is simply the well-known natural frequency of the Helmholtz resonator, i.e. in the dimensional form, while the latter is a natural frequency of a half-wavelength tube with both ends open. For a gentle temperature gradient 0<*λ*≪1, the natural angular frequency of the Helmholtz resonator increases with the temperature as , which is smaller than corresponding to .

### (b) Marginal condition for the dumbbell-shaped tube

We now solve the frequency equation (3.50) for the antisymmetric mode by taking account of *b* to derive a marginal condition of instability where *σ* takes a real value. To do this, we first examine a *lossless limit* as *b*→0. In the lossless case with *b*=0, the frequency may take any value on the curve in figure 4. But when *b* is taken into account, though infinitesimally small, the frequency is singled out on the curve. This point is determined as follows.

Rewriting equation (3.50) into
4.5
where *W* designates , the right-hand side is expanded up to the first order of *b* as
4.6
with given by
4.7
Noting that *b* is proportional to 1−i, and separating equation (4.5) into real and imaginary parts, it follows from the imaginary one that the coefficient of the terms proportional to *b* should vanish as
4.8
Thus, the remaining terms independent of *b* should also vanish. This relation is simply the equation for the frequency of neutral oscillations. Thus, equation (4.8) determines a frequency on the neutral curve, which gives the lossless limit.

Using this frequency as a starting point, we solve equation (3.50) numerically to seek a pair of real solutions of *σ* and *λ* for a *finite* value of *b*. To do this, we set *b*=(1−i)*ζ* with
4.9
where *ζ*_{0} may be interpreted as a parameter representing the tube length or the aspect ratio *R*/*L*. Increasing the magnitude of *ζ*_{0} step-by-step, the sets of the solutions are sought successively. Figure 5 shows the angular frequency *σ* against the temperature ratio *T*_{L}/*T*_{0} [=(1+*λ*)^{2}] for the various values of *κ* labelled, where the dots indicate the lossless limit. Here, air at 300 K and 0.1 MPa is assumed with *γ*=1.40 and *P**r*=0.717, so that *C*=1.47 and *C*_{T}=1.14. The broken curves represent the relations of the neutral oscillations in the antisymmetric modes.

As the magnitude of *ζ*_{0} is increased, the frequency and the temperature ratio move from the dot to yield the curve. While *ζ*_{0} is small, each curve is located close to the one for the neutral oscillations, but as it becomes larger, the curve tends to deviate greatly from it. In figure 5, however, no information on *ζ*_{0} is read.

So taking in place of *σ*, i.e. the tube radius referenced to a typical thickness of the boundary layer at *x*=0, figure 6 shows the marginal curves of the temperature ratio against the tube radius thus defined for the values of *κ* labelled. Because is expressed as or , the information on *ζ* rather than *ζ*_{0} may be read. The lossless limit corresponds to infinity of the horizontal axis, where the temperature ratio remains finite. Thus, this plane is divided into two regions.

To identify which side of the marginal curve corresponds to instability, the imaginary part of *σ* (≡*σ*_{r}+i*σ*_{i}) is sought. By seeking complex solutions to equation (3.50) with the temperature ratio *T*_{L}/*T*_{0} fixed at 3,3.38,4,7.53 and 10, figure 7 depicts graphs of *σ*_{i} for the case with *κ*=0.2, where the temperature ratios 3.38 and 7.53 are the extremal values in figure 6.

Because a negative imaginary part implies instability, it is found that the upper region above the marginal curve corresponds to an unstable regime. Similarly, figure 8 depicts graphs of *σ*_{i} for several values of *κ* (=0.2,0.5,1,2 and 4) with the temperature ratio *T*_{L}/*T*_{0} fixed at 3. It is also found that the upper regions of the marginal curves in figure 5 are unstable.

When the value of *κ* vanishes, the marginal curve is almost the same as the one previously obtained for the Taconis oscillations (Sugimoto & Yoshida 2007). The temperature ratio takes the minimum 7.83(4) at *σ*=1.98 for *ζ*_{0}=0.465, which is very slightly lower than the previous value 7.84. This lower value results from no account of the boundary layer on the end wall in the present case.

For *κ*=0, the marginal curve consists of right and left branches with respect to this minimum. As the value of *κ* becomes large, it is found that the temperature ratio for instability becomes low and that the left branch does not go upward so that the unstable region spreads leftward. This is different from the marginal curves obtained by Rott & Zouzoulas (1976), where the left branches always exist. A reason for this difference might be attributed to the use of the boundary-layer approximation because it tends to break down towards a smaller value of the horizontal axis. In view of the facts, however, that the left branches exist while the value of *κ* is small, and that a large cavity lowers a frequency significantly, we would not be led immediately to the above conclusion, but no definite reasons would be known.

The marginal curve obtained is not the only solution to equation (3.50). This is the lowest branch of solutions, but there are other branches. We examine solutions of the second branch. Figure 9 shows graphs of the imaginary part *σ*_{i} of the angular frequency against its real part *σ*_{r} for several temperature ratios *T*_{L}/*T*_{0} (=2, 3, 4, 5 and 10) in two cases with *κ*=1 and *κ*=0.2, represented by the solid and broken curves, respectively. These curves are sought by starting from a frequency on the horizontal axis equal to a second smallest neutral frequency (higher than those shown in figure 4 and not drawn) where |*b*| vanishes, and by increasing the magnitude of *ζ*_{0} successively.

Each curve ends on the horizontal axis at a lower frequency. This frequency is given by *λ*/2 where |*b*| diverges with *ψ**b*→*π*. Of course, the result for such a large value has no physical significance. But since the imaginary part is always positive for any value of *b*, no instability occurs for the second branch of the antisymmetric mode. While |*b*|≪1 near the upper frequency, each curve presents a damping rate *σ*_{i} as well as an angular frequency *σ*_{r} shifted down from one of the neutral oscillations when the thermoviscous effects come into play.

Next, we examine a case of the symmetric mode. Proceeding in a similar way to that demonstrated for the antisymmetric mode, equation (3.54) is rewritten in the same form as equation (4.5) with *W* given by . Expanding the right-hand side with respect to *b*, it follows that
4.10
where
4.11
Thus it follows from the balance of the terms proportional to *b* that
4.12
If *C*_{b} is positive, i.e. *C*<2, then is positive so that there are no real solutions of *σ* to equation (4.12). This implies no lossless limit. In case *C*_{b} is negative, however, there still remains a possibility of existence of the lossless limit.

Although no existence of the marginal curve is suggested, we now check complex solutions to equation (3.54). The procedure is the same as the one used in seeking the complex solutions shown in figure 9. In figure 10, graphs of *σ*_{i} are drawn against *σ*_{r} for several temperature ratios *T*_{L}/*T*_{0} (=2, 3, 4, 5 and 10) and for the case with *κ*=1. For one ratio of *T*_{L}/*T*_{0}, solid and broken curves represent, respectively, the lowest and second branches of the solutions. Each curve meets the horizontal axis at two points. The upper frequency corresponds to the frequency of neutral oscillations of the symmetric mode. The lower frequency is also given by *λ*/2 with *ζ*_{0} diverging but *ψ**b*→*π* and 2*π*, respectively, for the lowest and second branches. For the symmetric mode, hence, no instability occurs in the case with *κ*=1. This holds for the other values of *κ*.

In passing, it is noted that Yazaki *et al.* (1980*b*)1 derived a marginal curve for a symmetric mode in a half-wavelength tube corresponding to the case with *κ*=0. They extended Rott’s theory to obtain the curve similar to the one for the antisymmetric mode. It consists of right and left branches. The left branch was confirmed experimentally but the right one was not observed because it is located in the unstable region of the antisymmetric mode. But it is also reported that the mode is not observed if the hot part prevails over more than half of the tube length. Thus, this mode will depend not only on the temperature ratio, but also on the explicit profile of the temperature distribution. To seek instability in the symmetric mode, the present model may be extended to include a plateau in the temperature distribution given as
4.13
For this distribution, the stability analysis would become a little complicated but still straightforward.

### (c) Marginal condition for the Sondhauss tube

With the marginal conditions available for the dumbbell-shaped tube, we examine the case of the Sondhauss tube by taking account of radiation. The admittance at the open end is very difficult to evaluate, even in the case of a tube without temperature gradient. But when a tube without a flange is placed in free space unbounded externally, there is a famous result derived by Levine & Schwinger (1948). This is the only result available analytically so far.

Borrowing their result, though again no temperature distribution is taken into account, the inverse of the admittance, i.e. impedance, is given by
4.14
with *s*=*R**σ*/*L*, where denotes a reflection coefficient with |ℛ| given approximately by for *s*≪1. Here, *χ* is a famous constant 0.6133 for the end correction at the open end of an unflanged tube, with the tube being lengthened by Δ*L* (=*χ**R*). Note that the sign of *k* in Levine & Schwinger (1948) is reversed is consistent with the temporal factor in the present paper. Expanding the last term of equation (4.14) with respect to *s*, it follows that
4.15
where the real term responsible for radiation damping is retained, though quadratic in *s*, to see its effect. For the impedance of the open end flanged by a large baffle, see the book by Crighton *et al.* (1992).

Adding equation (4.15) to the frequency equation, the marginal condition is sought by solving equation (3.25) numerically. Figure 11 shows the marginal curves for the same values of *κ* used in figure 6 and for the aspect ratio *R*/*L*=1/100. For comparison with the experimental data by Rott & Zouzoulas (1976), nitrogen gas is assumed with *γ*=1.4 and *P**r*=0.74, so that *C* and *C*_{T} take the values 1.46 and 1.12, respectively. However, air is close to nitrogen in composition, and so are these values. The solid and broken curves represent, respectively, the case of the Sondhauss tube with the radiation taken into account and the one without it, i.e. the lowest branch of the antisymmetric mode in the dumbbell-shaped tube.

If the angular frequency is drawn against the temperature ratio just as in figure 5, the frequency is lowered slightly due to the end correction. In figure 11, however, the temperature ratio in the solid curves is also lowered slightly, especially for a small value of *κ*. Although this result seems to be reversed, it may be understood as follows. When the radiation is taken into account, *σ* is lowered. For the case with *κ*=0.5, for example, *σ* is decreased from 1.28 to 1.27 at *T*_{L}/*T*_{0}=4, i.e. 0.4 per cent smaller than 0.6 per cent due to the end correction *χ**R*/*L*, whereas *ζ*_{0} is greatly decreased from 0.243 to 0.237, i.e. 3 per cent. Because of this competition, the value of the horizontal axis becomes large.

Imagine a case in which the tube radius is variable with its length fixed. By radiation, *σ* is lowered so that the boundary layer becomes thicker everywhere. The above result means that the oscillations may occur in an even wider tube. To find effects of radiation damping, on the other hand, marginal curves are sought by setting *χ* to be zero. It turns out that there are no visible differences for *R*/*L*<50 and . In the range , the marginal curves are shifted upwards, as expected, than those without the radiation.

Here, a comparison is made with the experimental data, which are summarized in table 1 in the paper by Rott & Zouzoulas (1976). The data are available only for the case with *κ*=4 and at the temperature ratio 3.8. While the tube radius is fixed, tubes of various lengths are used. The greatest aspect ratio is 1/350 and is smaller than 1/100 assumed here. Because the impedance also increases with the aspect ratio, the marginal curve for *R*/*L*=1/350 appears to be little different from the broken curves without the radiation. Hence, it may be concluded that the radiation has almost no effects on the marginal curve in the experiments.

The data measured are plotted in figure 11 as open circles and a cross, which represent, respectively, the unstable case with spontaneous oscillations observed and the stable case without them. All points are located within the unstable region of the present theory. The cross is found to be located close to the marginal curve, but above it.

A few reasons for this may be pointed out. Firstly, the temperature distribution is different, though the temperature ratio in the case of the step is expected to be lower than the present one. Secondly, the present model would not take full account of real dissipative effects in the cavity due to the formation of a jet at the junction, the formation of a thermal boundary layer on the cavity wall and so on. In addition, no temperature dependence of the shear viscosity is considered here. Since it usually increases as (*β*>0), *ν*_{e} increases as (*β*=0.85 for N_{2}, see Rott & Zouzoulas (1976)). But, this would decrease the temperature ratio. Thirdly, the cavity used in the experiments is not a sphere but a cylinder whose length is comparable to that of the neck. Thus, no uniform state in the cavity would be assumed in the experiments. However, no clear reasons for discrepancy are identified.

## 5. Conclusions

This paper has examined, in detail, the marginal conditions for the onset of thermoacoustic oscillations in a Sondhauss tube and a dumbbell-shaped tube within the linear acoustic theory based on a first-order boundary-layer approximation. When the wall temperature varies along the neck in the form of a parabola, the frequency equation has been derived analytically. For both tubes, the marginal conditions have been obtained for the lowest branch corresponding to the antisymmetric mode. But they are unavailable not only for the second branch, but also for the lowest and second branches of the symmetric mode.

It has been revealed that the marginal curves for the Sondhauss tube are almost the same as those for the antisymmetric mode. Effects of radiation may be negligible if the aspect ratio is less than 1/100. The end correction makes the frequency lower while the radiation damping is negligible. It has also been revealed that as the cavity’s volume becomes large, the temperature ratio required for the onset of oscillations is lowered so that oscillations will occur in many other gases. When the cavity is absent, the marginal curve consists of the right and left branches for a given temperature ratio. As the cavity becomes large, however, the left branch is not distinguished from the right one in the present case. This result contradicts the marginal curves obtained by Rott & Zouzoulas (1976).

Along the right branch, dissipative effects are small so that the boundary-layer approximation may be appropriate. In view of the present result and the one for the step temperature distribution, it is conjectured that this branch would be available in any case only if the temperature increases from the open end towards the cavity monotonically, irrespective of explicit profiles of its distribution. Along the left branch, however, dissipative effects are prominent over the cross section of the tube. It might be concluded that the present boundary-layer theory cannot cover the left one. But since the present theory can give the left branch for a small value of *κ*, this conclusion should be left open. Rott & Zouzoulas (1976) noted the difficulty in identifying the marginal conditions for a classical Sondhauss tube with a spherical cavity.

## Footnotes

↵1 They observed a second harmonic of fundamental oscillations and used it in the title (T. Yazaki

*et al.*1980*b*, personal communication). Although not explicitly described, it corresponds to a symmetric mode in the present context.- Received May 25, 2009.
- Accepted July 31, 2009.

- © 2009 The Royal Society