2.1.1. The axial wave structure within an SBH

Mironov & Pislyakov (Reference Mironov and Pislyakov2002) first proposed a novel SBH duct design by inserting a series of annular rings that gradually reduce the effective sound speed along the waveguide, as shown in figure 1(a). The inner radius of the rings decreases to a finite (non-zero) value, forming a retarding structure that occupies the region between $x=l$ and $x=L$ , where $l\gt 0$ and it is here called the ‘terminal truncation’.

Figure 1.

An SBH configuration with finite terminal truncation. (a) Waveguide with annular rings of varying inner radius $r(x)=xR/L, x\in [l,L]$ . (b) Two-dimensional numerical model for the SBH.

Figure 1. Long description

Panel A: A diagram of a waveguide with annular rings of varying inner radius. The waveguide extends from 0 to L along the x-axis. The rings have varying inner radii, and the diagram shows incident and reflected waves within the waveguide. Panel B: A two-dimensional numerical model for the sonic black hole. The model includes a non-reflecting boundary at x = l, an incident wave, and a reflected wave. The wall admittance is indicated as Y = ik0((R^2 - r^2)/2r). The model also includes a perfectly matched layer (PML) on the right side.

The acoustic wave propagation within the SBH structure can be mathematically described using the generalised Webster equation for sound pressure $p$ , derived from (A1)–(A5) in Appendix A and is reproduced as

(2.1) \begin{align} p^{\prime \prime }+p^{\prime }(\ln S)^{\prime }+p k_0^2 G=0, \quad G=1+\frac {2 Y }{\mathrm{i} k_0 r},\quad Y=\frac {\rho _0c_0v}{p}, \end{align}

where $S=\pi r^2$ is the area of the cross-section, $Y$ is the varying dimensionless admittance of the side-wall of the waveguide, $v$ is the radial velocity entering the annular rings, $k_0=\omega /c_0$ is the free-space wavenumber and primes indicate differentiation with respect to $x$ .

Substituting Mironov’s low-frequency approximation for wall admittance (see (A12) in Appendix A)

(2.2) \begin{align} \left .\ Y\right |_{k_0\rightarrow 0}=\mathrm{i} k_0\frac {R^2-r^2}{2r} \end{align}

into (2.1), we obtain $G=R^2/r^2$ . In the particular case of a linear SBH with radius $r(x)=Rx/L$ , (2.1) yields a set of exact solutions in the form of power-law functions,

(2.3) \begin{align} p(x)=A_1\left (\frac {x}{L}\right )^{\alpha _1}+A_2\left (\frac {x}{L}\right )^{\alpha _2}, \end{align}

where $A_1,A_2$ are constants and $\alpha _{1,2}$ are two eigenvalues:

(2.4) \begin{align} \alpha _{1,2}= \left \{ \begin{alignedat}{2} &\frac {1}{2}\left [-1\pm \sqrt {1-\left (2 k_0 L\right )^2}\right ]\!,\quad & 2k_0L\lt 1, \\[5pt]&\frac {1}{2}\left [-1\pm \mathrm{i}\sqrt {\left (2k_0L\right )^2-1}\right ]\!,\quad & 2k_0L\gt 1. \end{alignedat} \right . \end{align}

Below the frequency that satisfies $k_0L=1/2$ , $\alpha _{1,2}$ are real numbers and the solution only has evanescent waves, indicating complete reflection for the downstream entry region, $x\gt L$ . Above the cut-on frequency, $k_0L\gt 1/2$ , $\alpha _1$ and $\alpha _2$ are complex conjugates and the solution represents two waves propagating in opposite directions, as illustrated later. For complex exponents $\alpha _{1,2}$ , we use the fundamental identity $ (x/L )^\alpha =e^{\alpha \ln { (x/L )}}$ and obtain

(2.5) \begin{align} p(x) =A_1\left (\frac {x}{L}\right )^{-\tfrac {1}{2}} e^{\mathrm{i} \int _L^x \sqrt {\left (2 k_0 L\right )^2-1} \tfrac {\text{d} x}{2 x}}+A_2\left (\frac {x}{L}\right )^{-\tfrac {1}{2}} e^{-\mathrm{i} \int _L^x \sqrt {\left (2 k_0 L\right )^2-1} \tfrac {\text{d} x}{2 x}} . \end{align}

The integrand shows a variation with respect to $x$ indicating a varying wavenumber $k(x)$ and phase accumulation of the sound wave from $L$ to $x$ ,

(2.6) \begin{align} k(x)=\frac {1}{2 x} \sqrt {\left (2 k_0 L\right )^2-1},\quad \phi (x)=\int _L^x k(x)\, \mathrm{d} x=\int _L^x \sqrt {\left (2 k_0 L\right )^2-1}\,\frac {\text{d} x}{2 x}, \end{align}

we can conclude that (2.5) represents a standing wave with a wavenumber, $k(x)$ , that varies with respect to $x$ . These waves have spatially varying wavenumbers and will be referred to as inhomogeneous waves. We will now use this basic theory of wave propagation in SBH to further analyse the nature of reflection that occurs in such waveguides under various boundary conditions at the terminal of the waveguide, $x=l$ .

2.1.2. Reflection within SBH

To derive the reflection coefficients in SBH, we first define $W=A_2/A_1$ as the inhomogeneous wave reflection coefficient at the terminal ( $x=l$ ), derived using the boundary conditions given in (B2) in Appendix B,

(2.7) \begin{align} W=-\left (\frac {l}{L}\right )^{\alpha _1-\alpha _2}\frac {\mathrm{i}k_0lY_{cs}-\alpha _1}{\mathrm{i}k_0lY_{cs}-\alpha _2},\quad Y_{cs}=\left .\frac {\rho _0 c_0 u_n}{p}\right |_{x=l}, \end{align}

where $Y_{cs}$ is the dimensionless terminal axial admittance in the cross-section, $x=l$ , and $u_n$ is the outward normal velocity (axial to the left).

The downstream region ( $x\ \geqslant \ L$ ) is modelled by plane waves, with a reflection coefficient $R_L$ at $x=L$ defined as the ratio of the reflected wave $R_L\ e^{-\mathrm{i}k_0x}$ to the unit incident wave $e^{\mathrm{i}k_0x}$ . The derivation can be found from (B1)–(B3) in Appendix B,

(2.8) \begin{align} R_L=\frac {1+W-\frac {1}{\mathrm{i}k_0 L}\left (\alpha _1+W\alpha _2\right )}{1+W+\frac {1}{\mathrm{i}k_0 L}\left (\alpha _1+W\alpha _2\right )}e^{2\mathrm{i}k_0L}. \end{align}

We will now study three different axial boundary conditions at $x=l$ and investigate their significance.

2.1.3. Numerical verification and discussion of wave propagation in SBH

Two-dimensional, axisymmetric analysis of the SBH is considered and numerically simulated in COMSOL Multiphysics. The geometry is shown in figure 1(b). An incident wave is modelled at the inlet with its reflection wave absorbed by a perfectly matched layer (PML). It should be noted that no damping is added in this calculation. The small truncation at the end of the SBH is set at $x=l$ . We introduce a dimensionless axial impedance $Z_{\textit{out}}$ at the SBH exit boundary $\varGamma$ shown in figure 1(b),

(2.17) \begin{align} Z_{\textit{out }}=\frac {1}{Y_{c s} }=\frac {\mathrm{i} k_0 l}{\alpha _1}=\underbrace {\frac {l}{L} \sqrt {1-\left (2 k_0 L\right )^{-2}}}_{z_r}+\frac {l}{\underbrace {2 \textrm{i} k_0 L^2}_{z_i}}, \end{align}

where the real part $z_r$ features 0 damping at the cut-off frequency of $2k_0L=1$ , and an asymptotic value of $l/L$ when $k_0\to \infty$ . The imaginary part, $z_i$ , features a pure compliance term, for which a specific implementation is proposed towards the end of § 2.

Figure 2(a) compares the one-dimensional analytical theory (solid curve) with three two-dimensional simulations showing good general agreements. Persistent reflections arise from wave shortening. The red dash-dotted and the green dotted curves share the same outer radius $R=0.15$ m, but has different termination values of $l=0.1$ mm and 1 mm. The fact that these two curves are similar in their magnitudes of deviation from the analytical solution (2.14) means that $l$ does not matter much as long as the terminal impedance $Z_{{out}}$ is implemented. The black dashed curve has a much reduced outer radius of $R=0.05$ m, which shows minimal deviation from the analytical solution. The latter is expected since when $R\rightarrow 0$ , the configuration approaches a one-dimensional geometry. In summary, the two-dimensional solution deviates from the one-dimensional solution, but the trend shown in the one-dimensional analytical solution is valid.

Figure 2.

Comparison of the linear reflection coefficient $|R_L|$ with results of various two-dimensional axisymetric numerical simulations with $L=0.5$ m. (a) Numerical analysis under non-reflecting boundary condition $Z_{\textit{out}}$ in (2.17) at the exit terminal $x=l$ , and Mironov wall admittance in (2.2) with varying $R$ and $l$ . (b) Pressure magnitude distribution $|p|$ along the axis of SBH and the pressure contour at the frequency of $k_0L=4.58 (500 \,\text{Hz})$ with $R=0.15$ m and $l=1$ mm.

Figure 2(b) illustrates an increasing pressure magnitude distribution $|p|$ along the axis of the SBH as $x\rightarrow 0$ , along with a pressure contour that displays an increasing wavenumber and a correspondingly decreasing phase velocity, consistent with the analytical solution. Both figures 2(a) and 2(b) demonstrate good alignment between the two-dimensional axisymmetric numerical model with the one-dimensional analytical solution.

The idea presented by the Minorov solution that wave absorption can be postponed and focused towards a terminal, and gets treated collectively by a single exit impedance is analytically inspiring, as explained later. A waveguide is essentially an impedance transformer. For all incident sound to be absorbed at $x=L$ , a perfect dissipative impedance of $Z=p/(\rho _0c_0u_n)=1$ is needed, which has zero reactance and is very hard to achieve by any practical design. When waves are guided towards $x=l$ , the required impedance becomes $Z_{\textit{out}}$ given in (2.17). The terminal impedance of (2.17) ensures the anechoic condition and the only propagating wave is the $A_1$ term in (2.3). The total sound energy flux along this wave, $S(x) ( {1}/{2}) \textrm{Re} (p u_n^*)$ , can be easily derived and is found to be independent of $x$ , where superscript ^* denotes complex conjugate and the outward normal velocity $u_n$ is derived from the momentum equation as $(\partial p/\partial n)/(-\rho _0\mathrm{i}\omega )$ . The constancy of sound energy flux implies no wave energy reflection along the way, and all reflection occurs at the entrance of $x=L$ where the local impedance can be easily derived from (2.17) by allowing $x=l$ to be replaced by $L$ . Thus,

(2.18) \begin{align} Z_L=\sqrt {1-\left (2 k_0 L\right )^{-2}}+\frac {1}{2 \textrm{i} k_0 L}. \end{align}

It can be verified that the reflection coefficient $R_L$ given in (2.14) is exactly $(Z_L-1)/ (Z_L+1)$ . In other words, the SBH device with an anechoic terminal has an entrance impedance of $Z_L$ . The reactance term is equivalent to that of a uniform cavity of depth $2L$ at low frequencies. In other words, the wave slowing down has effectively doubled the useful space. Note that, for an actual cavity of depth $L$ , the reactance is $1/[\mathrm{i}\tan (k_0L)]$ . The denominator $\text{tan}(k_0L)$ is approximated by $k_0L$ for low frequencies. However, when $k_0L\gt 1.17$ , $\text{tan}(k_0L)$ already exceeds $2k_0L$ . Since the SBH absorption only begins from $k_0L=1/2$ and descent performance is only available from approximately $k_0L\sim 1$ . That space-saving bandwidth is really narrow. It could have been very useful if its effective frequency goes below $k_0L=1/2$ . The reason why it does not do so at $k_0L=1/2$ can also be seen from the complete mismatch between the air impedance of unity and $Z_L=-\mathrm{i}$ at $k_0L=1/2$ .

The alternative to this idea of sudden absorption at the terminal is the classic design of acoustic wedge where sound is absorbed by gradually increasing density of porous material (Jiang et al. Reference Jiang, Zhang and Huang2016). Comparison of the performance of different types of gradient absorbers will be made in § 4.

2.2.1. Exact admittance at the entry of the annular-rings

Many existing literature considers the structure discussed in the previous section to achieve ‘perfect’ absorption (black-hole phenomenon) and attribute its failure to do so to the fact that $l$ can never truly be zero, essentially treating this as an issue of engineering inaccuracy. This theoretical conclusion is found to have limited applicability when Mironov’s low-frequency wall admittance assumption (2.2) is not adopted. We will now further analyse the wall cavity in figure 1(a) by solving the wave equation in cylindrical coordinates to yield the following exact admittance $Y(r,k_0)$ at its entry, with details in Appendix A (A6)–(A10):

(2.19) \begin{align} Y\left (r,k_0\right )=\frac {\rho _0c_0v}{p}=\frac {1}{\mathrm{i}}\frac {J_1\left (k_0r\right )-\frac {J_1\left (k_0R\right )}{Y_1\left (k_0R\right )}Y_1\left (k_0r\right )}{J_0\left (k_0r\right )-\frac {J_1\left (k_0R\right )}{Y_1\left (k_0R\right )}Y_0\left (k_0r\right )}. \end{align}

When the frequency approaches zero ( $k_0\rightarrow 0$ ), we can apply the small-argument approximations in (A11), and the admittance $Y(r,k_0)$ reduces to the Mironov approximation given in (2.2). However, we already know that waves will be cut-off below frequency $k_0L\lt 1/2$ , meaning that the device length is $1/(4\pi )$ times of the wavelength, not too far from the general ‘compact device’ threshold of $1/6$ of a wavelength. So, it is more interesting to explore what happens for a fixed and finite frequency while the terminal inner radius $r(x)\rightarrow 0$ . Details of this analysis are given in Appendix A leading to (A13). When $J_1 (k_0R )\neq 0$ , we have the following for the admittance approaching zero radius:

(2.20) \begin{align} \left .Y\right |_{r \rightarrow 0 \& J_1\left (k_0 R\right ) \neq 0} \sim \frac {\mathrm{i}}{k_0 r \ln \left (k_0 r/2\right )} . \end{align}

Substituting this expression into (2.1), we obtain

(2.21) \begin{align} p^{\prime \prime }+p^{\prime }(\ln S)^{\prime }+p k_0^2 G=0, \quad G=1+\frac {2}{\left (k_0 r\right )^2 \ln \left (k_0 r/2\right )}\lt 0 . \end{align}

Sturm’s comparison theorem shows that, for an equation of the form

(2.22) \begin{align} \left [A\left (x\right )p^\prime \right ]^\prime +B\left (x\right )p=0 \quad \mathrm{with} \, A\left (x\right )\gt 0\,\text{ and }\,B\left (x\right )\lt 0 \quad \mathrm{on} \, \left [a,b\right ], \end{align}

the solution $p (x )$ does not oscillate and vanishes at most once on $ [a,b ]$ . Application of this theorem to (2.21) means that the pressure field does not exhibit wave-like behaviour as $r\rightarrow 0$ for any finite frequency that satisfies $J_1 (k_0R )\neq 0$ . The no-wave behaviour is more obvious when $G=-a^2\lt 0$ in (2.21) and the cross-section is uniform, $(\ln S)^{\prime }=0$ ; hence, $p^{\prime \prime }=(ak_0)^2p$ leading to $p=e^{\pm ak_0x}$ . Only the exponentially decaying solution is admissible and there is no phase angle variation. Specifically, as the parameter $a$ tends to infinity, the pressure $ p$ tends to 0 and thus satisfies the definition of a soft boundary.

For the special case $J_1 (k_0R )=0$ as $r\rightarrow 0$ ,

(2.23) \begin{align} \left .Y\right |_{r \rightarrow 0 \& J_1\left (k_0 R\right )=0} \sim \frac {k_0 r}{2\mathrm{i}} . \end{align}

Substituting this expression of $Y$ into the Webster equation (2.1), we have $G=0$ . Given $(\ln S)^{\prime }=2/x$ for the linear radius profile $r(x)=Rx/L$ , we obtain

(2.24) \begin{align} p^{\prime \prime }+\frac {2}{x}p^\prime =0, \end{align}

with the exact general solution of

(2.25) \begin{align} p(x)=\frac {D_1}{x}+D_2, \end{align}

where $D_1$ and $D_2$ are arbitrary constants. It means that, for frequencies satisfying $J_1 (k_0R )=0$ , much like the case of $J_1 (k_0R )\neq 0$ , the solution also does not exhibit wave-like behaviour as $r\rightarrow 0$ . In fact, the first non-zero frequency for $J_1(k_0R)=0$ is $k_0R=3.83$ and is in the rather high region of frequency (for a typical design of $R/L=0.3$ , it means $k_0L=12.8$ ), which is not the primary frequency region of concern for SBH devices.

Since the sign of the coefficient $G=1+2Y/(ik_0r)$ in (2.1) controls the solution behaviour, oscillating or non-oscillating, we now examine where $G$ changes sign. Around the condition where the side-wall impedance vanishes, $1/Y (r,k_0 )=0$ , resonance occurs inside the side-wall cavity. The side-wall acoustic admittance $Y$ changes between $-\infty$ and $\infty$ . The radius $r$ at which the sign of $G$ changes is near the resonance location, but not coincided due to the presence of unity in the expression of $G=1+2Y/(ik_0r)$ . It is nevertheless adequate for our purpose to examine the occurrence of the side-wall resonance. The physical significance of the side-wall resonance is the following. All axially propagating waves in the waveguide will be fully reflected. When there is damping, the reflection is less complete, but the waveguide function is still considered a failure. For a rectangular channel of depth $R$ , the admittance is $Y=\mathrm{i}\tan (k_0R)$ and the first resonance is when $R$ is equal to a quarter-wavelength. For such rectangular case, one can always find a small $R$ such that the resonance occurs at a frequency higher than all our frequencies of interest and the low-frequency approximation becomes satisfactory: $G$ does not change sign. Interestingly, such resonance cannot be avoided for any frequency when $r\rightarrow 0$ in the cylindrical waveguide featuring Bessel functions in the admittance expression, (2.19). This is so because the high curvature gives a large and frequency-dependent virtual mass when $r\rightarrow 0$ . This result implies that the scenario of slow waves propagating and coming to a complete rest at $r=0$ , or that it takes infinite time, cf. (2.16), to reach the sharp terminal, is theoretically impossible for this SBH configuration. Taking the frequency of $k_0 L = 2.29$ (250 Hz) as an example, the resonance point lies around $x=3$ mm when $R=0.15\,\text{m},\,L=0.5\,\text{m}$ . When the side-wall has zero impedance, waves enter the side-branch without any ‘effort’. The waves reach the outer hard wall, get fully reflected back and form a pressure-release condition there, $p=0$ , despite the fact that the incident wave has a high amplitude scaled with $x^{-1/2}$ due to the narrowing passage. Such reflection is called the ‘soft reflection’, contrasting with the hard-wall reflection if a rigid wall terminal is placed at $x=l$ .

In earlier literature, the side-wall resonance at different locations by different wall depth was characterised as ‘rainbow trapping’. Here, ‘trapping’ has to be understood in the sense that waves are prohibited from crossing the resonance location instead of its energy flux being ‘trapped’ by absorption or other unspecified ‘black hole’ behaviour. It should also be noted that complete sound absorption by a single side-branch resonance is impossible even when suitable damping is present at the side-branch.

To summarise, the sign of coefficient $G$ dictates the wave propagation type. For $G\gt 0$ (Mironov’s approximation, (2.2)), the side-wall is compliance-dominated and the wave is oscillatory above the cut-on frequency $k_0L\gt ({1}/{2})$ . When accurate admittance (2.19) is used, which reduces to (2.20) when $r\rightarrow 0$ , the wall is dominated by inertia for any finite frequency. The solution is of the no-wave type. The incident waves are totally reflected where $G\gt 0$ breaks down. While waves cannot cross the point of side-wall resonance when there is no damping, some waves can still infiltrate the resonance location when there is side-wall damping. Therefore, we shall first illustrate numerically the two solution types before giving physical explanations on the effect of side-wall on wave propagation.

Figure 3(a) shows a sketch of a one-dimensional segment of length $L_{1D}$ , over which the Webster equation, (2.1), is solved numerically using the coefficient setting in the COMSOL one-dimensional mathematical module. The left end of the waveguide is defined as a rigid boundary ( $\partial p/\partial x=0$ ), while the inlet at the right-end, $x=L_{1D}$ , is set as a source ( $\partial p/\partial x\neq 0$ ). Figure 3(b) is a sketch to be used for the next subsection. The numerical results are given in figures 3(c) and 3(d) using the parameters given in the caption.

Figure 3.

Illustration of the different solutions for compliance-dominated $(G\gt 0)$ and inertia-dominated $(G\lt 0)$ regions by solving the one-dimensional Webster equation, (2.1), over a small segment. (a) A sketch of the one-dimensional mathematical model. (b) A sketch of a duct with side-wall admittance and speed $c=\Delta x/ \Delta t$ defined as the speed of pressure disturbance from left to right across an element of width $\Delta x$ over a time period of $\Delta t$ . (c) Instantaneous pressure calculated using the Mironov side-wall admittance (2.2) which gives $G\gt 0$ . (d) Pressure calculated using the exact side-wall admittance (2.19) which gives $G\lt 0$ . For panels (c) and (d), the segment length is $L_{1D}=0.09$ m and the frequency calculated is $k_0L=4.58$ (500 Hz). The overall device sizes are $R=0.3\,\text{m},\, L=0.5\,\text{m}$ . The pressure magnitude is normalised such that $|p|=1$ at the right end.

Figure 3. Long description

Panel A: A sketch of the one-dimensional mathematical model showing a rigid end, a segment labeled L1D, and a source with wall admittance assumption. Panel B: A sketch of a duct with side-wall admittance and speed defined as the speed of pressure disturbance from left to right across an element of width over a time period. Panel C: A line graph showing instantaneous pressure calculated using the Mironov side-wall admittance. The x-axis is labeled x (m) and the y-axis is labeled Re(p). The graph shows oscillating pressure values. Panel D: A line graph showing pressure calculated using the exact side-wall admittance. The x-axis is labeled x (m) and the y-axis is labeled Re(p). The graph shows a smooth increase in pressure values.

Figure 3(c), which uses a logarithmic coordinate scale based on the Mironov assumption, shows an almost constant wave-crest spacing. This indicates that the wavelength decreases rapidly towards the left end while the amplitude increases. Figure 3(d) shows an evanescent wave with exponential decay of pressure magnitude.

2.2.2. Physical explanation of the effect of compliance versus inertial sidewalls

The physical effect of a sidewall on wave propagation may be illustrated by looking at the special case with constant cross-section, as shown in figure 3(b), in which the mass conservation in (A2) may be combined with the momentum equation to yield

(2.26) \begin{align} \frac {1}{c_0^2}\left (1+\frac {2Y}{\mathrm{i}k_0r}\right )\frac {\partial ^2p}{\partial t^2}=\frac {\partial ^2p}{\partial x^2}. \end{align}

The effective sound speed $c_e$ can be expressed as

(2.27) \begin{align} c_e=\frac {c_0}{\sqrt {1+2Y/(\mathrm{i}k_0 r)}}. \end{align}

When the sidewall is a spring, as shown by the low-frequency approximation with $r\rightarrow 0$ , $Y (r )\approx \mathrm{i}k_0(( {R^2-r^2})/({2r}))$ , the effective sound speed is determined by

(2.28) \begin{align} c_{e0}=\frac {c_0}{\sqrt {1+(R / r)^2}} \xrightarrow {r \rightarrow 0} \frac {r}{R} c_0 . \end{align}

The speed slows down to zero when $r\rightarrow 0$ , which is the essential theory behind Mironov’s low-frequency assumption. When the side-wall behaves as an inertia boundary, as indicated by the accurate solution shown in (2.20) and (2.21), the effective sound speed becomes

(2.29) \begin{align} \begin{gathered} c_{e1}=\frac {c_0}{\sqrt {1+\frac {2}{\left (k_0 r\right )^2 \ln \left (k_0 r / 2\right )}}}. \end{gathered} \end{align}

The effective sound speed $c_{e1}$ becomes purely imaginary as $r\rightarrow 0$ , indicating that the wave-retarding structure cannot support propagating waves. Instead, the evanescent field exhibits exponential attenuation.

As illustrated in figure 3(b), for a plane wave propagating in a duct with wall admittance $Y$ , the speed of pressure disturbance is defined as $c={\Delta x}/{\Delta t}$ , where $\Delta x$ and $\Delta t$ represent the spatial and temporal increments of the wave, respectively. Following this line of logic, we will analyse the governing equations using the changes of acoustic properties across the element, namely the pressure rise ( $\Delta p$ ), the change in axial velocity ( $\Delta u$ ) and the change in density ( $\Delta \rho$ ). From the linearised momentum equation ${\partial p}/{\partial x}+\rho _0{\partial u}/{\partial t}=0$ , we obtain the relation ${\Delta p}\ = \rho _0c{\Delta u}$ . The thermodynamic equation provides the isentropic relation ${\Delta p}\ =\ c_0^{2}{\Delta \rho }$ , with $c_0$ representing the intrinsic sound speed in the unbounded fluid. The wall dynamics are characterised by a stiffness (per unit axial distance) parameter $\kappa$ , defined through the relation $\kappa = {\Delta p 2 \pi r}/{\Delta r}$ , where $r$ is the duct radius and $\Delta r$ is the radial wall displacement caused by the pressure perturbation $\Delta p$ .

Mass conservation in cylindrical geometry necessitates the consideration of axial and radial contributions to the mass flow rate. In the case of a constant cross-section, the mass conservation (A1) can be simplified to

(2.30) \begin{align} \rho _0\frac {\partial u}{\partial x}+\rho _0\frac {2v}{r}+\frac {\partial \rho }{\partial t}=0, \end{align}

where the first term represents net axial mass flux, the second term accounts for the motion of the radial wall and the third term denotes density variations with time.

Substituting the fundamental relations $v = {\mathrm{\Delta } r}/{\mathrm{\Delta } t}$ , $c ={\mathrm{\Delta } x}/{\mathrm{\Delta }t}$ and $\mathrm{\Delta }p =\rho _0c\mathrm{\Delta }u$ , and multiplying (2.30) by $\Delta t$ , gives

(2.31) \begin{align} -\mathrm{\Delta }p\ +\ \rho _0\left (\frac {2c^2\mathrm{\Delta }r}{r}\right )+\ c^2\mathrm{\Delta \rho }\ =\ 0. \end{align}

Due to the radial spring-like wall displacement caused by the pressure perturbation $\mathrm{\Delta } p$ , the second term and the third term of (2.31) have the same sign, indicating that the radial mass exchange through the walls increases axial particle velocity since $\Delta u=\mathrm{\Delta }p/ (\rho _0c )$ . Using the wall stiffness definition to express $\mathrm{\Delta }r\ =\Delta p 2\pi r \kappa$ and substituting the thermodynamic relation $\mathrm{\Delta }\rho \ = {\mathrm{\Delta } p}/{c_0^{2}}$ , we obtain

(2.32) \begin{align} -\mathrm{\Delta }p\ +\ \rho _0\left (\frac {4\pi c^2}{\kappa }\right )\mathrm{\Delta }p\ +\ \left (\frac {c^2}{c_0^{2}}\right )\mathrm{\Delta }p\ =\ 0. \end{align}

Dividing through by $\mathrm{\Delta }p$ yields the final expression

(2.33) \begin{align} \frac {1}{c^2}=\rho _0\left (\frac {4\pi }{\kappa }\right )+\frac {1}{c_0^2}, \end{align}

which relates the sound speed $c=\mathrm{\Delta } x/{\mathrm{\Delta } t}$ to the wall stiffness $\kappa$ and the intrinsic sound speed $c_0$ . This equation demonstrates how wall compliance reduces the effective propagation speed below the intrinsic sound speed.

In contrast, when the wall dynamics are characterised by a mass per unit axial distance $m$ , the sound speed $c$ in a cylindrical duct with inertial walls can be derived by analysing the coupled dynamics between the acoustic wave propagation and wall motion. The wall dynamics are governed by ${m\partial ^2r}/{\partial t^2}= p 2\pi r$ and under harmonic motion becomes $-m\omega ^2\Delta r = \mathrm{\Delta }p 2\pi r$ , thus yielding the wall displacement $\Delta r = -(\Delta p2\pi r)/(m \omega ^2)$ .

For an inertia-dominated boundary in an oscillatory system, the force acting on the wall always points towards the equilibrium position, meaning the force on the wall ( $\Delta p$ ) is opposite to its displacement from equilibrium ( $\Delta r$ ). Consequently, the radial wall displacement induced by the pressure perturbation $\Delta p$ results in the second and third terms of (2.30) having opposite signs. This suggests that the radial mass transfer through the walls reduces the variation of the axial change in velocity $\Delta u$ in the case of a mass-like boundary.

Substituting the relations $\Delta r\ =\ -(\Delta p2\pi r)/(m \omega ^2)$ into the continuity equation (2.31) and simplifying this expression leads to the key dispersion relation

(2.34) \begin{align} -\frac {\omega }{c_0^{2}}-\frac {\omega }{c^2}+\frac {4\pi \rho _0}{m\omega }=\ 0, \end{align}

which can be rearranged to show the frequency-dependent nature of the effective sound speed

(2.35) \begin{align} \frac {1}{c^2}=\frac {1}{c_0^{2}}-\frac {4\pi \rho _0}{m\omega ^2}. \end{align}

This result reveals several important physical insights. First, the inertial wall introduces a frequency-dependent modification to the phase velocity that differs fundamentally from the compliance-dominated case. The $m\omega ^2$ term in the denominator represents an effective negative stiffness that arises from the wall inertia, leading to accelerated sound speed ( $c \gt c_0$ ). It should be noted that the sound speed discussed in this section refers to the phase velocity in the frequency domain.

When the mass is further reduced to ${{4\pi \rho _0c}_0^2}\,/{\omega ^2}$ and below, the phase velocity gradually increases to infinity, exhibiting incompressible-flow behaviour. As shown by the exact solution for CWIG sidewall admittance $Y$ in (2.19), the boundary admittance $Y\rightarrow + ({\infty }/{\mathrm{i}\omega })$ as $r\rightarrow 0$ behaves as an inertia tending to zero. The fact that phase speed can exceed the speed of sound in air when there is a moving boundary is consistent with established physical principle and demonstrated experiments by our earlier studies. In our earlier studies (Huang et al. Reference Huang, Choy, So and Chong2000), both subsonic and supersonic eigen waves are identified experimentally in different frequency regimes. We can see from (2.35) that as $m\rightarrow 0$ , the phase velocity continuously increases to infinity, forming a point of incompressible flow for all frequencies as $r\rightarrow 0$ .

2.2.3. Numerical analysis of wave propagation in CWIG

Having clarified the concept that waves cannot get ‘trapped’ by slowing down to zero phase speed at $x=l=0$ and that the wave speed increases like incompressible-flow in the inertia-dominated narrow region, we will now focus on practical sound absorption performance of an SBH device with a small $l\gt 0$ . If $R$ is small enough or when $l$ is large enough, the whole frequency region of interest, such as $k_0L\in (1/2,5)$ , may lie below the terminal resonance frequency and the Mironov approximation can be mostly valid for the purpose of deriving the reflection coefficient $R_L$ , (2.14) provided that the no-reflection condition of $W=0$ is satisfied at $x=l$ .

Two-dimensional axisymmetric configuration of CWIG is now numerically simulated in COMSOL Multiphysics using the exact side-wall admittance condition, (2.19). The results are compared with the one-dimensional analytical solution in figure 4.

Figure 4.

Comparison of reflection coefficients $R_L$ obtained from the one-dimensional (1-D) analytical solution (solid line) and numerical analysis (dotted and dashed lines), all with $L=0.5\,\text{m}$ and the exact side-wall admittance given in (2.19). (a) Numerical results obtained with a fixed $l$ = 5 mm, various $R$ , a non-reflecting boundary condition $Z_{\textit{out}}$ given in (2.17) at the exit terminal. (b) Numerical result obtained for a proposed terminal configuration in which a micro-perforated panel (aperture diameter $d$ = 0.1 mm, thickness $t_p$ = 0.1 mm and porosity $\sigma$ = 0.058) is used as terminal plate at $x=l=0.15\,\text{m}$ to which a closed cavity of volume $V = 2\pi R^2l$ is attached to approximate the required exit boundary impedance $Z_{\textit{out}}$ , (2.17). A small outer radius of $R=0.05\,\text{m}$ is used to ensure no side-wall resonance for $x\in (l,L)$ occurring within the frequency range of $k_0L\lt 5$ .

In figure 4(a), we use the exact axial exit impedance $Z_{\textit{out}}$ , (2.17), so that we can focus on the impact of the Minorov approximation for the side-wall. The lengths of $L=0.5$ m and terminal truncation $l$ = 5 mm are fixed. For the outer radius $R=0.15$ m (red dash-dotted curve), the $R_L=1$ is found around frequency $k_0L=2.3$ , indicating a full reflection by the side-branch resonance. The Mironov approximation fails completely for $k_0L\gt 2.3$ . The validity range is extended somewhat when $R=0.09$ m (green dotted curve) is used. When $R=0.03$ m is adopted, the result (black dashed curve) follows the one-dimensional analytical (approximate) solution rather well over the whole frequency range shown ( $k_0L\lt 5$ ). The occurrence of full reflection corresponds to the ‘rainbow trapping’ phenomenon.

As a practical implementation of the above-mentioned case, the required impedance $Z_{\textit{out}}$ in (2.17) features a reactance term which can be easily implemented by a compact cavity. We test a rigid-wall cylindrical cavity with volume $V = 2\pi R^2 l$ , whose overall shape approximates a cube to minimise higher-order mode effects, as shown in figure 4(b). Sound waves are decelerated according to the exact side-wall admittance and propagate towards the boundary at $x = l$ , where a micro-perforated panel (MPP) introduces damping. The acoustic impedance of the MPP follows Maa’s theory (Maa Reference Maa1998), with aperture diameter $d = 0.1$ mm and panel thickness $t_p = 0.1$ mm. The porosity $\sigma = 0.058$ is estimated from $\operatorname {Re}(Z_{\textit{MPP}})/\sigma =z_r$ , where $z_r$ is given in (2.17). The cavity essentially acts as a Helmholtz resonator, for which the neck reactance is $z_n = p/(\rho _0 c_0 u_n) = A_n/(i k_0 V)$ , where the neck area is $A_n = \pi (l R / L)^2$ . This $z_n$ turns out to be identical to $z_i$ given in (2.17). However, the virtual mass around the neck and the mass contributed by the MPP degrade the reactance matching at high frequencies. Figure 4(b) presents a numerical example of such an exit-cavity implementation with a large terminal truncation of $l = 0.15$ m. The black dashed curve agrees well with the one-dimensional analytical solution at low frequencies but, as expected, diverges at higher frequencies. In summary, under exact wall admittance, especially when $R$ is small and with a certain frequency range, one can achieve sound absorption similar to that predicted by the Mironov analytical solution.