2.1. Linearised basic equations

Suppose a quiescent gas under a uniform pressure $p_0$ in a circular duct of radius $R$ and of infinite length, where the duct wall is rigid and smooth. An ideal gas is considered and gravity is ignored. The wall temperature $T_w$ is assumed to vary in the axial direction only, i.e. $T_w(x)$, and so gently that the following inequalities may be satisfied:

(2.1)\begin{equation} \frac{R^{2}}{T_w} \left| \frac{{\rm{d}}^{2} T_w}{{\rm{d}}\kern0.06em x ^{2}} \right| \ll \frac{R}{T_w} \left| \frac{{\rm{d}} T_w}{{\rm{d}}\kern0.06em x} \right| \ll 1. \end{equation}

In developing a theory, the non-uniformity in $T_w$ is taken up to the first-order derivative ${\mathrm {d}} T_w/{\mathrm {d}}\kern0.06em x$ giving rise to a heat flow, and the second-order one, i.e. the curvature effect of $T_w$, is neglected. As long as this is assumed, the gas temperature $T_e$ is set equal to $T_w$, and is uniform over a duct cross-section, i.e. $T_e(x)$ (Sugimoto Reference Sugimoto2010). This is also the case with the gas density $\rho _e(x)$, because $\rho _e T_e$ is constant by the ideal-gas law under a constant pressure, and set to be $\rho _0 T_0$. Here and hereafter, the subscript $0$ is used to designate quantities in the quiescent reference state. It is assumed that the heat capacity of the solid wall is so large that the wall temperature does not to change even when the gas is in motion.

Consider infinitesimally small disturbances to such a quiescent state by neglecting thermoviscous diffusions. Then a gas particle is subjected to adiabatic change even in a non-uniform gas. The disturbances must satisfy fluid dynamical equations which stipulate the conservation of mass, momentum and energy together with the equation of state for the ideal gas. Linearising these equations with respect to the disturbances around the quiescent state, they are given, respectively, as follows:

(2.2)\begin{gather} \frac{1}{\rho_e} \left( \frac{\partial \rho^{\prime}}{\partial t} + v_x^{\prime} \frac{{\rm{d}} \rho_e}{{\rm{d}}\kern0.06em x} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot} \boldsymbol{v}^{\prime} = 0, \end{gather}

(2.3)\begin{gather}\rho_e \frac{\partial \boldsymbol{v}^{\prime}}{\partial t} ={-} \boldsymbol{\nabla} p^{\prime}, \end{gather}

(2.4)\begin{gather}\rho_e c_p \left( \frac{\partial T^{\prime}}{\partial t} + v_x^{\prime} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) = \frac{\partial p^{\prime}}{\partial t}, \end{gather}

(2.5)\begin{gather}\frac{p^{\prime}}{p_0} = \frac{\rho^{\prime}}{\rho_e} + \frac{T^{\prime}}{T_e}, \end{gather}

where the prime designates the disturbance, $\rho$, $p$, $T$ and $\boldsymbol {v}$ denote, respectively, the density, pressure, temperature and velocity vector of the gas, $v_x$ being the axial component of $\boldsymbol {v}$ and $c_p$ and also $c_v$ below (2.6) the specific heats at constant pressure and constant volume, respectively.

The quantities in the parentheses in (2.2) and (2.4) represent the linearised Lagrangian derivatives of the density and the temperature, respectively. In the adiabatic process, the enthalpy change $\rho _e c_p T^{\prime }$ per volume is equal to the pressure change $p^{\prime }$. Equation (2.4) is simply the equality of the rates of both quantities. Using (2.5) in (2.4) to eliminate $T^{\prime }/T_e$, the Lagrangian derivative of the density is written simply as

(2.6)\begin{equation} \frac{1}{\rho_e} \left( \frac{\partial \rho^{\prime}}{\partial t} + v_x^{\prime} \frac{{\rm{d}} \rho_e}{{\rm{d}}\kern0.06em x} \right) = \frac{1}{\gamma p_0} \frac{\partial p^{\prime}}{\partial t}, \end{equation}

where $\gamma$ denotes the ratio of specific heats $c_p/c_v$, and use has been made of the equation of state $p_0 = {\mathcal {R}}\rho _e T_e$, ${\mathcal {R}}$ being a gas constant and equal to $c_p-c_v$, and $\rho _e^{-1} \,{\mathrm {d}} \rho _e/{\mathrm {d}}\kern0.06em x = - T_e^{-1}\,{\mathrm {d}} T_e/{\mathrm {d}}\kern0.06em x$.

Noting that $\sqrt {\gamma p_0/\rho _e}$ represents the adiabatic sound speed $a_e$, as will be shown below, (2.6) indicates the adiabatic relation for a gas particle when the temperature (density) gradient is present. In the absence of the gradient, in fact, (2.6) is reduced to the well-known relation $p^{\prime } = a_0^{2} \rho ^{\prime }$. Thanks to (2.6), the equation of continuity (2.2) is rewritten as

(2.7)\begin{equation} \frac{1}{\gamma p_0} \frac{\partial p^{\prime}}{\partial t} ={-} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v}^{\prime}, \end{equation}

where $1/\gamma p_0$ is equal to $1/\rho _e a_e^{2}$, which is the adiabatic compressibility of the gas at $p_0$. As (2.7) holds in a uniform gas, it is simpler than (2.2) because there is no advection in pressure. Using (2.7), (2.4) is rewritten as

(2.8)\begin{equation} \frac{\partial T^{\prime}}{\partial t} + {v_x^{\prime}} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} ={-} \frac{\gamma p_0}{\rho_e c_p} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{v}^{\prime}, \end{equation}

where $\gamma p_0/\rho _e c_p = (\gamma - 1 )T_e$.

Differentiating (2.7) with respect to $t$, (2.3) is used to eliminate $\boldsymbol {v}^{\prime }$. Then $p^{\prime }$ is governed by

(2.9)\begin{equation} \frac{\partial^{2} p^{\prime}}{\partial t ^{2}} = \boldsymbol{\nabla} \boldsymbol{\cdot} (a_e^{2} \boldsymbol{\nabla} p^{\prime}), \end{equation}

where $a_e(x)$ denotes the local adiabatic sound speed given by $\sqrt {\gamma p_0/\rho _e}\,[ = \sqrt {(\gamma -1)c_p T_e}]$. This is the wave equation for the sound pressure in a gas that is non-uniform in temperature.

Here, the equation for the acoustic energy is considered. Multiplying (2.7) by $p^{\prime }$, and taking the inner product of (2.3) with $\boldsymbol {v}^{\prime }$, addition of both equations leads to the conservation equation for the acoustic energy

(2.10)\begin{equation} \frac{\partial E}{\partial t} + \boldsymbol{\nabla} \boldsymbol{\cdot} (p^{\prime} \boldsymbol{v}^{\prime}) = 0, \end{equation}

where $E$ represents the acoustic energy density given by the sum of the potential energy density $p^{\prime 2}/2 \rho _e a_e^{2}$ and the kinetic one $\rho _e \boldsymbol {v}^{\prime }\boldsymbol {\cdot}\boldsymbol {v}^{\prime }/2$, while $p^{\prime } \boldsymbol {v}^{\prime }$ represents the acoustic energy flux density vector called the intensity vector.

When (2.10) is averaged over the duct cross-section, it follows that

(2.11)\begin{equation} \frac{\partial \overline{E}}{\partial t} + \frac{\partial \overline{I}}{\partial x} = 0, \end{equation}

with $I = p^{\prime } v_x^{\prime }$ where the overbar designates the mean over the cross-section defined by

(2.12a,b)\begin{equation} \overline{E} \equiv \frac{1}{A} \int_A E \,{\mathrm{d}} A \quad \mathrm{and} \quad \overline{I} \equiv \frac{1}{A} \int_A I \,{\mathrm{d}} A, \end{equation}

with $A$ ($ = {\rm \pi}R^{2}$) and ${\mathrm {d}} A$ being, respectively, the cross-sectional area of the duct and its area element, and the boundary condition $\boldsymbol {v}^{\prime } \boldsymbol {\cdot} \boldsymbol {n} = 0$ on the duct wall has been used, $\boldsymbol {n}$ being a normal to the wall surface directed into the gas.

2.2. Dispersion relations for non-diffusive propagation in a uniform gas

At the outset, the simplest case where the temperature is uniform, $T_e = T_0$, is recapitulated (Rienstra Reference Rienstra2016). Taking the cylindrical coordinates $(r, \theta, x)$ with $x$ along the duct axis, (2.9) is expressed for $p^{\prime } (r, \theta, x, t)$ as

(2.13)\begin{equation} \frac{\partial^{2} p^{\prime}}{\partial t ^{2}} = a_0^{2} \left( \frac{\partial^{2} p^{\prime}}{\partial r ^{2}} + \frac{1}{r} \frac{\partial p^{\prime}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} p^{\prime}}{\partial \theta ^{2}} + \frac{\partial^{2} p^{\prime}}{\partial x ^{2}} \right). \end{equation}

When a wave travelling along the $x$ axis and spinning about it is considered in the form of $p^{\prime } = P(r)$ $\cos (kx \pm m\theta - \omega t)$ $(m = 1, 2, 3, \dots )$, a bounded solution of $P$ is obtained from (2.13) as

(2.14)\begin{equation} P = {\mathcal{A}} \,{\rm J}_m(\alpha r/R), \end{equation}

with $\alpha \equiv (\omega ^{2} /a_0^{2} - k^{2})^{1/2}R$ for $0 \leqslant r < R$, where ${\mathcal {A}}$ is an arbitrary constant, and ${\rm J}_m$ denotes the Bessel function of the first kind; $k$ and $\omega$ denote, respectively, an axial wavenumber and an angular frequency. The choice of ${\pm }m \theta$ allows $p^{\prime }$ to spin about the $x$ axis left or right handedly. By addition of these, non-spinning waves proportional to $\cos (m \theta )$ or $\sin (m\theta )$ are constructed.

Application of the boundary condition ${\mathrm {d}} P/{\mathrm {d}} r = 0$ on the wall leads to the relation $\alpha \dot {{\rm J}}_m(\alpha ) = 0$ for ${\mathcal {A}}$ to be non-trivial. Here and hereafter the dot over ${\rm J}_m$ designates the derivative with respect to its argument. A trivial solution $\alpha = 0$ represents a planar sound only when $m = 0$, and this case is excluded. Zeros of $\dot {{\rm J}}_m(\alpha )$, designated by $\alpha _{m, j}$ $(j=1, 2, 3, \dots )$, are ordered as $0 < \alpha _{m,1} < \alpha _{m,2} < \alpha _{m,3} < \dots$. For moderate values of $m = 0, 1, 2, \dots, 8$, the zeros up to $j = 20$ are given in table 9.5 in Abramowitz & Stegun (Reference Abramowitz and Stegun1972). For other values of $m$ and $j$, see Tyler & Sofrin (Reference Tyler and Sofrin1962), Michalke (Reference Michalke1989) and Rienstra (Reference Rienstra2016). It is noted that $\alpha _{0,j}$ for $j = 1,2$ and 3 are greater than $\alpha _{1,j}$ and $\alpha _{2,j}$. For a large value of $m$, $\alpha _{m,1}$ is given asymptotically by $\alpha _{m,1} = m + \chi m^{1/3} + O(m^{-1/3})$, $\chi = 0.80861\ldots$ and $\alpha _{m,1}$ exceeds $m$ (Abramowitz & Stegun Reference Abramowitz and Stegun1972).

The solutions (2.14) with $\alpha = \alpha _{m,j}$ $(j = 1, 2, 3, \dots )$ constitute eigenfunctions in the radial direction, where two indices $m$ and $j$ designate the azimuthal and radial mode numbers, respectively. For the $(m,j)$ mode, the dispersion relation is given by

(2.15)\begin{equation} \left( \frac{\omega R}{a_0} \right)^{2} = \alpha_{m,j}^{2} + (kR)^{2}. \end{equation}

The wave is propagated in the axial direction only when $\omega R/a_0 > \alpha _{m,j}$. At $\omega R/a_0 = \alpha _{m,j}$, the cutoff occurs, below which $kR$ becomes imaginary and the wave becomes evanescent in the axial direction, although it is propagated circumferentially. Another characteristic of the non-planar sound is dispersive propagation in contrast to the planar one. The phase velocity $V_p~(\equiv \omega /k)$ differs from the group velocity $V_g\ ({\equiv }{\mathrm {d}} \omega /{\mathrm {d}} k)$ for the energy transfer given by

(2.16)\begin{equation} V_g = \frac{a_0^{2} k}{\omega}. \end{equation}

This relation shows $V_g V_p = a_0^{2}$. It is easily found that $V_p$ is always faster than $a_0$, whereas $V_g$ is slower than $a_0$.

Figure 1 shows how the radial and azimuthal distributions of ${\rm J}_m (\alpha _{m,1}r/R) \cos (m \theta )$ normalised by ${\rm J}_m(\alpha _{m,1})$ in the first radial mode $j=1$ change with $m$, as $m$ takes values of $0, 1, 2, 4, 8$ and 16. Blank line(s) through the centre for $m \ne 0$ show the node line(s) where the pressure vanishes, while the node circle appears for $m = 0$. For a large value of $m$, ${\rm J}_m(\alpha _{m,1}r/R)/{\rm J}_m(\alpha _{m,1})$ tends to vanish rapidly away from the wall ($r/R \to 0$) so that the sound tends to be confined in a narrow band adjacent to the duct wall, outside of which silence prevails. This is a so-called whispering gallery mode (Wright Reference Wright2012). As $m$ increases, the band width becomes narrower but slowly of the order of $m^{-2/3}$ (see Appendix A).

Figure 1.

Contour plots of ${\rm J}_m(\alpha _{m,1}r/R) \cos (m \theta )$ normalised by ${\rm J}_m(\alpha _{m,1})$ in the $(\kern0.7pt y, z)$-plane with $y = r \cos \theta$ and $z = r \sin \theta$ for $m = 0, 1, 2, 4, 8$ and 16 where $\alpha _{0,1}=3.8317$, $\alpha _{1,1}=1.8411$, $\alpha _{2,1}= 3.0542$, $\alpha _{4,1} = 5.3175$, $\alpha _{8,1}=9.6474$ and $\alpha _{16,1}=18.0632$. By periodicity in $\theta$, $m$ node lines (in blank) where $p^{\prime } = 0$ are visible except for the case with $m = 0$ where one node circle (in blank) is visible.

The Bessel function ${\rm J}_m(\alpha )$ oscillates around zero, while decaying slowly, and it takes extrema at $\alpha _{m,j}$ $(j = 1, 2, 3, \dots )$. Because it necessarily vanishes at a point between $\alpha _{m,j}$ and $\alpha _{m,j+1}$, ${\rm J}_m(\alpha _{m,j}r/R)$ makes zero crossings $j-1$-times in $0 < r < R$ except for the case with $m = 0$ where one more crossing is counted. Thus the $j-1$ node circles appear for $m > 0$. Figure 2 shows the radial and azimuthal distributions of ${\rm J}_m(\alpha _{m,2}r/R) \cos (m \theta )$ normalised by ${\rm J}_m(\alpha _{m,1})$ in the case of the second radial mode $j = 2$ where $m$ takes the same values as in figure 1. It is common that the sound is confined in a narrow band as $m$ increases.

Figure 2.

Contour plots of ${\rm J}_m(\alpha _{m,2}r/R) \cos (m \theta )$ normalised by ${\rm J}_m(\alpha _{m,1})$ in the $(\kern0.7pt y, z)$-plane with $y= r \cos \theta$ and $z = r \sin \theta$ for $m = 0, 1, 2, 4, 8$ and 16 where $\alpha _{0,2}=7.0155$, $\alpha _{1,2}=5.3314$, $\alpha _{2,2}=6.7061$, $\alpha _{4,2}=9.2823$, $\alpha _{8,2}=14.1155$ and $\alpha _{16,2}=23.2642$. Besides $m$ node lines (in blank) where $p^{\prime } =0$, one node circle is visible except for the case with $m=0$ where two node circles (in blank) are visible.

For the $(m, j)$ mode, the disturbances in the density and the temperature are given, respectively, by $\rho ^{\prime } = p^{\prime }/a_0^{2}$ and $T^{\prime } = p^{\prime }/\rho _0 c_p$ in the adiabatic process. The velocity vector is obtained from (2.3) as

(2.17a)\begin{gather} v_r^{\prime} = \frac{{\mathcal{A}}}{\rho_0 a_0}\frac{\alpha_{m,j}}{ \varOmega } \dot{{\rm J}}_m (\alpha_{m,j} r/R)\sin \psi, \end{gather}

(2.17b)\begin{gather}v_\theta^{\prime} ={\pm} \frac{{\mathcal{A}}}{\rho_0 a_0} \frac{~m R}{\varOmega r} {\rm J}_m (\alpha_{m,j} r/R) \cos \psi, \end{gather}

(2.17c)\begin{gather}v_x^{\prime} = \frac{{\mathcal{A}}}{\rho_0 a_0}\frac{K}{\varOmega} {\rm J}_m(\alpha_{m,j} r/R) \cos \psi, \end{gather}

with $\psi = kx \pm m \theta - \omega t$ and the sign $\pm$ in (2.17b) vertically ordered as the one in $\psi$, where $\varOmega$ and $K$ represent, respectively, the dimensionless angular frequency $\omega R/a_0$ and axial wavenumber $kR$. It is noted that only $v_r^{\prime }$ differs in phase of $\psi$ from the other variables by ${\rm \pi} /2$.

Using these solutions, the means of the acoustic energy density $E$ and the axial acoustic energy flux density $I\ ({\equiv}\,p^{\prime } v_x^{\prime })$ are calculated. The former is given by

(2.18)\begin{equation} \overline{E} = \frac{1}{{\rm \pi} R^{2}} \int_0^{R} r \,{\mathrm{d}} r \int_0^{2{\rm \pi}} \left( \frac{p^{\prime 2}}{2 \rho_0 a_0^{2}} + \frac{\rho_0}{2} \boldsymbol{v}^{\prime} \boldsymbol{\cdot} \boldsymbol{v}^{\prime} \right) {\mathrm{d}} \theta = \frac{{\mathcal{A}}^{2}}{\rho_0 a_0^{2} R^{2}} \int_0^{R} r \,{\rm J}_m^{2}(\alpha_{m,j}r/R) \,{\mathrm{d}} r, \end{equation}

where the potential and kinetic energies contribute equally to $\overline{E}$, while the latter is given by

(2.19)\begin{equation} \overline{I} = \frac{1}{{\rm \pi} R^{2}} \int_0^{R} r \,{\mathrm{d}} r \int_0^{2{\rm \pi}} p^{\prime} v_x^{\prime} \,{\mathrm{d}} \theta = \frac{{\mathcal{A}}^{2} k}{\rho_0 \omega R^{2}} \int_0^{R} r\, {\rm J}_m^{2}(\alpha_{m,j}r/R) \,{\mathrm{d}} r, \end{equation}

where the integral in common is executed analytically to be $R^{2}/ c_{m,j}$ by using (2.23) and (2.24) below. Note that the dependence on $x$ and $t$ is smeared out on integration with respect to $\theta$. It is confirmed from (2.18) and (2.19) that the acoustic energy density $\overline{E}$ is transferred in the axial direction by the group velocity because ${\overline{I}}/{\overline{E}} = {a_0^{2} k}/{\omega } = V_g$.

2.3. Effects of non-uniform temperature

Consider (2.9) when the temperature $T_e$ is non-uniform axially. Using a complex Fourier series expansion in $\theta$, $p^{\prime }$ is decomposed into $P_m(r,x,t) {\mathrm {e} }^{{\mathrm {i}} m \theta }$ ($m = 0, 1, 2, \dots$) where $P_m$ may be complex, in general, but is taken as real here. In such a complex notation, the real part is understood to be taken here and hereafter. It follows from (2.9) that $P_m$ satisfies

(2.20)\begin{equation} \frac{\partial^{2} P_m}{\partial t ^{2}} = a_e^{2} \left[ \frac{1}{r}\frac{\partial \ }{\partial r}\left( r \frac{\partial P_m}{\partial r} \right) - \frac{m^{2}}{r^{2}}P_m \right] + \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial P_m}{\partial x} \right). \end{equation}

Making use of a Fourier–Bessel series expansion also known as Dini series, $P_m$ is expanded into the following series:

(2.21)\begin{equation} P_m(r,x,t)= \sum_{j=1}^{\infty} c_{m,j} \bar{P}_{m,j}(x,t){\rm J}_m (\alpha_{m,j} r/R), \end{equation}

where $\bar {P}_{m,j}$ $(j = 1, 2, 3, \dots )$ are defined by

(2.22)\begin{equation} \bar{P}_{m,j}(x, t) \equiv \frac{1}{R^{2}} \int_0^{R} P_m(r,x,t) r \,{\rm J}_m(\alpha_{m,j} r/R) \,{\mathrm{d}} r, \end{equation}

with $c_{m,j}$ given by (Abramowitz & Stegun Reference Abramowitz and Stegun1972)

(2.23)\begin{equation} c_{m,j} = \frac{2 \alpha_{m,j}^{2}}{(\alpha_{m,j}^{2} - m^{2}) {\rm J}_m^{2}(\alpha_{m,j})}, \end{equation}

and the orthogonality relation holds

(2.24)\begin{equation} \frac{1}{R^{2}} \int_0^{R} r \,{\rm J}_m \left( \frac{\alpha_{m,i}r}{R} \right) {\rm J}_m \left( \frac{\alpha_{m,j} r}{R} \right) {\mathrm{d}} r = \left\{\begin{array}{@{}ll} 1/c_{m,j} & \mathrm{if} \ i = j,\\ 0 & \mathrm{if} \ i \ne j, \end{array}\right. \end{equation}

for $i, j = 1, 2, 3, \dots$. Here, $\bar {P}_{m,j}$ are weighted means of $P_{m}$ by $r \,{\rm J}_m (\alpha _{m,j}r/R)$. Although the overbar $\overline {(\,\cdot\,)}$ has already been used to designate the mean over the cross-section, no confusion would occur with the short bar used here.

Taking the weighted means of (2.20) for $j = 1, 2, 3, \dots$, this is transformed into

(2.25)\begin{equation} \frac{\partial^{2} \bar{P}_{m,j} }{\partial t ^{2}} - \frac{\partial \ }{\partial x} \left(a_e^{2} \frac{\partial \bar{P}_{m,j} }{\partial x} \right) + \left( \frac{\alpha_{m,j} }{R} \right)^{2} a_e^{2} \bar{P}_{m,j} = 0, \end{equation}

where the relation $\dot {{\rm J}}_m(\alpha _{m,j}) = 0$ and the Bessel differential equation have been used. Note that, when (2.25) holds, each term in the series of (2.21) satisfies (2.20) separately. When $a_e$ is constant, (2.25) are the telegraphic equation or the one-dimensional Klein–Gordon equation. Thanks to the Fourier–Bessel series expansion, (2.20) is reduced to the one-dimensional dispersive wave equation for each $\bar {P}_{m,j}$. As long as the temperature is non-uniform axially only, the radial structure in any cross-section stipulated by the Bessel functions in (2.21) is kept unchanged from that in a uniform case, and the non-uniform effects appear only in $\bar {P}_{m,j}$. Therefore, the cross-sectional configuration in each mode displayed in figures 1 and 2 is also unchanged in the non-uniform case.

2.4. Conservation of acoustic energy

Next, consider the acoustic energy equation in the azimuthal mode $m$. Being consistent with $p^{\prime }$, $\boldsymbol {v}^{\prime }$ is also decomposed into

(2.26)\begin{equation} \boldsymbol{v}^{\prime} \equiv (v_r^{\prime}, v_\theta^{\prime}, v_x^{\prime}) = (V_m, {\mathrm{i}} W_m, U_m) {\mathrm{e} }^{{\mathrm{i}} m \theta}. \end{equation}

Then, (2.7) and (2.3) are written, respectively, as

(2.27a)\begin{gather} \frac{1}{\rho_e a_e^{2}} \frac{\partial P_m}{\partial t} ={-}\frac{1}{r} \frac{\partial \ }{\partial r}(r V_m) + \frac{m}{r} W_m - \frac{\partial U_m}{\partial x}, \end{gather}

(2.27b)\begin{gather}\rho_e \frac{\partial V_m}{\partial t} ={-} \frac{\partial P_m}{\partial r}, \end{gather}

(2.27c)\begin{gather}\rho_e \frac{\partial W_m}{\partial t} ={-} \frac{m}{r} P_m, \end{gather}

(2.27d)\begin{gather}\rho_e \frac{\partial U_m}{\partial t} ={-} \frac{\partial P_m}{\partial x}. \end{gather}

Multiplying (2.27a) with $r P_m$, and also multiplying (2.27b) to (2.27d) with $r V_m$, $r W_m$ and $r U_m$, respectively, and adding them together to integrate from $r=0$ to $r=R$, it follows that

(2.28)\begin{equation} \frac{\partial E_m}{\partial t} + \frac{\partial {I}_m}{\partial x} = 0, \end{equation}

where $E_m$ and ${I}_m$ are defined, respectively, by

(2.29a)\begin{gather} E_m = \frac{1}{R^{2}} \int_0^{R} \left[ \frac{P_m^{2}}{2 \rho_e a_e^{2}} + \frac{\rho_e}{2} \left( V_m^{2} + W_m^{2} + U_m^{2} \right) \right] r \,{\mathrm{d}} r, \end{gather}

(2.29b)\begin{gather}{I}_m = \frac{1}{R^{2}} \int_0^{R} P_m U_m r \,{\mathrm{d}} r, \end{gather}

where the boundary condition $V_m = 0$ at $r = R$ has been used, and all variables are assumed to be real. Here, $E_m$ and ${I}_m$ denote, respectively, the acoustic energy density and the axial acoustic energy flux density averaged over $R$. Equation (2.28) dictates the local conservation of the acoustic energy in the azimuthal mode $m$.

Using (2.21) and setting $\bar {P}_{m,j}$ to be $\partial \varPhi _{m,j}/\partial t$, the potential energy in $E_m$ is expressed as

(2.30)\begin{equation} \frac{1}{R^{2}} \int_0^{R} \frac{P_m^{2}}{2 \rho_e a_e^{2}} r \,{\mathrm{d}} r = \frac{1}{2\rho_e a_e^{2}} \sum_{j=1}^{R} c_{m,j} \left( \frac{\partial \varPhi_{m,j}}{\partial t} \right)^{2}, \end{equation}

where the orthogonality relation (2.24) has been used. In terms of $\varPhi _{m,j}$, the velocity components are expressed from (2.27b) to (2.27d) as

(2.31a)\begin{gather} V_m ={-} \frac{1}{\rho_e} \sum_{j=1}^{\infty} c_{m,j} \frac{\alpha_{m,j}}{R} \varPhi_{m,j} \dot{{\rm J}}_m (\alpha_{m,j} r/R), \end{gather}

(2.31b)\begin{gather}W_m ={-} \frac{1}{\rho_e} \sum_{j=1}^{\infty} c_{m,j} \frac{m}{r} \varPhi_{m,j} {\rm J}_m( \alpha_{m,j}r/R), \end{gather}

(2.31c)\begin{gather}U_m ={-} \frac{1}{\rho_e} \sum_{j=1}^{\infty} c_{m,j} \frac{\partial \varPhi_{m,j}}{\partial x} {\rm J}_m(\alpha_{m,j} r/R). \end{gather}

Using these expressions, the kinetic energy in $E_m$ is expressed as

(2.32)\begin{equation} \frac{1}{R^{2}} \int_0^{R} \frac{\rho_e}{2} \left( V_m^{2} + W_m^{2} + U_m^{2} \right) r \,{\mathrm{d}} r = \frac{1}{2\rho_e} \sum_{j=1}^{\infty} c_{m,j} \left[ \left( \frac{\alpha_{m,j}}{R} \right)^{2} \varPhi_{m,j}^{2} + \left( \frac{\partial \varPhi_{m,j}}{\partial x} \right)^{2} \right], \end{equation}

where the following relations have been used:

(2.33)\begin{align} &\int_0^{R} \left[ \frac{\alpha_{m,i}}{R} \frac{\alpha_{m,j}}{R} \dot{{\rm J}}_m \left( \frac{\alpha_{m,i} r}{R} \right) \dot{{\rm J}}_m \left( \frac{\alpha_{m,j} r}{R} \right) + \frac{m^{2}}{r^{2}} {\rm J}_m \left( \frac{\alpha_{m,i} r}{R} \right) {\rm J}_m \left( \frac{\alpha_{m,j}r}{R} \right) \right] r \,{\mathrm{d}} r \nonumber\\ &\quad = \left\{ \begin{array}{ll} {\alpha_{m,j}^{2}}/{c_{m,j}} & \mathrm{if} \ i = j,\\ 0 & \mathrm{if} \ i \ne j, \end{array} \right. \end{align}

for $i, j = 1, 2, 3,\dots$. In (2.32), the kinetic energy due to the radial and azimuthal velocity is counted simply in $(\alpha _{m,j}/R)^{2} \varPhi _{m,j}^{2}$, which yields the dispersion.

Adding (2.30) and (2.32), $E_m$ is given by

(2.34)\begin{equation} E_m = \sum_{j=1}^{\infty} c_{m,j} E_{m,j}, \end{equation}

with

(2.35)\begin{equation} E_{m,j} = \frac{1}{2 \rho_e a_e^{2}} \left[ \left( \frac{\partial \varPhi_ {m,j}}{\partial t} \right)^{2} + \left( \frac{\alpha_{m,j}}{R} \right)^{2} a_e^{2} \varPhi_{m,j}^{2} + a_e^{2} \left( \frac{\partial \varPhi_{m,j}}{\partial x} \right)^{2} \right]. \end{equation}

On the other hand, ${I}_m$ is given by

(2.36)\begin{equation} {I}_m = \sum_{j=1}^{\infty} c_{m,j} {I}_{m,j}, \end{equation}

with

(2.37)\begin{equation} {I}_{m,j} ={-} \frac{1}{\rho_e} \frac{\partial \varPhi_{m,j}}{\partial t} \frac{\partial \varPhi_{m,j}}{\partial x}. \end{equation}

The relation (2.28) with (2.34) and (2.36) shows that the conservation of the acoustic energy holds for any radial variations, although restricted in the azimuthal mode $m$. This also suggests that the conservation holds each of the $(m, j)$ modes independently.

3.1. Boundary-layer approximation

Figure 3(a) displays a cross-sectional configuration of the duct where the central core region is surrounded by the boundary layer (drawn exaggerated) on the duct wall, while figure 3(b) blows up the boundary layer in a three-dimensional configuration. To capture properly the phenomena occurring in the thin layer, the boundary-layer approximation is introduced by following the procedure demonstrated in Sugimoto & Tsujimoto (Reference Sugimoto and Tsujimoto2002). A radial coordinate $n$ normal to the duct wall is taken to be $n = R-r$, where $n$ covers a narrow domain comparable to $\delta$ ($0 < n \sim \delta \ll R$). A circumferential coordinate $\eta$ along the duct wall is taken to be $\eta = R \theta$. In the boundary layer, $n$ and $\eta$ are used instead of $r$ and $\theta$, while $x$ is used in common.

Figure 3.

(a) A cross-sectional configuration of the duct of radius $R$ consisting of the central core region and the boundary layer (drawn exaggerated) on the duct wall where the cylindrical coordinates $(r, \theta, x)$ are taken for the core region, the $x$-coordinate normal to the sheet, while the coordinates $n\ ( = R-r)$ and $\eta \ ( = R \theta )$ are taken for the boundary layer normal to the duct wall and directed inward, and along its periphery, respectively, with the $x$-coordinate in common. (b) Blow-up of the boundary layer in a three-dimensional configuration where the pressure is unchanged over the thickness from $p^{\prime }$ at the edge of the boundary layer; $\tilde {\boldsymbol {v}}\ [ = (\tilde {v}, \tilde {u}, \tilde {w})]$ represents the velocity vector; $s_x$, $s_\eta$ and $q$ represent, respectively, the $x$- and $\eta$-components of the shear stress on the duct wall and acting on the gas, and the heat flux density flowing into the gas; $\check {v}_b$ represents the defect radial velocity directed inward at the edge of the boundary layer as $n \to \infty$.

By the constraints due to the boundary conditions on the duct wall, some physical variables must vary steeply in the radial direction over the scale of $\delta$, but slowly in the axial and circumferential directions. This means that $\partial /\partial x \sim \partial /\partial \eta \sim R^{-1} \ll \partial /\partial n \sim \delta ^{-1}$. At the scale of $\delta$, the curvature of the duct wall is invisible so a term with $1/R$ is ignored in comparison with $\partial /\partial n$, and $r^{-1}\partial /\partial \theta$ is approximated to be $R^{-1}\partial /\partial \theta \ ( = \partial /\partial \eta )$ by ignoring $n/R\ ({\sim }\delta /R \ll 1)$.

Using the boundary-layer approximation, the fluid dynamical equations are linearised around the quiescent state. Disturbances in the boundary layer are designated by attaching a tilde. The equation of continuity is then given by

(3.2)\begin{equation} \frac{1}{\rho_e} \left( \frac{\partial \tilde{\rho}}{\partial t} + \tilde{u} \frac{{\rm{d}} \rho_e}{{\rm{d}}\kern0.06em x} \right) + \tilde{\boldsymbol{\nabla}} \boldsymbol{\cdot} \tilde{\boldsymbol{v}} = 0, \end{equation}

with

(3.3)\begin{equation} \tilde{\boldsymbol{\nabla}}\boldsymbol{\cdot} \tilde{\boldsymbol{v}} \equiv \frac{\partial \tilde{v}}{\partial n} + \frac{\partial \tilde{u}}{\partial x} + \frac{\partial \tilde{w}}{\partial \eta}, \end{equation}

where $\tilde {\boldsymbol {v}}$ has the components $\tilde {v}$, $\tilde {u}$ and $\tilde {w}$ in the $n$-, $x$- and $\eta$-directions, respectively. Since the positive $n$ is taken opposite to that of $r$, note that $\tilde {v}$ corresponds to $-v_r$, and that $\partial /\partial r$ is replaced by $- \partial /\partial n$. Letting a typical axial length to be $L$, which is assumed to be comparable to $R$, this equation means that $\tilde {\boldsymbol {\nabla }} \boldsymbol {\cdot} \tilde {\boldsymbol {v}}$ balances with $\omega \tilde {\rho }/\rho _e$ or $\tilde {u}/L$, and that $\tilde {v}/\delta \sim \tilde {u}/L \sim \tilde {w}/R$. Therefore, it follows that $\tilde {v}$ is small of $(\delta /L) \tilde {u}$ or $(\delta /R) \tilde {w}$.

Next, the equation of motion is considered. When viscosities vary with the temperature and the pressure, the full Navier–Stokes equations for compressible fluids are given in Appendix B. Noting that $R^{-2}\ll R^{-1}\partial /\partial n \ll \delta ^{-2}$ and $\partial /\partial x \sim L^{-1} \sim R^{-1}$, the Laplacian $\varDelta$ may be approximated to be $\partial ^{2}/\partial n^{2} ({\sim }\delta ^{-2})$. After the linearisation, the leading terms in (B1a) to (B1c) are given, respectively, by

(3.4a)\begin{gather} \rho_e \frac{\partial \tilde{v}}{\partial t} ={-} \frac{\partial \tilde{p}}{\partial n} + \mu_e \frac{\partial^{2} \tilde{v}}{\partial n ^{2}} + \left( \mu_{ve} + \frac{\mu_e}{3} \right) \frac{\partial }{\partial n} \tilde{\boldsymbol{\nabla}} \boldsymbol{\cdot} \tilde{\boldsymbol{v}} + \frac{\partial \mu_e}{\partial x} \frac{\partial \tilde{u}}{\partial n}, \end{gather}

(3.4b)\begin{gather}\rho_e \frac{\partial \tilde{u}}{\partial t} ={-} \frac{\partial \tilde{p}}{\partial x} + \mu_e \frac{\partial^{2} \tilde{u}}{\partial n ^{2}} , \end{gather}

(3.4c)\begin{gather}\rho_e \frac{\partial \tilde{w}}{\partial t} ={-} \frac{\partial \tilde{p}}{\partial \eta} + \mu_e \frac{\partial^{2} \tilde{w}}{\partial n ^{2}} , \end{gather}

where $\mu _e$ and $\mu _{{v}e}$ denote, respectively, the shear and bulk viscosities at $T=T_e$.

In (3.4a), the terms associated with the velocity are comparable by noting $\delta \sim (\nu _e/\omega )^{1/2}$. The excess pressure $\tilde {p}$ is estimated by using a typical acoustic impedance ${\rho}_{0} a_0$ to be ${\rho}_0 a_0 \tilde {u}$ or $\rho _0 a_0 \tilde {w}$ (see (2.17b) and (2.17c)). Comparing $\mu _e \partial ^{2} \tilde {v}/\partial n^{2}$ with $\partial \tilde {p}/\partial n$, it is found that the former is smaller by $(\delta /R)^{2}$, where the relations $\omega R \sim a_0$ and $\tilde {v}/\tilde {w} \sim \delta /R$ have been used. Hence it turns out that (3.4a) is substantially given by $\partial \tilde {p}/\partial n = 0$, which means no pressure gradient in the radial direction. Then, the pressure at the edge of the boundary layer penetrates into it so that $\tilde {p}(n, x, \eta, t) = p^{\prime } (R-\delta, x, \eta, t)$. This is the well-known outcome of the boundary-layer approximation. In (3.4b) and (3.4c), the terms with $\tilde {\boldsymbol {\nabla }} \boldsymbol {\cdot} \tilde {\boldsymbol {v}}$ and $\partial \mu _e /\partial x$ in (B1b) and (B1c) have been ignored because they are smaller by $(\delta /R)^{2}$.

Equation (2.4) is augmented by thermal diffusion as

(3.5)\begin{equation} \rho_e c_p \left( \frac{\partial \tilde{T}}{\partial t} + \tilde{u} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) = \frac{\partial \tilde{p}}{\partial t} + k_e \frac{\partial^{2} \tilde{T}}{\partial n ^{2}} , \end{equation}

where $k_e$ denotes a thermal conductivity at $T_e$. The temperature dependence of $k_e$ and of $\mu _e$ and $\mu _{{v}e}$ is taken into account in a power law of the form (Sugimoto Reference Sugimoto2010)

(3.6)\begin{equation} \frac{\mu_e}{\mu_0} = \frac{\mu_{{v}e}}{\mu_{v0}} = \frac{k_e}{k_0} = \left( \frac{T_e}{T_0} \right)^{\beta}, \end{equation}

where $\beta$ is a constant between 0.5 and 0.7 for air. Finally the equation of state is given by

(3.7)\begin{equation} \frac{\tilde{p}}{p_0} = \frac{\tilde{\rho}}{\rho_e} + \frac{\tilde{T}}{T_e}. \end{equation}

In the boundary layer, it is convenient to introduce defects, designated by a check as $\check {(\, \cdot\, )}$, from the variables of the core region at the edge of the boundary layer as

(3.8)\begin{equation} (\tilde{\rho}, \tilde{v}, \tilde{u}, \tilde{w}, \tilde{T}) = (\rho^{\prime} + \check{\rho}, -v_r^{\prime} + \check{v}, v_x^{\prime} + \check{u}, v_\theta^{\prime} + \check{w}, T^{\prime} + \check{T}). \end{equation}

If $\tilde {p}$ is set to be $p^{\prime } + \check {p}$, $\check {p}$ vanishes throughout the boundary layer. Since the variables with prime vary little over the thin boundary layer, they may be approximated by those at $r = R$ to the lowest order. For example, $v_x^{\prime }$ at $r = R-\delta$ is expanded around $r = R$ in terms of $n$ as $v_x^{\prime } = v_x^{\prime }|_R - \partial v_x^{\prime }/\partial r|_R n + \dots$. The second term is small of $v_x^{\prime } \delta /R$ and is of higher order. This is also the case with the other variables but $v_r^{\prime }$. As $v_r^{\prime }$ vanishes at $r = R$, it is expanded into $\partial v_r^{\prime } /\partial r|_R n + \dots$ and is small of $v_r^{\prime } \delta /R$. This appears to be negligible. However, it will turn out to be comparable to $\check {v}$.

Using the replacements (3.8) in (3.2) and making use of (2.2) at the edge of the boundary layer where $r^{-1}\partial v_\theta ^{\prime }/\partial \theta \approx R^{-1} \partial v_\theta ^{\prime }/\partial \theta$ and $v_r^{\prime }/r$ is negligible, (3.2) is written as

(3.9)\begin{equation} \frac{1}{\rho_e} \left( \frac{\partial \check{\rho}}{\partial t} + \check{u} \frac{{\rm{d}} \rho_e}{{\rm{d}}\kern0.06em x} \right) + \frac{\partial \check{v}}{\partial n} + \frac{\partial \check{u}}{\partial x} + \frac{\partial \check{w}}{\partial \eta} = 0. \end{equation}

Substituting (3.8) into (3.4b) and (3.4c) and using (2.3), similarly, $\check {u}$ and $\check {w}$ are described by the following equations:

(3.10a)\begin{gather} \frac{\partial \check{u}}{\partial t} = \nu_e \frac{\partial^{2} \check{u}}{\partial n ^{2}} , \end{gather}

(3.10b)\begin{gather}\frac{\partial \check{w}}{\partial t} = \nu_e \frac{\partial^{2} \check{w}}{\partial n ^{2}} , \end{gather}

where $\nu _e$ is the kinematic viscosity defined by $\mu _e/\rho _e$. In (3.4b), $\partial ^{2} v_x^{\prime } /\partial n^{2}$ is negligible in comparison with $\partial ^{2} \check {u}/\partial n^{2}$ of order $v_x^{\prime }/\delta ^{2}$, because $\partial ^{2} v_x^{\prime }/\partial n^{2}~(= \partial ^{2} v_x^{\prime }/\partial r^{2})$ is of order $v_x^{\prime }/R^{2}$. This is also the case with (3.4c).

In a similar fashion, (3.5) is reduced, by using (2.4), to the heat conduction equation for $\check {T}$ with advection as

(3.11)\begin{equation} \frac{\partial \check{T}}{\partial t} + \check{u} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} = \kappa_e \frac{\partial^{2} \check{T}}{\partial n ^{2}} , \end{equation}

where $\kappa _e$ is the thermal diffusivity defined by $k_e/\rho _e c_p$, and $\partial ^{2} T^{\prime }/\partial n^{2}$ has been neglected by the same reason as $\partial ^{2} v_x^{\prime }/\partial n^{2}$ in (3.10a). Equation (3.7) is reduced to

(3.12)\begin{equation} 0 = \frac{\check{\rho}}{\rho_e} + \frac{\check{T}}{T_e}, \end{equation}

because $\check {p}$ vanishes in the boundary layer. This means that the defects of the density and the temperature are subjected to the isobaric change. Making use of (3.12), (3.9) is rewritten as

(3.13)\begin{equation} \frac{1}{T_e} \left( \frac{\partial \check{T}}{\partial t} + \check{u} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) = \frac{\partial \check{v}}{\partial n} + \frac{\partial \check{u}}{\partial x} + \frac{\partial \check{w}}{\partial \eta}. \end{equation}

This relation is compared with (2.8) for the adiabatic change.

Equations (3.10) and (3.11) are supplemented by the no-slip and isothermal boundary conditions on the duct wall as

(3.14)\begin{equation} (\tilde{u}, \tilde{w}, \tilde{T}) = (0, 0, 0) \quad \mathrm{at} \ n = 0. \end{equation}

On the other hand, the edge of the boundary layer is viewed to be located at $n/\delta \to \infty$ in the coordinate $n$, but $n/R = (R-r)/R \ll 1$, even if $n$ is taken to be infinity. Then the matching conditions with the core region are given by

(3.15)\begin{equation} (\tilde{u}, \tilde{w}, \tilde{T}) \to (v_x^{\prime}, v_\theta^{\prime}, T^{\prime}) \quad \text{as }n \to \infty, \end{equation}

where the right-hand side is evaluated at $r = R$. In terms of the defects, the boundary conditions and the matching conditions are given, respectively, by

(3.16)\begin{equation} (\check{u}, \check{w}, \check{T}) = (- v_x^{\prime}, -v_\theta^{\prime}, - T^{\prime}) \quad \mathrm{at} \ n = 0, \end{equation}

and

(3.17)\begin{equation} (\check{u}, \check{w}, \check{T}) \to (0, 0, 0) \quad \mathrm{as} \ n \to \infty. \end{equation}

Thus the boundary-value problems for (3.10a), (3.10b) and (3.11) are posed and solved for $\check {u}$, $\check {w}$ and $\check {T}$. The defects $\check {\rho }$ and $\check {v}$ that remain are to be determined after these problems are solved. The defect $\check {\rho }$ is available immediately by (3.12), while $\check {v}$ is obtained by integrating (3.13) with respect to $n$ where the left-hand side is replaced by (3.11).

3.2. Shear stress and heat flux on the wall

Analytical solutions to (3.10a) and (3.10b) are available by following the procedure used in Sugimoto & Tsujimoto (Reference Sugimoto and Tsujimoto2002). Applying the method of Fourier transform defined, e.g. for $\check {u}$, by

(3.18)\begin{equation} {\mathcal{F}}\{ \check{u}\} \equiv \frac{1}{\sqrt{2 {\rm \pi}}} \int_{-\infty}^{\infty} \check{u}(n, x, \eta, t) {\mathrm{e} }^{{\mathrm{i}} \omega t} \,{\mathrm{d}} t \equiv \hat{\check{u}}(n, x, \eta, \omega), \end{equation}

and using the boundary conditions (3.16) and the matching conditions (3.17), $\hat {\check {u}}$ and $\hat {\check {w}}$ are obtained, respectively, as follows:

(3.19a)\begin{gather} \hat{\check{u}} ={-} \hat{v}_x^{\prime} \exp({- (\sigma/\nu_e)^{1/2} n}), \end{gather}

(3.19b)\begin{gather}\hat{\check{w}} ={-} \hat{v}_\theta^{\prime} \exp({- (\sigma/\nu_e)^{1/2} n }), \end{gather}

with $\sigma = -{\mathrm {i}} \omega$, where the real part of $\sigma ^{1/2}$ is taken positive. Using (3.19a) in (3.11), $\hat {\check {T}}$ is solved as

(3.20)\begin{equation} \hat{\check{T}} \!=\!{-} \hat{T}^{\prime} \exp({- (\sigma/\kappa_e)^{1/2} n }) -\! \frac{{Pr}}{1\!-\!{Pr}} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \left[ \exp({- (\sigma/\nu_e)^{1/2} n }) \!-\! \exp({- (\sigma/\kappa_e)^{1/2} n }) \right] \sigma^{{-}1} \hat{v}_x^{\prime}. \end{equation}

Detailed structures of the defects in the boundary layer are obtained by effecting the inverse Fourier transforms, but they are less interesting now. Rather it is of importance to have the shear stress and the heat flux on the duct wall.

The shear stress acting on the gas is decomposed into components $s_x$ and $s_\eta$ in the $x$- and $\eta$-directions, respectively, as follows:

(3.21a)\begin{gather} s_x ={-} \mu_e \left.\frac{\partial \tilde{u}}{\partial n}\right|_{n=0} ={-} \mu_e \left.\frac{\partial \check{u}}{\partial n}\right|_{n=0}, \end{gather}

(3.21b)\begin{gather}s_\eta ={-} \mu_e \left.\frac{\partial \tilde{w}}{\partial n}\right|_{n=0} ={-} \mu_e \left.\frac{\partial \check{w}}{\partial n}\right|_{n=0}, \end{gather}

where $\partial v_x^{\prime }/\partial n$ and $\partial v_\theta ^{\prime } /\partial n$ vanish because of (2.3) in the $x$- and $\theta$-directions, and $\partial p^{\prime } /\partial r = 0$ at $r = R$. Using (3.19a) and (3.19b), $\hat {s}_x$ and $\hat {s}_\eta$ transformed are given, respectively, as follows:

(3.22a)\begin{gather} \hat{s}_x ={-}\rho_e ({\nu_e} \sigma)^{1/2} \hat{v}_x, \end{gather}

(3.22b)\begin{gather}\hat{s}_\eta ={-}\rho_e ({\nu_e} \sigma)^{1/2} \hat{v}_\theta. \end{gather}

Here, use is made of the formulae of the inverse Fourier transforms of $\sigma ^{\pm 1/2}\hat {f}(\sigma )$ by

(3.23)\begin{equation} {\mathcal{F}}^{{-}1} \left\{ \sigma^{{\pm} 1/2} \hat{f} (\sigma) \right\} = \frac{ {\mathrm{d}}^{{\pm} 1/2} f }{ {\mathrm{d}} t^{{\pm} 1/2} }, \end{equation}

where the right-hand side represents fractional derivatives of plus and minus half-order of a function $f(t)$ defined by

(3.24)\begin{equation} \frac{ {\mathrm{d}}^{{\pm} 1/2}\!f}{ {\mathrm{d}} t^{{\pm} 1/2} } \equiv \frac{1}{\sqrt{\rm \pi}} \int_{-\infty}^{t} \frac{1}{\sqrt{t-\tau}} \frac{{\mathrm{d}}^{1/2 \pm 1/2} f(\tau)} {{\mathrm{d}} \tau^{1/2 \pm 1 /2} } {\mathrm{d}} \tau, \end{equation}

with the sign $\pm$ vertically ordered (Sugimoto Reference Sugimoto1989, Reference Sugimoto2017). It is obvious from (3.23) that the addition law, e.g. ${\mathrm {d}}^{1/2}f/{\mathrm {d}} t^{1/2} = ({\mathrm {d}}^{-1/2}/{\mathrm {d}} t^{-1/2})\, {\mathrm {d}} f/{\mathrm {d}} t$ holds. Thanks to the formulae, $s_x$ and $s_\theta$ are given immediately as follows:

(3.25a)\begin{gather} s_x ={-}\rho_e \sqrt{\nu_e} \frac{\partial^{1/2} v_x^{\prime}}{\partial t^{1/2}} = \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left( \frac{\partial p^{\prime}}{\partial x}\right), \end{gather}

(3.25b)\begin{gather}s_\eta ={-}\rho_e \sqrt{\nu_e} \frac{\partial^{1/2} v_\theta^{\prime}}{\partial t^{1/2}} = \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left( \frac{1}{R} \frac{\partial p^{\prime}}{\partial \theta} \right), \end{gather}

where (2.3) at $r = R$ has been used.

Next, the heat flux $q$ from the duct wall into the gas is considered. This is given in the transformed form as

(3.26)\begin{equation} \hat{q} = \left. - k_e \frac{\partial \hat{\check{T}}}{\partial n} \right|_{n=0}, \end{equation}

where $\partial \hat {T}^{\prime }/\partial n$ vanishes because $T^{\prime }$ is proportional to $p^{\prime }$. Substituting (3.20) into (3.26), $\hat {q}$ is obtained as

(3.27)\begin{equation} \hat{q} ={-} \rho_e c_p T_e \sqrt{\nu_e} \sigma^{{-}1/2} \left( \frac{1}{\sqrt{{Pr}}}\frac{\sigma \hat{T}^{\prime}}{T_e} + \frac{1}{1 + \sqrt{{Pr}}}\frac{1}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \hat{v}_x^{\prime} \right), \end{equation}

where the quantities in the parentheses are evaluated at $r=R$. Eliminating $\hat {T}^{\prime }$ by using (2.8) transformed, and applying the formulae (3.23), $q$ is expressed in terms of $\boldsymbol {v}^{\prime }$ at $r = R$ as

(3.28)\begin{equation} q = \rho_e c_p T_e \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left[ \frac{\gamma-1}{\sqrt{{Pr}}} \left( \frac{\partial {v}_r^{\prime}}{\partial r} + \frac{1}{R} \frac{\partial {v}_\theta^{\prime}}{\partial \theta} + \frac{\partial {v}_x^{\prime}}{\partial x} \right) + \frac{1}{\sqrt{{Pr}}+{Pr}}\frac{1}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} {v}_x^{\prime} \right], \end{equation}

where $v_r^{\prime }/r$ in $\boldsymbol {\nabla }\boldsymbol {\cdot}\boldsymbol {v}^{\prime }$ vanishes at $r = R$. It is estimated from this that $q/\rho _e c_p T_e$ is of order $\delta v_x^{\prime }/L$ or $\delta v_\theta ^{\prime }/R$. The relation (3.28) is the generalisation to the non-planar case (see (5.19) in Sugimoto Reference Sugimoto2010). For later use, $\partial q/\partial t$ is calculated. Using (2.7) with $\rho _e a_e^{2} = (\gamma - 1)\rho _e c_p T_e$, and (2.3), it is expressed in terms of $p^{\prime }$ as

(3.29)\begin{equation} \frac{\partial q}{\partial t} ={-} \frac{\sqrt{\nu_e}}{\gamma - 1} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left( \frac{\gamma - 1}{\sqrt{{Pr}}} \frac{\partial^{2} p^{\prime}}{\partial t ^{2}} + \frac{1}{\sqrt{{Pr}} + {Pr}} \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial p^{\prime}}{\partial x} \right). \end{equation}

In case no temperature gradient is present, (3.29) gives simply

(3.30)\begin{equation} q = \sqrt{\kappa_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left( - \frac{\partial p^{\prime}}{\partial t} \right). \end{equation}

This forms a canonical pair with the shear stress (3.25). The thermoviscous diffusions appear in the form of the memory integral. The shear stress is expressed in terms of the minus half-order derivative of the spatial gradient of the pressure, and the heat flux in terms of that of the temporal gradient of the pressure with sign reversed, with difference of the diffusivities in the coefficients $\sqrt {\nu _e}$ and $\sqrt {\kappa _e}$. When the temperature gradient is present, unfortunately, such a canonical relation is destroyed. The shear stress is determined by the spatial pressure gradient only and independently of the temperature gradient, whereas the heat flux depends on the temperature gradient. In the rate of the heat flux (3.29), the second term in the parentheses is associated with the shear stress by (3.25a). Thus the shear stress influences the heat flux when the temperature gradient is present, but not vice versa.

3.3. Radial velocity at the edge of the boundary layer

Although the defects decay as the edge of the boundary layer is approached, only the defect of the radial velocity remains there. This is found by integrating (3.13) over $n$. Designating $\check {v}$ at the edge by $\check {v}_b$, it follows from (3.13) with (3.11) that

(3.31)\begin{equation} \check{v}_b = \int_0^{\infty} \frac{\partial \check{v}}{\partial n} {\mathrm{d}} n ={-} \int_0^{\infty} \left( \frac{\partial \check{u}}{\partial x} + \frac{\partial \check{w}}{\partial \eta} \right) {\mathrm{d}} n - \left.\frac{\kappa_e}{T_e} \frac{\partial \check{T}}{\partial n} \right|_{n=0}, \end{equation}

where $\check {v} = 0$ at $n = 0$ because $v_r^{\prime }$ vanishes on the wall. This velocity at the edge affects directly the propagation in the core region, while the shear stress and the heat flux on the wall influence it indirectly through the boundary layer. There is a relation among them.

Integrating (3.10a) and (3.10b) over the thickness to note that

(3.32a)\begin{gather} \rho_e \int_0^{\infty} \frac{\partial \check{u}}{\partial t} {\mathrm{d}} n = {\mu_e} \int_0^{\infty} \frac{\partial^{2} \check{u}}{\partial n ^{2}} {\mathrm{d}} n ={s_x}, \end{gather}

(3.32b)\begin{gather}\rho_e \int_0^{\infty} \frac{\partial \check{w}}{\partial t} {\mathrm{d}} n = {\mu_e} \int_0^{\infty} \frac{\partial^{2} \check{w}}{\partial n ^{2}} {\mathrm{d}} n = {s_\eta}, \end{gather}

$\partial \check {v}_b /\partial t$ is expressed in terms of the shear stress and the heat flux as

(3.33)\begin{equation} \frac{\partial \check{v}_b}{\partial t} ={-} \frac{\partial \ }{\partial x} \left( \frac{s_x}{\rho_e} \right) - \frac{\partial \ }{\partial \eta} \left( \frac{s_\eta}{\rho_e} \right) + \frac{1}{\rho_e c_p T_e} \frac{\partial q}{\partial t}. \end{equation}

Since $\rho _e a_e^{2}$ is the constant, this is written alternatively as

(3.34)\begin{equation} \rho_e a_e^{2} \frac{\partial \check{v}_b}{\partial t} ={-} \frac{\partial \ }{\partial x} (a_e^{2} s_x ) - \frac{\partial \ }{\partial \eta}(a_e^{2} s_\eta) + \frac{a_e^{2}}{c_p T_e} \frac{\partial q}{\partial t}, \end{equation}

where $a_e^{2}/c_p T_e = \gamma - 1$. This relates $v_b$ to the shear stress and the heat flux on the wall. It is found that the circumferential shear stress is added simply to the planar case. It is interesting to note in (3.33) that, if the shear stress is relatively small, i.e. ${Pr} \ll 1$, $v_b$ is proportional to $q$, which means that the heat flux is linked with $v_b$.

Equation (3.33) is expressed in terms of $\boldsymbol {v}^{\prime }$ or $p^{\prime }$ at $r = R$. Substitution of (3.25) and (3.28) into (3.33) yields

(3.35)\begin{equation} \frac{\partial \check{v}_b}{\partial t} = \sqrt{\nu_e} \frac{\partial^{1/2} \ }{\partial t^{1/2}} \left[ C \left( \frac{\partial v_r^{\prime}}{\partial r} + \frac{1}{R} \frac{\partial v_\theta^{\prime}}{\partial \theta} + \frac{\partial v_x^{\prime}}{\partial x} \right) - \frac{\partial v_r^{\prime}}{\partial r} + \frac{C_T}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} v_x^{\prime} \right], \end{equation}

where use has been made of the relation $\sqrt {\nu _e}^{\,-1} \,{\mathrm {d}} \sqrt {\nu _e}/{\mathrm {d}}\kern0.06em x = [(1+\beta )/2] T_e^{-1} \,{\mathrm {d}} T_e/{\mathrm {d}}\kern0.06em x$, and $C$ and $C_T$ are constants defined, respectively, by

(3.36a,b)\begin{equation} C = 1 + \frac{\gamma-1}{\sqrt{{Pr}}} \quad \mathrm{and} \quad C_T = \frac{1+\beta}{2} + \frac{1}{\sqrt{{Pr}} + {Pr}}. \end{equation}

In planar sound, $v_r^{\prime }$ and $v_\theta ^{\prime }$ vanish over the core region. Setting $v_x^{\prime }$ to be $u^{\prime }$, (3.35) is reduced to (1.1) after integration with respect to $t$ (see (5.17) in Sugimoto Reference Sugimoto2010). By (3.35), the magnitude of $\check {v}_b$ is estimated. Noting $({\mathrm {d}}^{\pm 1/2}/{\mathrm {d}} t^{\pm 1/2})\,{\mathrm {e} }^{{\mathrm {i}} \omega t} = ({\mathrm {i}} \omega )^{\pm 1/2} \,{\mathrm {e} }^{{\mathrm {i}} \omega t}$ for a time-harmonic disturbance, it is confirmed that $\check {v}_b$ is of $\delta v_r^{\prime } /R$, $\delta v_\theta ^{\prime } /R$ and $\delta v_x^{\prime }/L$. This is comparable to $v_r^{\prime }$ at the edge, as estimated below (3.8).

The relation (3.35) is expressed in terms of $p^{\prime }$. Using (2.7) and (2.3) at $r = R$, (3.35) multiplied with $\rho _e a_e^{2}$ is expressed in terms of $p^{\prime }$ at $r = R$ as

(3.37)\begin{equation} \rho_e a_e^{2} \frac{\partial \check{v}_b}{\partial t} ={-} \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left[(C-1) \frac{\partial^{2} p^{\prime}}{\partial t ^{2}} + \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial p^{\prime}}{\partial x} \right) + \frac{a_e^{2}}{R^{2}} \frac{\partial^{2} p^{\prime}}{\partial \theta ^{2}} + C_T \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial p^{\prime}}{\partial x} \right], \end{equation}

where (2.9) at $r = R$ has been used. Here, it is remarked that the velocity and the pressure on the right-hand sides of (3.35) and (3.37), respectively, are those in the core region substantially at $r = R$. This is different from the case of the planar sound where they are uniform in a cross-section of the core region. For a time-harmonic disturbance in a uniform gas, (3.37) provides the acoustic impedance at the edge of the boundary layer on a cylindrical rigid wall.

3.4. Matching between the core region and the boundary layer

So far the analysis has been made separately for the core region and the boundary layer. To incorporate the effects due to the boundary layer into the core region, both regions are to be matched. This is done through the equation of continuity. To this end, the tilde used so far to indicate the disturbances in the boundary layer is extended to designate the disturbances in the whole domain. In the core region, the defects are taken to vanish and the variables with the tilde are identified with the primed ones.

The Lagrangian derivative of the density is expressed, without any approximations, in terms of those of the pressure and of the temperature as $\rho ^{-1} \,{\mathrm {D}} \rho /{\mathrm {D}} t = p^{-1} \,{\mathrm {D}} p/{\mathrm {D}} t - T^{-1} \,{\mathrm {D}} T/{\mathrm {D}} t$ by using the equation of state $p = {\mathcal {R}}\rho T$, where ${\mathrm {D}}/{\mathrm {D}} t = \partial /\partial t + \boldsymbol {v}\boldsymbol {\cdot}\boldsymbol {\nabla }$. Using the linearised version of this, the equation of continuity is written as

(3.38)\begin{equation} \frac{1}{p_0} \frac{\partial \tilde{p}}{\partial t} - \frac{1}{T_e} \left( \frac{\partial \tilde{T}}{\partial t} + \tilde{v}_x \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\tilde{\boldsymbol{v}} = 0. \end{equation}

In the core region, the second term on the left-hand side is expressed in terms of $p^{\prime }$ by (2.4) for the adiabatic process as

(3.39)\begin{equation} \frac{1}{T_e} \left( \frac{\partial \tilde{T}}{\partial t} + \tilde{v}_x \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) = \frac{1}{\rho_e c_p T_e} \frac{\partial p^{\prime}}{\partial t}, \end{equation}

where use of $p^{\prime }$ is retained. In the boundary layer, on the other hand, it is given by (3.5) as

(3.40)\begin{equation} \frac{1}{T_e} \left( \frac{\partial \tilde{T}}{\partial t} + \tilde{v}_x \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \right) = \frac{1}{\rho_e c_p T_e} \frac{\partial p^{\prime}}{\partial t} + \frac{k_e}{\rho_e c_p T_e} \frac{\partial^{2} \tilde{T}}{\partial n ^{2}} , \end{equation}

and the right-hand side consists of the adiabatic part as in the core region and the heat conduction in the boundary layer.

Each variable with a tilde in (3.38) is written as the sum of the one with the prime and the defect, i.e. $\widetilde {(\,\cdot\, )}= (\,\cdot\, )^{\prime } +\check {(\,\cdot\, )}$ except for the pressure disturbance ($\check {p} = 0$ so that $\tilde {p} = p^{\prime }$ everywhere). Replacing the variables in (3.38) with the sums, and differentiating it with respect to $t$, it follows that

(3.41)\begin{equation} \frac{1}{\gamma p_0} \frac{\partial^{2} p^{\prime}}{\partial t ^{2}} - \boldsymbol{\nabla}\boldsymbol{\cdot} \left( \frac{1}{\rho_e} \boldsymbol{\nabla} p^{\prime} \right) - \frac{k_e}{\rho_e c_p T_e} \frac{\partial \ }{\partial t} \left( \frac{\partial^{2} \check{T}}{\partial n ^{2}} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot} \frac{\partial \check{\boldsymbol{v}}}{\partial t} = 0, \end{equation}

where the second term stems from $\boldsymbol {\nabla }\boldsymbol {\cdot}(\partial \boldsymbol {v}^{\prime } /\partial t)$ by use of (2.3). Note that the thermal diffusion is ignored in the core region because the acoustic Reynolds number is large. The last two terms reflect the effects of the boundary layer and vanish in the core region.

The disturbances have so far been assumed to be of arbitrary form. In view of the periodicity in $\theta$, it is pertinent to consider a Fourier component proportional to ${\mathrm {e} }^{{\mathrm {i}} m \theta }$ $(m=1,2,3,\dots )$. The disturbances in the core region are set in the form of

(3.42)\begin{equation} (v_r^{\prime}, v_\theta^{\prime}, v_x^{\prime}, p^{\prime}, T^{\prime}) = (V_m, {\mathrm{i}} W_m, U_m, P_m, \varTheta_m) {\mathrm{e} }^{{\mathrm{i}} m \theta}, \end{equation}

where $V_m$, $W_m$, $U_m$, $P_m$ and $\varTheta _m$ depend on $r$, $x$ and $t$. In accordance with this, the defects are also set in the form of

(3.43)\begin{equation} (\check{v}, \check{w}, \check{u}, \check{T}) = (\check{V}_m, {\mathrm{i}} \check{W}_m, \check{U}_m, \check{\varTheta}_m) \exp({{\mathrm{i}} m \eta/R}), \end{equation}

with $\eta /R = \theta$ where $\check {V}_m$, $\check {W}_m$, $\check {U}_m$ and $\check {\varTheta }_m$ depend on $n$, $x$ and $t$.

Substituting (2.21) into (3.41), the mean of (3.41) in the sense of (2.22) is taken for $j = 1, 2, 3, \dots$. The means of the terms associated with $p^{\prime }$ are easily available and given by (2.25) divided by $\rho _e a_e^{2}$, while the means of the defects need to be noted. Since the defects except for $\check {v}$ tend to vanish exponentially as $n$ increases, the integral is taken substantially over the thin boundary layer only. Because $r\,{\rm J}_m(\alpha _{m,j}r/R)$ varies little in the boundary layer, it may be set approximately to $R\,{\rm J}_m(\alpha _{m,j})$. Then, the mean of the defect is evaluated by

(3.44)\begin{equation} \frac{1}{R^{2}} \int_0^{R} \check{(\,\cdot\,)} r \,{\rm J}_m (\alpha_{m,j}r/R) \,{\mathrm{d}} r \approx\frac{{\rm J}_m (\alpha_{m,j} )}{R} \int_0^{\infty} \check{(\,\cdot\,)} \,{\mathrm{d}} n. \end{equation}

In passing, errors due to the truncation may be of concern. The function $r \,{\rm J}_m(\alpha _{m,j}r/R)$ is expanded around $r=R$ in terms of the power series in $n\ ( = R-r)$. Since the defect decays exponentially in the form of ${\mathrm {e} }^{- n/\varepsilon }$ for $|\varepsilon | \ll 1$, $\varepsilon$ implying the ratio $(\nu _e/\sigma )^{1/2} /R$ (see, e.g. (3.19)), the integral of $n^{N} {\mathrm {e} }^{-n/\varepsilon }$ decays as $\varepsilon ^{N+1} \Gamma (N+1)$ for $N = 0, 1, 2, \dots$, $\Gamma$ being the gamma function. Thus the leading error (for which $N = 1$) is found to be quadratic in $\varepsilon$, i.e. $(\delta /R)^{2}$. Since this is neglected in the present analysis, the truncation in (3.44) is justified.

Using (3.44) to evaluate the mean of the defects, the third term in (3.41) is evaluated by using (3.26) and setting $q = Q_m {\mathrm {e} }^{{\mathrm {i}} m \eta /R}$ to be

(3.45)\begin{equation} - \frac{k_e}{\rho_e c_p T_e R^{2}} \int_0^{R} \frac{\partial \ }{\partial t}\left( \frac{\partial^{2} \check{T}}{\partial n ^{2}} \right) r \,{\rm J}_m(\alpha_{m,j} r/R) \,{\mathrm{d}} r \approx{-} \frac{{\rm J}_m(\alpha_{m,j})}{\rho_e c_p T_e R} \frac{\partial Q_m}{\partial t}, \end{equation}

where $\partial Q_m/\partial t$ is written by using (3.29) in terms of $P_m$ at $r = R$ as

(3.46)\begin{equation} \frac{\partial Q_m}{\partial t} ={-}\frac{\sqrt{\nu_e}}{\gamma - 1} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left( \frac{\gamma - 1}{\sqrt{{Pr}}} \frac{\partial^{2} P_m}{\partial t ^{2}} + \frac{1}{\sqrt{{Pr}}+{Pr}} \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial P_m}{\partial x} \right). \end{equation}

On the other hand, the mean of the defect velocity on the last term is calculated term-by-term as follows:

(3.47a)\begin{gather} \frac{1}{R^{2}} \int_0^{R} \frac{1}{r} \frac{\partial \ }{\partial r} \left( r \frac{\partial \check{V}_m}{\partial t} \right) r \,{\rm J}_m (\alpha_{m,j} r/R) \,{\mathrm{d}} r ={-} \frac{ \alpha_{m,j} }{R^{3}} \int_0^{R} \frac{\partial \check{V}_m}{\partial t} r\, \dot{{\rm J}}_m (\alpha_{m,j} r/R)\,{\mathrm{d}} r\nonumber\\ \approx{-} \frac{\alpha_{m,j} \dot{{\rm J}}_m(\alpha_{m,j})}{R^{2}} \int_0^{\infty} \frac{\partial \check{V}_m}{\partial t} {\mathrm{d}} n = 0, \end{gather}

(3.47b)\begin{gather} \frac{1}{R^{2}} \int_0^{R} \left( - \frac{m}{r} \frac{\partial \check{W}_m}{\partial t} \right) r\, {\rm J}_m (\alpha_{m,j} r/R) \,{\mathrm{d}} r \approx \frac{{\rm J}_m (\alpha_{m,j})}{R} \int_0^{\infty} \frac{\partial \ }{\partial t} \left(- \frac{m}{R} \check{W}_m \right) {\mathrm{d}} n, \end{gather}

(3.47c)\begin{gather} \frac{1}{R^{2}} \int_0^{R} \frac{\partial \ }{\partial x} \left( \frac{\partial \check{U}_m}{\partial t} \right) r\, {\rm J}_m (\alpha_{m,j} r/R) \,{\mathrm{d}} r \approx \frac{{\rm J}_m (\alpha_{m,j})}{R} \int_0^{\infty} \frac{\partial \ }{\partial t} \left( \frac{\partial \check{U}_m}{\partial x} \right) {\mathrm{d}} n, \end{gather}

where the boundary condition $\check {V}_m = 0$ at $r=R$ and $\dot {{\rm J}}_m(\alpha _{m,j}) = 0$ have been used in (3.47a). However, (3.47a) is subtle because $\partial \check {V}_m /\partial t$ does not decay exponentially but remains as $n \to \infty$, where $\dot {{\rm J}}_m(\alpha _{m,j} r/R)$ tends to deviate from zero. It is justified rather because $\check {V}_m$ is smaller than $\check {W}_m$ and $\check {U}_m$ by $\delta /R$ and $\delta /L$, and the product with $\dot {{\rm J}}_m(\alpha _{m,j}r/R)$ of $\delta /R$ makes the term small, of order $(\delta /R)^{2}$.

Hence, the means of the last two terms in (3.41) are written in view of (3.31) as

(3.48)\begin{equation} \frac{1}{R^{2}} \int_0^{R} \left[ - \frac{k_e}{\rho_e c_p T_e} \frac{\partial \ }{\partial t} \left( \frac{\partial^{2} \check{T}}{\partial n ^{2}} \right) + \boldsymbol{\nabla}\boldsymbol{\cdot}\frac{\partial \check{\boldsymbol{v}}}{\partial t} \right] r\, {\rm J}_m (\alpha_{m,j}r/R) \,{\mathrm{d}} r ={-} \frac{{\rm J}_m(\alpha_{m,j})}{R} \frac{\partial \check{V}_{bm}}{\partial t}, \end{equation}

with $\check {v}_b = \check {V}_{bm} {\mathrm {e} }^{{\mathrm {i}} m \theta }$ where $\partial V_{bm}/\partial t$ is given by

(3.49)\begin{equation} \frac{\partial \check{V}_{bm}}{\partial t} = \frac{1}{\rho_e c_p T_e} \frac{\partial Q_m}{\partial t} - \frac{\partial \ }{\partial t} \int_0^{\infty} \left( - \frac{m}{R} {\check{W}_m} + \frac{\partial \check{U}_m}{\partial x} \right) {\mathrm{d}} n. \end{equation}

Summing up the mean of each term in (3.41) leads to the one-dimensional wave equations for $\bar {P}_{m,j}$ taking account of the thermoviscous diffusions as

(3.50)\begin{equation} \frac{\partial^{2} \bar{P}_{m,j}}{\partial t ^{2}} - \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial \bar{P}_{m,j}}{\partial x} \right) + \left( \frac{\alpha_{m,j}}{R} \right)^{2} a_e^{2} \bar{P}_{m,j} = \frac{{\rm J}_m(\alpha_{m,j})\,}{R} \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t}, \end{equation}

for $j = 1, 2, 3, \dots$ where $\rho _e a_e^{2} \partial \check {V}_{bm}/\partial t$ is expressed by (3.37) in terms of $P_m$ at $r = R$ as

(3.51)\begin{align} & \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t}\nonumber\\ & \quad ={-} \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left[ (C-1) \frac{\partial^{2} P_m}{\partial t ^{2}} + \frac{\partial }{\partial x} \left( a_e^{2} \frac{\partial P_m}{\partial x} \right) - \frac{m^{2}}{R^{2}} a_e^{2} P_m + C_T \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial P_m}{\partial x} \right]. \end{align}

Note again that the diffusive terms on the right-hand side of (3.50) are small, of order $\delta /R$, in comparison with the non-diffusive terms on the left-hand side.

4.1. Diffusive wave equation for a single mode

Equations (3.50) with (3.51) are the diffusive wave equations for the weighted means $\bar {P}_{m,j}$. Since (3.51) is determined by $P_m$ at $r = R$, which is related to all $\bar {P}_{m,j}$ by (2.21) as

(4.1)\begin{equation} P_m(R,x,t) = \sum_{j=1}^{\infty} c_{m,j} \bar{P}_{m,j} {\rm J}_m (\alpha_{m,j}), \end{equation}

(3.50) are coupled with all $\bar {P}_{m,j}$ through the boundary layer. Hence, (3.50) provide a system of an infinite number of equations for $\bar {P}_{m,j}$ $(j=1,2,3,\dots )$. In this sense, the independence of each mode in the non-diffusive case is lost.

Because (4.1) is difficult to solve, of concern here is a case in which $P_m$ consists of a single mode only, say the $j$th mode, and the other modes vanish as

(4.2)\begin{equation} P_m(r, x, t) = c_{m,j}\,f(x,t){\rm J}_m(\alpha_{m,j}r/R), \end{equation}

where $\bar {P}_{m,j}$ is set simply to be $f$ without the bar and the subscripts. Then it follows from (3.50) that

(4.3)\begin{equation} \frac{\partial^{2} f}{\partial t ^{2}} - \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial f}{\partial x} \right) + \left( \frac{\alpha_{m,j}}{R} \right)^{2} a_e^{2} f = \frac{{\rm J}_m(\alpha_{m,j}) }{R} \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t}, \end{equation}

with

(4.4)\begin{align} & \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t}\nonumber\\ &\quad ={-} c_{m,j} {\rm J}_m (\alpha_{m,j}) \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left[ (C-1) \frac{\partial^{2} f}{\partial t ^{2}} + \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial f}{\partial x} \right) - \frac{m^{2}}{R^{2}} a_e^{2} f + C_T \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial f}{\partial x} \right], \end{align}

where $P_m(R,x,t) = c_{m,j} {\rm J}_m(\alpha _{m,j}) f$. Using the lowest equation of (4.3) with the right-hand side neglected, (4.4) may alternatively be expressed as

(4.5)\begin{equation} \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t} ={-} c_{m,j} {\rm J}_m(\alpha_{m,j}) \sqrt{\nu_e} \frac{\partial^{{-}1/2} \ }{\partial t^{{-}1/2} } \left[ C \frac{\partial^{2} f}{\partial t ^{2}} + \frac{(\alpha_{m,j}^{2} - m^{2})}{R^{2}} a_e^{2} f + C_T \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{\partial f}{\partial x} \right]. \end{equation}

For the planar sound, it is easily verified that (4.3) with (4.4) or (4.5) is reduced to (1.2) by setting $m =0$ and then $\alpha _{m,j} = 0$ so that $c_{m,j} {\rm J}^{2}_m(\alpha _{m,j})/R$ becomes $2/R$, and using the lowest relation $\partial ^{2} f/\partial t^{2} - (\partial /\partial x)(a_e^{2} \partial f/\partial x) = 0$ and ${\mathrm {d}} a_e^{2}/{\mathrm {d}}\kern0.06em x = a_e^{2} T_e^{-1} \,{\mathrm {d}} T_e/{\mathrm {d}}\kern0.06em x$.

4.2. Damping due to thermoviscous diffusions in a uniform gas

To examine the diffusive effects in a uniform gas, a dispersion relation of (4.3) with (4.5) is sought. Assuming $f = X \exp ({{\mathrm {i}}( \omega t - kx)})$, $X$ being a complex amplitude, $\omega$ and $k$ must satisfy the following dispersion relation for $X$ to be non-trivial:

(4.6)\begin{equation} \varOmega^{2} = \alpha_{m,j}^{2} + K^{2} - \varepsilon ({\mathrm{i}} \varOmega)^{{-}1/2} \left[ (C-1) \varOmega^{2} + K^{2} + m^{2} \right], \end{equation}

with $\varepsilon$ defined by

(4.7)\begin{equation} \varepsilon \equiv c_{m,j} {\rm J}_m^{2} (\alpha_{m,j}) \sqrt{\frac{\nu_0}{a_0 R}} = \frac{2 \alpha_{m,j}^{2}}{(\alpha_{m,j}^{2} - m^{2})}\sqrt{\frac{\nu_0}{a_0 R}}, \end{equation}

$\varOmega = \omega R/a_0$ and $K = kR$, where the real parts of $\omega$ and $k$ are taken positive, and ${\mathrm {i}}^{-1/2}$ is chosen to be $(1-{\mathrm {i}})/\sqrt {2}$. Here, the factor $\sqrt {\nu _0/a_0 R}$ takes a very small value because this measures the ratio of the boundary-layer thickness $\sqrt {\nu _0/\omega }$ to $R$, if $\omega$ were taken to be $a_0/R$, i.e. $\varOmega = 1$. In an air-filled duct of radius $R = 0.5$ m, for example, $\sqrt {\nu _0/a_0 R} \approx 2.92 \times 10^{-4}$ for $a_0 = 340\ {\rm m}\ {\rm s}^{-1}$ and $\nu _0 = 1.45 \times 10^{-5}\ {\rm m}^{2}\ {\rm s}^{-1}$. Note, however, that, as $m$ becomes large, the coefficient $2 \alpha ^{2}_{m,j}/(\alpha ^{2}_{m,j} - m^{2})$ increases as $m^{2/3}/\chi$ in the radial mode $j = 1$.

Equation (4.6) is difficult to solve for $\varOmega$ as it is. However, making use of the smallness of $\varepsilon$, $\varOmega$ is sought asymptotically to its first order. To the lowest order, $\varOmega$ is given by $(\alpha _{m,j}^{2} + K^{2})^{1/2}$, denoted by $\varOmega _K$. Approximating $\varOmega$ in the last term of (4.6) by $\varOmega _K$, $\varOmega$ is obtained as $\varOmega \approx \varOmega _K - \varepsilon (1 - {\mathrm {i}}) \sqrt {{\varOmega _K}/{8}} [ C - (\alpha _{m,j}^{2} - m^{2})/\varOmega _K^{2} ]$. Here, $C$ is greater than unity, e.g. $C = 1.47$ for air, and $(\alpha _{m,j}^{2} - m^{2})/\varOmega _K^{2} < 1$. Hence, the positive imaginary part suggests damping. Setting $\omega = \omega _r + {\mathrm {i}} \omega _i$, the decay rate $\omega _i$ in ${\mathrm {e} }^{-\omega _i t}$ is given in dimensionless form by

(4.8)\begin{equation} {\omega_i R}/{a_0} \equiv \varOmega_i \approx \varepsilon \sqrt{ {\varOmega_K}/{8}} \left[ C - {(\alpha_{m,j}^{2} - {m^{2}})}/{\varOmega_K^{2}} \right]. \end{equation}

For $K \gg \alpha _{m,j}$, $\varOmega _K \approx K$ so that $\varOmega _i \approx \varepsilon C \sqrt {K/8}$. In passing, the decay rate of the planar sound is given by $\varepsilon _0 C \sqrt {K/8}$, $\varepsilon _0 = 2 \sqrt {\nu _0/a_0 R}\ ({\ll }\varepsilon )$, and is much smaller than (4.8). From the real part of $\varOmega$, on the other hand, $\varOmega - \varOmega _K \approx - \varOmega _i$. Hence, it is found that the cutoff frequency as $K \to 0$ lowers slightly from $\alpha _{m,j}$ by an amount equal to $\varOmega _i$.

Given a frequency above the cutoff, on the other hand, a spatial decay rate is sought. From (4.6), $K$ is obtained as $K \approx {K}_\varOmega + \varepsilon (1 - {\mathrm {i}}) ( C \varOmega ^{2} - \alpha _{m,j}^{2} + m^{2})/ \sqrt {8 \varOmega } {K}_\varOmega$ with ${K}_\varOmega = (\varOmega ^{2} - \alpha _{m,j}^{2})^{1/2}$. Setting $k = k_r - {\mathrm {i}} k_i$, the spatial decay rate $k_i$ in ${\mathrm {e} }^{-k_i x}$ is given in dimensionless form by

(4.9)\begin{equation} k_i R \equiv {K}_i \approx \varepsilon \left( C\varOmega^{2} - \alpha_{m,j}^{2} + m^{2} \right)/\sqrt{8 \varOmega} {K}_\varOmega. \end{equation}

For $\varOmega \gg \alpha _{m,j}$, ${K}_\varOmega \approx \varOmega$ so that ${K}_i \approx \varepsilon C \sqrt {\varOmega / 8}$ and the spatial decay rate is proportional to $\sqrt {\varOmega }$. For the planar sound, ${K}_i$ is given by $\varepsilon _0 C \sqrt {\varOmega /8}$.

When $\varOmega$ is close to $\alpha _{m,j}$ such that $|\varOmega - \alpha _{m,j}| \ll O(\varepsilon )$, the approximation of $K$ is invalid as ${K}_\varOmega$ vanishes. It is then found from (4.6) that $K$ is approximated by $K \approx \sqrt {\varepsilon }\, {\mathrm {i}}^{-1/4} {b_{m,j}}$with $b_{m,j} = \{[(C-1) \alpha _{m,j}^{2} + m^{2}]/\sqrt {\alpha _{m,j}}\}^{1/2}$, and that the diffusive effects are enhanced to be of order $\sqrt {\varepsilon }$. Hence, the spatial decay rate ${K}_i$ is given by $\sqrt {\varepsilon } \sin ({\rm \pi} /8) {b_{m,j}}$. At the same time, the real part of $K$ is given by $\sqrt {\varepsilon } \cos ({\rm \pi} /8) b_{m,j}$. In the non-diffusive case, the phase velocity becomes infinite at the cutoff, whereas the diffusive effects suppress it to be large but finite.

4.3. Acoustic energy equation

When the temperature gradient is present, it is not so easy to have such analytical solutions and decay rates as obtained in the preceding subsection. So a different strategy is taken to consider the acoustic energy.

Setting $f = \partial \varPhi /\partial t$ and integrating (4.3) with respect to $t$, it follows that

(4.10)\begin{equation} \frac{1}{\rho_e a_e^{2}} \frac{\partial^{2} \varPhi}{\partial t ^{2}} - \frac{\partial \ }{\partial x} \left( \frac{1}{\rho_e} \frac{\partial \varPhi}{\partial x} \right) + \left( \frac{\alpha_{m,j}}{R} \right)^{2} \frac{\varPhi}{\rho_e} = \frac{{\rm J}_m(\alpha_{m,j}) }{R} \check{V}_{bm}. \end{equation}

Multiplication with $\partial \varPhi /\partial t$ leads to

(4.11)\begin{equation} \frac{\partial E}{\partial t} + \frac{\partial {I}}{\partial x} = \frac{{\rm J}_m (\alpha_{m,j})}{R} \frac{\partial \varPhi}{\partial t} \check{V}_{bm}, \end{equation}

where $E$ and $I$ are equal to $E_{m,j}$ and ${I}_{m,j}$ given, respectively, by (2.35) and (2.37) with $\varPhi _{m,j} = \varPhi$, This corresponds to the acoustic energy equation (2.28) for the single mode, when $c_{m,j}$ is multiplied. Then $c_{m,j} {\rm J}_m(\alpha _{m,j}) \partial \varPhi /\partial t$ gives the pressure $P_m (R,x,t)$, and the product with $\check {V}_{bm}$ gives the power density done by the edge of the boundary layer on the core region. The velocity at the edge is given by the sum of $\check {V}_{bm}$ and $-V_m$ at $r = R - \delta$. It is obvious physically, however, that the latter does not contribute to this power density.

When the boundary layer is ignored, the right-hand side of (4.11) drops so that the conservation of the acoustic energy holds locally. To find the global conservation, a finite duct of length $L$ is considered in a domain $0 < x < L$. At a duct end, either the sound pressure or the axial velocity is usually taken to vanish, if no end effects are considered. In either case, then, $I$ vanishes at the ends. Integration of (4.11) over the duct leads to

(4.12)\begin{equation} \frac{{\rm{d}} \ }{{\rm{d}} t} \int_0^{L} E(x, t)\, {\mathrm {d}}\kern0.06em x = 0. \end{equation}

Thus the integral corresponding to the total energy is conserved at any instant.

When the boundary layer is taken into account, it is no longer conserved but is subjected to change by

(4.13)\begin{equation} \frac{{\rm{d}} \ }{{\rm{d}} t} \int_0^{L} E \,{\mathrm {d}}\kern0.06em x = \int_0^{L} \frac{{\rm J}_m (\alpha_{m,j})}{R} f \check{V}_{bm}\,{\mathrm {d}}\kern0.06em x. \end{equation}

As will be shown below, the right-hand side is proportional to the small parameter $\varepsilon$ given by (4.7), and therefore the integral of $E$ changes slowly in time. To see this in detail, consider oscillations in the form of $f = X(x) {\mathrm {e} }^{{\mathrm {i}} \omega t}$ and $\check {V}_{bm} = Y(x) {\mathrm {e} }^{{\mathrm {i}} \omega t}$, $X$ and $Y$ being complex amplitudes, and $\omega \ ( = \omega _r + {\mathrm {i}} \omega _i)$ being complex. Since, as seen in (4.8), $\omega _i$ is proportional to $\varepsilon$ and is much smaller than $\omega _r$, the oscillations are substantially sinusoidal over a period of oscillations $({\approx }2{\rm \pi} /\omega _r)$, and the integral of $E$ over the period may be regarded as a constant. However, this constant contains $\omega _i t$ and changes slowly over a long time scale of $1/\omega _i\ ({\gg }2{\rm \pi} /\omega _r)$. According to the method of two timings $\omega _r t$ and $\omega _i t$ (Gupta, Lodato & Scalo Reference Gupta, Lodato and Scalo2017), (4.13) is integrated over one period of fast oscillations to yield

(4.14)\begin{equation} \frac{{\rm{d}} \ }{{\rm{d}} t} \int_0^{L} {\langle E \rangle} \,{\mathrm {d}} x = \int_0^{L} \frac{{\rm J}_m(\alpha_{m,j})}{R} \langle{f \check{V}_{bm}}\rangle \,{\mathrm {d}} x, \end{equation}

where ${\langle \,\cdot\,\rangle}$ denotes the mean over the period defined by

(4.15)\begin{equation} {\langle \,\cdot\,\rangle} \equiv \frac{\omega_r}{2 {\rm \pi}} \int_t^{t+2{\rm \pi}/\omega_r} (\,\cdot\,)\, {\mathrm{d}} t. \end{equation}

Here, the integral of ${\langle E \rangle}$ depends on $\omega _i t$, whose slow change is determined by the mean of $f \check {V}_{bm}$.

This mean is given by

(4.16)\begin{equation}{\langle f \check{V}_{bm} \rangle} = \text{Re} \{ X^{*} Y \}/2, \end{equation}

the asterisk denoting the complex conjugate, and $Y$ is given in terms of $X$ by (4.5) as

(4.17)\begin{equation} {Y} ={-} \frac{c_{m,j} {\rm J}_m(\alpha_{m,j}) \sqrt{\nu_e} ({\mathrm{i}} \omega)^{1/2}}{\rho_e a_e^{2}} \left[ C X - \left( {\alpha_{m,j}^{2}-m^{2}} \right) \frac{a_e^{2}}{\omega^{2} R^{2}} X - \frac{C_T}{\omega^{2}} \frac{a_e^{2}}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} \frac{{\rm{d}} X}{{\rm{d}}\kern0.06em x} \right]. \end{equation}

Here, ${\mathrm {d}} X/{\mathrm {d}}\kern0.06em x$ is related to the axial velocity. In accordance with $P_m$ in (4.2), $U_m$ in (2.26) is also set in the form of $U_m = c_{m,j} g(x, t) {\rm J}_m(\alpha _{m,j}r/R)$. Then it follows from (2.27d) that

(4.18)\begin{equation} \rho_e \frac{\partial g}{\partial t} ={-} \frac{\partial f}{\partial x}. \end{equation}

Setting $g = Z(x) {\mathrm {e} }^{{\mathrm {i}} \omega t}$, $Z$ being a complex amplitude, $Z$ is related to $X$ by

(4.19)\begin{equation} Z = \frac{{\mathrm{i}}}{\rho_e \omega} \frac{{\rm{d}} X}{{\rm{d}}\kern0.06em x}. \end{equation}

Using this, the integrand in (4.14) is set in the form of

(4.20)\begin{equation} \frac{{\rm J}_m( \alpha_{m,j} )}{ R} \langle{f \check{V}_{bm}}\rangle = \frac{\varepsilon}{2 \rho_0 a_0 R} D , \end{equation}

with $D$ defined by

(4.21)\begin{equation} {D} ={-} \text{Re} \left\{ N_e ({\mathrm{i}} \varOmega)^{1/2} \left[ C |X|^{2} - \left( {\alpha_{m,j}^{2} - m^{2}} \right) \frac{T_e}{T_0} \frac{|X|^{2}}{\varOmega^{2}} + \frac{{\mathrm{i}}}{\varOmega} \frac{C_T}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} R {\rho_0 a_0 X^{*} Z} \right] \right\}, \end{equation}

where $N_e(x)$ designates $\sqrt {\nu _e/\nu _0} = (T_e/T_0)^{(1+\beta )/2}$, and $\text{Re}\{\,\cdot\, \}$ means the real part of $\{\,\cdot\, \}$.

For planar sound in the absence of the temperature gradient where $m = 0$ and $\alpha _{m,j} = 0$, ${D}$ is obviously negative and the acoustic energy is damped. For non-planar sound, as shown in § 4.2, the second term due to the dispersion reduces $C$ substantially, in common with planar sound. A decay rate in energy is defined by the ratio of the left-hand side of (4.14) to the right-hand side with the sign reversed. This is twice the decay rate $\omega _i$ (see below (5.7)).

When the temperature gradient is present, the second term may overcome $-C$, if $T_e/T_0$ is high enough, and the last term appears to be crucial. Noting that $X/\rho _0 a_0 Z$ is the dimensionless acoustic impedance, and setting its phase angle to be $\phi$, i.e. $X/Z = |X/Z| {\mathrm {e} }^{{\mathrm {i}} \phi }$, $X^{*} Z$ is written as $X^{*} Z = |X|^{-1}|Z| (\cos \phi -{\mathrm {i}} \sin \phi ) |X|^{2}$. Taking the factor $|X|^{2}$ out of the square brackets in (4.21), the sign of the last term in $D$ is determined, the positive factors aside, by

(4.22)\begin{equation} \text{Re} \left\{ ({\mathrm{i}} \varOmega)^{{-}1/2} \frac{R}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} (\cos \phi - {\mathrm{i}} \sin \phi) \right\} \approx (2 |\varOmega|)^{{-}1/2} \frac{R}{T_e} \frac{{\rm{d}} T_e}{{\rm{d}}\kern0.06em x} (\cos \phi - \sin \phi). \end{equation}

Because this may change sign, the last term contributes not only to the dissipation of the energy but also to the production of it, depending on the phase and on the sign of the temperature gradient.

6.1. Expansion in terms of the radial modes

It has already been seen that, as $m$ becomes large, the sound tends to be confined in a narrow band so that silence prevails axially in a cylindrical column about the centre. This means that, even if another duct of radius $R_i\ ({<}R)$ were placed coaxially within it, the sound field would be unaffected. As $R_i/R$ approaches unity, however, the inner duct will influence the sound field greatly. In this section, extension to the case of the annular duct is described briefly, as long as the boundary layer is much thinner than a gap.

In the annular duct, the boundary condition ${\mathrm {d}} P/{\mathrm {d}} r = 0$ is added at $r = R_i$. In place of (2.14), then, $P$ is sought in the form of

(6.1)\begin{equation} P = {\mathcal{B}}_1 {\rm J}_m(\beta r/R) + {\mathcal{B}}_2 {\rm Y}_m(\beta r/R), \end{equation}

with $\beta \equiv (\omega ^{2}/a_0^{2} - k^{2})^{1/2} R$ for $R_i < r < R$, where ${\mathcal {B}}_1$ and ${\mathcal {B}}_2$ are arbitrary constants, and ${\rm Y}_m$ denotes the Bessel function of the second kind. The constants are determined by the boundary conditions at $r = R$ and at $r = R_i$, which are given for non-zero $\beta$ as follows:

(6.2a)\begin{gather} {\mathcal{B}}_1 \dot{{\rm J}}_m (\beta) + {\mathcal{B}}_2 \dot{{\rm Y}}_m (\beta) = 0, \end{gather}

(6.2b)\begin{gather}{\mathcal{B}}_1 \dot{{\rm J}}_m (\beta c) + {\mathcal{B}}_2 \dot{{\rm Y}}_m (\beta c ) = 0, \end{gather}

with $c = R_{i}/R\ ({<} 1)$. For ${\mathcal {B}}_1$ and ${\mathcal {B}}_2$ to be non-trivial, the following determinant must vanish:

(6.3)\begin{equation} \dot{{\rm J}}_m(\beta) \dot{{\rm Y}}_m(\beta c) - \dot{{\rm J}}_m(\beta c) \dot{{\rm Y}}_m(\beta) = 0. \end{equation}

Taking the limit as $c \to 0$, $\dot {{\rm J}}_m (\beta ) = 0$ is recovered by dividing (6.3) with $\dot {{\rm Y}}_m(\beta c)~(\to -\infty )$.

Zeros of (6.3) are designated as $\beta _{m,j}$ $(j = 1, 2, 3, \dots )$ and ordered as $0 < \beta _{m,1} < \beta _{m,2} < \beta _{m,3} < \dots$. Here, specific values of $\beta _{m,j}$ are not given but are available for $c = 0.25, 0.5$ and 0.75 in Tyler & Sofrin (Reference Tyler and Sofrin1962). For each of $\beta _{m,j}$, ${\mathcal {B}}_1$ and ${\mathcal {B}}_2$ are chosen as follows:

(6.4a,b)\begin{equation} {\mathcal{B}}_1 = {\mathcal{B}} \quad \mathrm{and} \quad {\mathcal{B}}_2 ={-}{\mathcal{B}} \,\dot{{\rm J}}_m(\beta_{m,j})/ \dot{{\rm Y}}_m (\beta_{m,j}), \end{equation}

with ${\mathcal {B}}$ being an arbitrary constant. Hence, the eigenfunctions ${\mathcal {J}}_{m,j}$ $(j = 1, 2, 3, \dots )$ are defined, except for ${\mathcal {B}}$, by

(6.5)\begin{equation} {\mathcal{J}}_{m,j}(r) \equiv {\rm J}_m(\beta_{m,j}r/R)- \dot{{\rm J}}_m(\beta_{m,j}) {\rm Y}_m(\beta_{m,j}r/R)/ \dot{{\rm Y}}_m(\beta_{m,j}). \end{equation}

It is straightforward to show that the eigenfunctions satisfy the orthogonality relation as

(6.6)\begin{equation} \frac{1}{R^{2}} \int_0^{R} r {\mathcal{J}}_{m,i} {\mathcal{J}}_{m,j} \,{\mathrm{d}} r = \left\{\begin{array}{ll} 1/d_{m,j} & \mathrm{if} \ i = j,\\ 0 & \mathrm{if} \ i \ne j, \end{array}\right. \end{equation}

for $i, j = 1, 2, 3,\dots$, where $1/d_{m,j}$ is given by

(6.7)\begin{equation} \frac{1}{d_{m,j}} = \frac{1}{2 \beta_{m,j}^{2}} \left[ \left( \beta_{m,j}^{2} - m^{2} \right) {\mathcal{J}}_{m,j}^{2}(R) - \left( \beta_{m,j}^{2} c^{2} - m^{2} \right) {\mathcal{J}}_{m,j}^{2} (R_i) \right], \end{equation}

with

(6.8a)\begin{gather} {\mathcal{J}}_{m,j}(R) = {\rm J}_m (\beta_{m,j})- \dot{{\rm J}}_m(\beta_{m,j}) {\rm Y}_m(\beta_{m,j})/ \dot{{\rm Y}}_m(\beta_{m,j}), \end{gather}

(6.8b)\begin{gather}{\mathcal{J}}_{m,j}(R_i) = {\rm J}_m (\beta_{m,j} c) - \dot{{\rm J}}_m(\beta_{m,j}) {\rm Y}_m( \beta_{m,j} c)/ \dot{{\rm Y}}_m(\beta_{m,j}). \end{gather}

In deriving (6.7), use has been made of the following integral:

(6.9)\begin{equation} \int_{R_i}^{R} r {\mathcal{X}}_m (br) {\mathcal{Y}}_m (br) \,{\mathrm{d}} r = \frac{r^{2}}{2} \left.\left[\left( 1 - \frac{m^{2}}{b^{2} r^{2}} \right) {\mathcal{X}}_m (br) {\mathcal{Y}}_m (br) + \dot{{\mathcal{X}}}_m (br) \dot{{\mathcal{Y}}}_m (br) \right]\right|_{r=R_i}^{r=R}, \end{equation}

where ${\mathcal {X}}_m$ and ${\mathcal {Y}}_m$ denote any two Bessel functions ${\rm J}_m$ and ${\rm Y}_m$, $b$ being a constant. This formula is easily verified by differentiating both sides with respect to $R$ (see also Abramowitz & Stegun Reference Abramowitz and Stegun1972).

In parallel with the analysis in § 2, $P_m$ is expanded similarly in terms of ${\mathcal {J}}_{m,j}$ as

(6.10)\begin{equation} P_m(r,x,t) = \sum_{j=1}^{\infty} d_{m,j} \bar{P}_{m,j} (x, t) {\mathcal{J}}_{m,j}(r), \end{equation}

for $R_i < r < R$ where $\bar {P}_{m,j}$ $(j = 1, 2, 3,\dots )$ are defined by

(6.11)\begin{equation} \bar{P}_{m,j} \equiv \frac{1}{R^{2}} \int_{R_i}^{R} P_m (r, x, t)r {\mathcal{J}}_{m,j}(r) \,{\mathrm{d}} r. \end{equation}

6.2. Effects of the boundary layer

Effects of the boundary layer on the inner duct are also taken into account by following the same procedures as in § 3. The only difference is in the choice of the coordinates $n$ and $\eta$ on the inner duct. Attaching the subscript $i$, $n_i$ and $\eta _i$ are taken to be $r- R_i$ and $-R_i \theta$, respectively, if $x$ is taken in common. Because $\eta _i$ is clockwise, $w$ is taken to be opposite as $-w$. The normal velocity at the edge of the boundary layer directed in the same sense as $n_i$ is given in the same form as (3.37), provided $R$ is replaced by $R_i$ and $p^{\prime }$ is taken at $r = R_i$.

The disturbances and the defects are decomposed into the Fourier mode in the azimuthal direction as (3.42) and (3.43). Using (6.10), the means of (3.41) are taken by multiplying with $r {\mathcal {J}}_{m,j}$ over the interval $R_i < r < R$ for $j = 1, 2, 3, \dots$. As in (3.44), the mean of the defect consists of two parts due to the boundary layers on the outer and inner duct walls as

(6.12)\begin{equation} \frac{1}{R^{2}} \int_{R_i}^{R} \check{(\,\cdot\,)} r {\mathcal{J}}_{m,j}(r) \,{\mathrm{d}} r \approx \frac{{\mathcal{J}}_{m,j}(R)}{R} \int_0^{\infty} \check{(\,\cdot\,)}_R \,{\mathrm{d}} n + \frac{c {\mathcal{J}}_{m,j}(R_i)}{R} \int_0^{\infty} \check{(\,\cdot\,)}_{R_i} \,{\mathrm{d}} n_i, \end{equation}

where the subscripts $R$ and ${R_i}$ denote the quantity pertaining to the boundary layer on the outer and inner walls, respectively. Each integral of the defect leads to $\check {V}_{bm}$ at the edge of the boundary layer. Hence, the equations for $\bar {P}_{m,j}$ $(j = 1, 2, 3, \dots )$ take the form of

(6.13)\begin{align} & \frac{\partial^{2} \bar{P}_{m,j}}{\partial t ^{2}} - \frac{\partial \ }{\partial x} \left( a_e^{2} \frac{\partial \bar{P}_{m,j}}{\partial x} \right) + \left( \frac{\beta_{m,j}}{R} \right)^{2} a_e^{2} \bar{P}_{m,j}\nonumber\\ & \quad = \frac{{\mathcal{J}}_{m,j}(R)}{R}\left( \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t} \right)_R + \frac{c {\mathcal{J}}_{m,j}(R_i)}{R} \left( \rho_e a_e^{2} \frac{\partial \check{V}_{bm}}{\partial t} \right)_{R_i}, \end{align}

with $\rho _e a_e^{2} \partial \check {V}_{bm}/\partial t$ given by (3.51) in which $P_m$ takes $P_m(R)$ on the outer wall and $P_m(R_i)$ on the inner wall.