How the waveguide acoustic black hole works: A study of possible damping mechanisms^a)

The acoustic black hole (ABH) effect is an efficient way to provide broadband control over structural vibrations and sound propagation. The mechanism behind this effect is to achieve a continuous reduction of the wave velocity through a specific retarding structure. Ideally, the speed of propagation should tend to zero as waves propagate toward the end of the ABH. The waves will then never reach the end, and therefore, there will be no reflections. This effect was first studied by Mironov (1988) for a beam with thickness decreasing smoothly to zero according to a power-law profile. Mironov showed that if a transverse wave propagates in such a beam, the wavenumber and the amplitude of the wave increase, while the speed of propagation tends to zero as the wave travels toward the tip of the beam. A few years later, Krylov demonstrated the same effect for a plate with a pit that has a power-law decrease in its thickness (Krylov, 1990, 2007; Krylov and Shuvalov, 2000). Since then, the number of studies on the ABH effect for beams and plates has grown a lot (Karlos et al., 2019; Lee and Jeon, 2019). Most works consider an imperfection in the power-law profile for the thickness of the plate, as it is not possible to reduce the thickness to zero in practice. The effect of the truncation of the ideal profile can be minimized, as proposed by Krylov and Tilman (2004), by using a small amount of damping material on the thin part of the plate where the speed of the waves is slowed down and the amplitude of the waves is increased. Thus, by using a small amount of damping material, the vibrations may be damped efficiently. Recently, the ABH effect has been studied in a variety of different applications, from controlling and absorbing vibrations (Georgiev et al., 2011; He and Ding, 2021; Ji et al., 2021; Lee and Jeon, 2021; McCormick and Shepherd, 2019; Raybaud et al., 2021; Shepherd et al., 2016; Tang et al., 2021; Tang et al., 2016) and wave propagation manipulation (Deng et al., 2021c; Deng et al., 2021e; Zhu and Semperlotti, 2017) to energy harvesting (Deng et al., 2021d; Maugan et al., 2019; Zhao et al., 2014, 2015) and vibro-acoustics (Conlon et al., 2015; Deng et al., 2021a,b; Ji et al., 2019; Ma and Cheng, 2019).

To date, most research in the field has been devoted to vibrations in beams and plates, and only a few works have been carried out on ABHs for sound absorption in air. Mironov and Pislyakov (2002) were the first to introduce such a device. They specified the design of a waveguide termination such that the speed of sound waves tends to zero, analogous to the transverse waves in the structural ABHs. Their design contains a cylindrical waveguide where the radius follows a power-law profile with a variable admittance of the walls. They also showed an example of such a waveguide in terms of a fixed-radius cylindrical duct with sound-hard walls containing a set of rings separated by cavities. The inner radius of these rings (ribs) decreases smoothly to zero, according to a power-law profile. Since it is not possible in practice to reduce the inner radius of the rings to zero, there will be a truncation of the power-law profile, just like for the structural counterpart. Since the waves will slow down and the amplitude increase toward the end of the waveguide in the presence of some damping, this design can still be used to reduce the reflection coefficient, particularly for higher frequencies. Inspired by these findings, El Ouahabi et al. (2015a,b) built two cylindrical ducts that contain rings with decreasing inner radii following linear as well as quadratic profiles. They referred to these two setups as linear ABH and quadratic ABH, respectively. Also, to reduce the effect of truncation of the power-law profile, they used damping materials at the end of the waveguide. The choice of the location of the damping materials was motivated by the method used in the structural ABHs to reduce the truncation error by adding damping material at the thin part of the domain. The authors measured the reflections from the end of the waveguide for different frequencies in both the linear and quadratic ABHs for different amounts and types of damping materials. The results showed an ABH effect that agreed reasonably well with the theoretical predictions of Mironov and Pislyakov (Mironov and Pislyakov, 2002), even without added damping materials. However, unlike the structural counterpart, additional damping material in the peripheral end of the device did not further reduce the reflection coefficient. Figure 1 shows the reflection coefficient measured by El Ouahabi et al. (2015a). Note that the addition of damping material has very little effect on the reflections.

Guasch et al. (2017) used the transfer matrix method to study effective parameters, such as the thickness and the number of rings, the length of the imperfection of the power-law profile, the minimum radius of the rings, and the amount of damping, on the reflection coefficient for linear and quadratic ABH. They considered the cavities between the rings to be filled with fibrous material and used empirical formulas to approximate the effects of the sound absorption inside the cavities. Also, in a recent work, Guasch et al. (2020) showed that if the number of rings and cavities is large enough, then the wave propagation in such a design can be treated as a continuous problem. Moreover, they showed that the governing equation for such a problem is identical to the equations that govern the wave propagation in a simple waveguide filled with a metafluid with a power-law-varying density, which allows the transfer matrix method to be used to solve the problem and calculate the reflection coefficients for different frequencies. Other than the transfer matrix method, Hollkamp and Semperlotti (2020) studied the application of fractional order operators in simulations of acoustic ducts with ABH terminations. As a variation of the ribbed design of an ABH considered in all of the above-mentioned works, Jiménez et al. (2017) used a cascade of Helmholtz resonators to achieve quasi-perfect sound absorption over a broad range of frequencies. They specified the resonance frequency of each resonator to obtain a perfect sound absorption at that specific frequency. That is, each resonator in the cascade is responsible for the absorption at a specified frequency. In a recent work, Mi et al. (2021) studied the transmission and reflection of an open-end quadratic ABH both analytically and experimentally. No damping material was used in their experiment, with the assumption that the visco-thermal losses are the main source of damping in the setup. Like most of the studies on modeling of the ABH effect in waveguides, Mi et al. (2021) considered a complex-valued speed of sound to model the damping inside the waveguide, motivated by the same assumption by Mironov and Pislyakov (2002). The complex-valued speed of sound is expected to be a rough estimation of the visco-thermal losses inside the cavities between the rings (Mi et al., 2021). Mironov and Pislyakov (2020) also used a similar ribbed design in an experimental study, where they considered the effect on the reflection coefficient of adding different damping materials to the last cavities. Their results show that the use of different absorption materials affects the reflection coefficients for low frequencies. However, at higher frequencies, the reflection coefficient is essentially independent of the choice of absorbing material. These findings are similar to the results reported by El Ouahabi et al. (2015a,b). Mironov and Pislyakov (2020) also built another ABH setup, where they consider a cylinder containing a metafluid with a density increase toward the end of the setup. The effective density of the medium inside the cylinder increases by adding layers of some dense material: paper, cardboard, and felt. The concentration of these layers grows toward the end of the cylinder. Again, the measured reflection coefficient agreed reasonably well with the predictions from the theory, however mainly in the lower frequencies (Mironov and Pislyakov, 2020).

It is well known that damping plays a crucial role in the performance of an ABH (Guasch et al., 2017). However, several experimental studies have shown that the addition of absorbing material to the end of the waveguide is insufficient, particularly at higher frequencies (El Ouahabi et al., 2015a,b; Mironov and Pislyakov, 2020). In other experimental studies, a significant reduction of the reflection coefficient is reported, even without considering any absorbing material, with the claim that the main mechanism of the absorption is the visco-thermal losses in cavities (Mi et al., 2021). In this study, we consider the setup used by El Ouahabi et al. (2015a,b), for which we systematically model and estimate the effects of various loss mechanisms. An exact reproduction of the experimental results is not our aim, nor is it possible, due to a lack of data about the experimental setup. However, we will see that the visco-thermal losses cannot be the main absorbing mechanism in this setup, particularly for the high frequencies, and we will suggest another likely candidate as the main damping mechanism. Preliminary results of this study were presented in Mousavi et al. (2021).

Figure 2 shows a 3D visualization of the linear ABH considered by El Ouahabi et al. (2015a,b). It comprises a cylindrical waveguide containing 18 rings (ribs) with retarding inner radius. The rings are made of steel, and the waveguide is sealed by a wooden wall, which functions as a damping layer at the end. In this study, we treat this problem as the axisymmetric problem illustrated in Fig. 3, where $R=115 mm$ is the radius of the waveguide, and $L=255 mm$ is the length of the ABH termination. The rings are $2 mm$ thick and mounted quasi-periodically with increasing distance between them. The rings are positioned according to $xn=5(n+1)+n(n−1)/2 mm$⁠, where x_n is the distance of the center of the nth ring from the wooden wall at the end, and $n=1,2,…,18$ is the ring number. The inner radius of the nth ring is $rn=115xn/255 mm$⁠. The operating frequency range for this setup is up to the cut-on frequency $fc=866 Hz$ of the first non-planar mode, which is circumferential. We thus consider incoming planar waves at the inlet in the frequency range 100–$866 Hz$⁠. We remark that the cut-on frequency for the first non-planar axisymmetric mode is $1803 Hz$⁠.

We consider linear wave propagation of acoustic waves in the air region inside the waveguide. Let P denote the acoustic pressure. Assuming time harmonic wave propagation, we may write $P(x,t)=Re{p(x)eiωt}$⁠, where ω is the angular frequency. The complex amplitude function p of the acoustic pressure satisfies the Helmholtz equation in cylindrical coordinates $x=(r,z)T$⁠,

where $∇=(∂/∂r,∂/∂z), Ω$ is the air region of the waveguide, c is the speed of sound, and $k=ω/c$ the wavenumber. In Eq. (1), we have not included any damping inside the domain. To model the effect of visco-thermal losses at the surface of the rings $Γw$⁠, we use the boundary condition devised by Berggren et al. (2018). This boundary condition is based on a boundary-layer analysis of the linearized, compressible Navier–Stokes equations and effectively compresses the effects of boundary layers into a diffusion–reaction problem on the surface. If the boundary layers are thin in comparison to the dimension of the domain Ω, a condition that is satisfied here, the boundary condition has been shown to very accurately approximate the visco-thermal losses in the audio regime (Berggren et al., 2018; Billard et al., 2021). The validity of this model for the current setup is further discussed in the  Appendix.

The models used for other loss mechanisms are discussed in Sec. II C. The waveguide is considered semi-infinite to the left. To numerically approximate this situation, the waveguide is truncated at an artificial boundary $ΓL$ so that the length of the waveguide inlet is $115 mm$ (see Fig. 3). Treating the wooden wall on the right as sound-hard material, we obtain the boundary-value problem

where n is the local, outward-directed, unit normal vector field. The artificial boundary condition (2c) ensures that left-going plane waves are absorbed and specifies a plane right-going wave with unit amplitude at $ΓL$ as the incoming wave. Boundary condition (2d) constitutes $O(δV)$ and $O(δT)$ perturbations of the sound-hard wall condition (2b) to take into account the visco-thermal losses at the ring surfaces. In condition (2d), $∇T$ is the tangential gradient operator defined by

and δ_V and δ_T are the viscous and thermal boundary-layer thicknesses, respectively, given by

where ν is the kinematic viscosity coefficient, κ the thermal conductivity, and c_p the specific heat capacity at constant pressure.

We apply the finite element method to discretize and solve boundary-value problem (2) using a uniform mesh of square elements on the domain Ω. Let $φj, j=1,2,…,N$⁠, be bi-quadratic shape functions, where N is the number of degrees of freedom in the finite element approximation, and $V=span{φ1,φ2,…,φN}$⁠. The finite element approximation of boundary-value problem (2) can then be written as follows:

The matrix form of problem (5) reads

where $p=[p1,p2,…,pN]T$ is the vector of nodal values of the complex acoustic pressure amplitude, and $1N=[1,1,…,1]T$ is a vector of length N. Also, the mass M and stiffness K matrices have components

respectively, and the boundary matrices $ML$ and $KL$ over $ΓL$ and $Mw$ and $KTw$ over $Γw$ have components

respectively.

Similarly as for the ABHs in beams and plates, it is not possible to manufacture the ideal design of an ABH for waveguides. Motivated by the success of the added damping material in structural ABHs, El Ouahabi et al. (2015a,b) added two different sources of damping to their setup to minimize the truncation error in their experiment. These are

• Damping material that fills the cavities between the last rings,

• A wooden wall that acts as an absorbing surface at the end of the waveguide.

The first of these introduces damping to the waves as they travel through the region filled with damping material. We use Rayleigh damping to model this effect. The second damps waves as they get reflected from a non-sound-hard surface, such as the wooden wall at the end of the ABH. To model this effect, we use a surface impedance boundary condition. In this section, we explain how these models of damping sources can be incorporated into state Eq. (6).

We need to simulate the effects of adding a layer of damping material in a part of the domain, $ΩD$⁠. Although not accurate as a model of damping effects in fibrous materials, a convenient way is to add a Rayleigh damping term. The inclusion of such a term in state Eq. (6) yields

The damping matrix C has components

where α and β are parameters to be specified. The values of α and β provide the mass and stiffness proportional contributions to the damping process, respectively.

The specific surface impedance is the quotient between the complex amplitudes of acoustic pressure and the normal velocity of a surface. It will be convenient for our purposes to nondimensionalize the impedance by the plane wave impedance ρc and define

where P is the acoustic pressure, u is the surface velocity, n is the outward unit normal vector on $ΓZ$⁠, and ρ and c are the density of air and the speed of sound, respectively. From here on, whenever the term surface impedance is used, it means the dimensionless specific surface impedance Z defined above.

The surface impedance boundary condition models the effects of a locally and linearly reacting surface. In preparation for the power balance analysis below, we consider boundary-value problem (2) with an added surface impedance condition on $ΓZ$ as

Since the walls considered here are acoustically passive, the real part of Z must be positive. As illustrated in Fig. 4, the values of the real and imaginary parts of Z are typically frequency-dependent. Here, we consider the wooden wall on the right to be equipped with surface impedance Z^R; that is, we let $ΓZ≡ΓR$ and $Z≡ZR$ in boundary condition (12e). Thus, the matrix form of boundary-value problem (12) reads

where $MR$ is the boundary mass matrix defined in the same way as the matrix $ML$ is defined in Eq. (8a), but with $ΓL$ replaced by $ΓR$⁠.

Here, we analyze how the input power is distributed among the constituents of the model when the Rayleigh damping term and the surface impedance boundary condition are included. Consider a slightly generalized version of the boundary-value problem (12), where boundary condition (12c) is replaced with

where $pin$ is the complex amplitude of an incoming planar wave at $ΓL$ [$pin=1$ in boundary condition (12c)]. Including also the proportional damping term, the corresponding finite element problem reads as follows:

Choosing $q=p¯$ in expression (15), where the bar signifies complex conjugate, we obtain

By the properties of complex numbers, it holds that

Taking the imaginary part of expression (16), dividing by k, and using formula (17) for the integrals on $ΓL$⁠, we find that

or, equivalently,

Equality (19) expresses that the input power provided through the boundary condition on $ΓL$ is split into a reflected portion and a portion dissipated in $ΩD$ and at $ΓZ$ and $Γw$⁠, respectively.

To appreciate and explain the somewhat surprising results we obtain in Sec. III, it is instructive to review and extend the model that was used by Mironov and Pislyakov (2002) to conceive the design of Fig. 2. Their waveguide design was based on the analysis of a one-dimensional model of the acoustics in a tube with a varying cross section and a varying acoustic admittance at the tube wall. Assuming that the acoustic pressure p varies only in the axial direction x, Mironov and Pislyakov (2002) show that the balance of momentum and conservation of mass lead to the generalized Webster equation

where $r̂(x)$ is the tube radius at x, and Y is the tube wall nondimensional specific admittance, that is, $Y=1/Z$⁠, with Z from definition (11) with u being the tube wall velocity u_W.

In the waveguide considered here, $x↦r̂(x)$ is a linear function, tracing out the inner radii of the rings (Fig. 3). Assuming that the dimensions of the setup are small with respect to the wavelength (low frequency assumption), Mironov and Pislyakov (2002) used $Y=Yc$ for the tube wall, where $Yc$ is the purely compliant wall admittance

This formula can be derived by considering two rings, both of inner radius $r̂$⁠, positioned close to each other, as in Fig. 5, forming a volume $V=π(R2−r̂2)t$⁠, where t is the distance between the rings. If the volume is small compared to the wavelength, then a massless membrane at the outlet $r̂$ of the volume moving radially with velocity u_W will cause an isentropic compression inside the volume. Treating air as an ideal gas, the quantity $PVγ$ will be constant, where P is the total pressure (static plus time-varying) and γ is the heat capacity ratio. Differentiating with respect to time and linearizing the isentropy law $PVγ$⁠, we obtain, after some algebra, expression (21). The equation obtained by substituting the admittance formula (21) into Eq. (20) is analyzed by Mironov and Pislyakov (2002), and the waveguide is shown to possess the ABH property of phase and group velocities decreasing toward the end of the waveguide.

Remark 1: The notation in this section is made consistent with the rest of the article, which is why it differs slightly from that used by Mironov and Pislyakov (2002); in particular, we use the opposite phase convention, $+iω$⁠, and a nondimensional admittance.

The purely compliant admittance of expression (21) constitutes a low frequency limit. We will see that it will be relevant to consider a more general admittance expression, not considered by Mironov and Pislyakov (2002), which also takes inertial effects into account and thus holds for higher frequencies. For such an analysis, consider again the geometry of Fig. 5. In the analysis leading up to admittance of expression (21), the acoustic pressure p was assumed to be constant within the gray-tinted region. Now we will allow a dependency of p on the radial direction r; however, we consider small enough t so we can ignore pressure variations in the axial direction. Assuming a radial wall velocity amplitude u_W applied at $r=r̂$⁠, the acoustic pressure inside the gray region will then satisfy the boundary-value problem

The general solution to Eq. (22a) is

where J₀ and Y₀ are the zeroth-order Bessel functions of the first and second kind. The coefficients A and B are determined by boundary conditions (22b) and (22c), which yield the linear system

where $Â$ and A, and $B̂$ and B are related by $A=ρcuWÂ$ and $B=ρcuWB̂$⁠, respectively. Equation system (24) is solvable as long as the determinant of the coefficient matrix does not vanish, a case that does not happen in the parameter regime considered here. Finally, the tube wall admittance emanating from model (22) will then become

which will be purely compliant and purely inertial below and above resonance, respectively. The same formula has been derived by Sharma (2017) and stated explicitly as (Sharma et al., 2017)

The differences between admittances (21) and (25) will be discussed in Sec. III.

In this section, we study the reflection coefficient as a function of the frequency of the incoming wave for different damping scenarios. Denote by $pin$ the complex amplitude of the incoming wave at $ΓL$⁠, as provided by boundary condition (14). (In practice, we set $pin=1$⁠.) Since only planar modes are propagating and due to the presence of an inlet section (Fig. 3), the acoustic pressure will be essentially constant at $ΓL$⁠, which is why we may define the reflection coefficient at $ΓL$ as

First, we simulate only the effect of the absorbing wooden wall at the end of the ABH. Then we add the term for the Rayleigh damping to model the effect of adding damping material to the last cavity in the waveguide. We vary the coefficients in the damping models to represent the effect of increasing damping at the end of the waveguide and observe how this affects the reflection coefficient.

Figure 6 shows the reflection coefficient for three different choices of surface impedance on $ΓR$⁠. Since data about the exact material used as the wooden wall by El Ouahabi et al. (2015a,b) are no longer available, we here consider the following three values for the impedance of the wooden wall on $ΓR$⁠. First, we consider the impedance of the commercial damping material in Fig. 4. Then we use a simplified model of a different commercial material from Pierce (2019), and at last we assume perfect planar wave absorption at $ΓR$⁠. Later, we show that the values of the impedance of the wooden wall at $ΓR$ do not affect the ABH effect in the waveguide. Thus, considering any other material with a different acoustic impedance at $ΓR$ will not affect our conclusion. In the first two cases, the impedance of the wall is frequency-dependent. The solid line in Fig. 6 shows the reflection coefficient for the particular frequency dependency of Fig. 4. The long dashed line corresponds to the surface impedance $ZR=3−i2000f−1$⁠, where f is the frequency. The densely dashed line is the reflection coefficient when $ZR=1$⁠, which yields perfect absorption of planar waves on $ΓR$⁠. We know that in a waveguide ABH, the critical frequency, that is, where the reflection coefficient starts to decrease rapidly, is roughly $fcrit=c/πL$⁠, where L is the ABH length (Mi et al., 2021). We thus expect that the reflection coefficient should decrease rapidly for frequencies above $fcrit=425$ Hz. However, Fig. 6 shows that for frequencies above $fcrit$⁠, there is essentially total reflection in all three cases. Even for the frequencies below $fcrit$⁠, the reflection coefficient is mostly above 0.85, except for a very small range of frequencies close to 100 Hz. Thus, the impedance of the wooden wall at $ΓR$ does not affect the reflection coefficient for a large band of frequencies. In particular, for frequencies above $fcrit$⁠, we obtain total reflection even in the case of a perfect absorber at $ΓR$⁠.

Next, we consider filling the last cavity of the waveguide with damping material modeled by Rayleigh damping. As noted above, the reflection coefficient is essentially insensitive to the choice of the surface impedance of the wall on $ΓR$⁠. We thus only consider the case $ZR=3−i2000f−1$⁠. Then we choose various coefficients for the Rayleigh damping model to study the effect of increasing the attenuation of waves in the damping material layer in the last cavity. Figure 7 shows that the added layer of damping material reduces the reflection coefficient for the frequencies below $fcrit$⁠, but it does not affect the reflections for any frequency above $fcrit$⁠. The graph remains essentially the same independent of the values of the coefficients in the Rayleigh damping or the impedance of the wall on $ΓR$⁠. Thus, the incoming waves with frequencies above $fcrit$ are affected by neither the wooden wall on the right nor the damping material added to the last cavity.

As discussed above, the choice of coefficient for the Rayleigh damping model and the impedance of the boundary condition at $ΓR$ has essentially no effect on the reflection coefficient, particularly at higher frequencies. Thus, in the next step, we only consider the case $ZR=3−i2000f−1,α=1,$ and $β=10−4$ and add the effect of the visco-thermal losses on the surface of the rings. Figure 8 shows that the visco-thermal losses are much more effective than the previous mechanisms and help reduce reflections for all frequencies. However, the effect gradually fades out toward higher frequencies. Also, for frequencies above $fcrit$⁠, the reflection coefficient starts to increase, which is the opposite behavior from what is expected in an ABH. This leads to a significant difference in the reflection coefficient compared to the experimental results by El Ouahabi et al. (2015a,b).

To further illustrate the effect of the different damping mechanisms considered so far, Fig. 9 shows the normalized power loss associated with the different loss mechanisms. The normalized power loss of each source of damping may be computed by dividing the corresponding term in power balance (19) by the input power. For frequencies above $fcrit$⁠, the contribution of the Rayleigh damping term and the surface impedance at $ΓR$ are essentially zero. Also, the contribution of the visco-thermal losses decreases for frequencies above $fcrit$⁠. In total, this is in stark contrast to substantial damping at higher frequencies that can be inferred from the experiments (Fig. 1). It is therefore reasonable to ask whether there is some other loss mechanism that helps with the attenuation. In the experimental setup, the rings are made of 2 mm thick steel (El Ouahabi et al., 2015a,b), so losses due to structural vibrations of the rings should be small.

Before stating any hypothesis about other sources of damping in the waveguide, we take a closer look at why the incoming waves at higher frequencies are not affected by the damping layer at the end of the waveguide. A first clue is given by Fig. 10, which shows the distribution of the absolute value of the complex acoustic pressure amplitude $|p|$ for seven different frequencies in the case where $ZR=3−i2000f−1$ on $ΓR$ and the Rayleigh coefficients α = 1 and $β=10−4$⁠. For higher frequencies, the pressure wave is reflected before it reaches the end of the waveguide. Thus, the addition of damping material in the last cavity and an absorbing wall at the end of the waveguide will not affect the propagation of the waves. In addition, we notice an apparent local frequency-dependent resonance in specific cavities.

Recall that Jiménez et al. (2017) studied a similar setup, but with clearly designed Helmholtz resonators, each responsible for attenuation at specific frequencies. The cavities between the rings in our setup also function as resonators, even though that effect was not intended in the original design based on the purely compliant wall impedance (21). Indeed, Fig. 10 shows that for higher frequencies, the distribution of pressure inside each cavity is no longer uniform in the radial direction, as assumed in the derivation of wall impedance (21); the maximum amplitude of pressure is at the top end of each cavity close to the outer tube.

The pressure amplitude plots of Fig. 10 are difficult to reconcile with the admittance model (21) that was originally used to analyze the current setup. The admittance model (21) is purely compliant and cannot support the apparent presence of resonances in the cavities in Fig. 10. Figure 11 compares the frequency responses of the wall admittance expressions (21) and (25) for $r̂=31$ mm. We note that these expressions agree well for low frequencies but that a resonance occurs around 772 Hz, where $|Yr|→+∞$ since $p(r̂)$ vanishes. This resonance is an analogue in radial symmetry of an open pipe resonance with a pressure peak and node at r = R and $r=r̂$⁠, respectively. By numerically solving the equation

with Newton's method, we can predict the inner radius $r̂$ of the cavity for which a resonance should occur for a given wavenumber. Table I shows the results of such a computation for the wavenumbers corresponding to the frequencies of Fig. 10 where clear resonances occur. We see that these numbers agree reasonably well with estimates obtained from the pressure plots.

The existence of these local resonances motivates the idea of an additional source of damping associated with the outer tube surface. Due to the high local pressure amplitudes, there may be significant losses associated with a finite surface impedance of the outer tube, associated with a term like the second last one in power balance (19). We note that a plastic material is used for the outer tube in the experiments (El Ouahabi et al., 2015a,b), which may be a source of structural damping as well as losses associated with sound radiation.

Thus, we examine the effect of including an outer tube $ΓT$ with a finite surface impedance Z^T. Here, we set $ZR=3−i2000f−1$ on $ΓR, ZT=5ZR$ on $ΓT$⁠, and the Rayleigh coefficients α = 1 and $β=10−4$⁠. Recall that a larger value of the surface impedance corresponds to a boundary condition closer to the lossless sound-hard wall condition. We thus consider the value of the impedance of the outer tube to be much larger than the value of the impedance of the wall on the right. Figure 12 shows that even for this large value of Z^T, which corresponds to the wall on the top being just slightly lossy, we obtain a significant reduction of the reflection coefficient. The effect of the finite surface impedance of the outer tube is more pronounced for higher frequencies, which is consistent with the experimental results, where the lowest values of the reflection coefficient occur at the higher operating frequencies (El Ouahabi et al., 2015a,b). We note that if we compare the results in Fig. 12 with the experimental results by El Ouahabi et al. (2015a,b) in Fig. 1, there is still a clear difference. This is because the small damping introduced to the sound waves by considering the impedance boundary condition Z^T is not an accurate model of the vibro-acoustic losses in the outer tube. However, in the current study, we do not aim to reproduce the experimental results, but rather we aim to investigate which of the possible sources of damping are likely to provide the low reflection coefficients at higher frequencies. Figure 13 shows the normalized power loss of the different loss mechanisms for the final experiment. The results show that most of the attenuation of the acoustic power is due to the finite impedance of the outer tube. Particularly at frequencies above $fcrit$⁠, the losses at $ΓT$ are the dominant loss mechanism.

Based on the above observation, we conclude that adding a small layer of damping material to the region close to the outer tube, at the end of all cavities, should significantly reduce the reflection coefficient. To provide further support for this conclusion, in the final experiment, we examine the effect of including a thin layer of porous material to this sensitive region close to the outer tube. First, we consider a sound-hard boundary condition at all walls, including the wooden wall at $ΓR$ and the outer tube $ΓT$⁠. This means that the visco-thermal losses are the only source of damping in the setup. Next, we add a layer of melamine foam with the acoustic properties given by Kino and Ueno (2008) (sample number 31) close to the outer tube with the thickness of 20 mm as in Fig. 14. The simulations for this final experiment are performed in COMSOL Multiphysics using the Johnson–Champoux–Allard poroacoustic model provided in the acoustics module. Figure 15 shows a significant reduction of the reflection coefficient especially at frequencies above $fcrit$⁠. We note that at a few frequencies, with the use of the ABH effect, close to perfect absorption is achieved using a subwavelength thickness of porous material (1/40 of the sound wavelength at $fcrit$ = 425 Hz).

We have numerically analyzed the contributions of various sources of damping to the function of a waveguide ABH. The results show that the addition of damping material to the end of the waveguide, motivated by the analogy to the ABH effect in beams, is not effective as a means to reduce the reflection coefficient. More precisely, the use of an absorbing wall at the end of the waveguide and added damping material in the last cavities only reduces the reflection coefficient for low frequencies, below the critical frequency for the ABH. In contrast, the simulations show that the visco-thermal losses in the cavities between the rings reduce the reflection coefficient at all frequencies. However, this effect diminishes with increasing frequency. Thus, at higher frequencies, taking only the above effects into account, the reflection coefficient is still much larger than what has been reported in the experiments by El Ouahabi et al. (2015a,b). We thus conclude that there should be another loss mechanism that causes the low reflection coefficient at higher frequencies.

Plots of the pressure for different frequencies reveal that at higher frequencies, the pressure distribution in each cavity is not uniform, and there are clear cavity resonances whose axial locations are frequency-dependent. We show that the location of these resonances as a function of frequency can be predicted, within an acceptable precision, by a model of wall admittance that allows for radial varying pressure in each cavity.

Moreover, the high localized pressure amplitudes in the cavities due to the resonances support the hypothesis that a finite surface impedance of the outer tube is the primary source of damping in the ABH. Thus, we consider a slightly absorbing wall on the top boundary of the axisymmetric model. The results show that this small amount of damping on the outer tube reduces the reflection coefficient very efficiently in the whole range of frequency, with a larger impact for the higher frequencies, just as in the experimental results. Based on our results, we suggest that any future studies on the ABH effect in waveguides, using the ribbed design considered here, should consider adding a small layer of damping material at the end of each cavity, close to the outer tube. In a final numerical experiment, we showed that by adding a relatively thin (with respect to the wavelength) layer of porous material close to the outer tube, the reflection coefficient can be efficiently reduced with almost perfect absorption achieved for few frequencies above $fcrit$⁠.

This work was supported by the Swedish strategic research programme eSSENCE and Swedish Research Council Grant No. 2018-03546.

Here, we discuss the validity of the acoustic boundary condition of Berggren et al. (2018) used in this study to model visco-thermal boundary-layer losses. As mentioned earlier, one of the main assumptions in this model is that the dimensions of the setup are such that the boundary layers are “thin,” in the sense that boundary layers at opposing walls should not overlap. We know that the thickness of the acoustic boundary layers increases for lower frequencies [see expressions (4)]. Therefore, to ensure non-overlapping boundary layers, we check the thickness at the lowest frequency, 100 Hz. Figure 16 shows that at this frequency, the boundary layers are indeed thin in comparison to the dimensions of the domain, and there is no risk for overlapping boundary layers here. Thus, we expect the acoustic boundary-layer model to be accurate.

Moreover, we numerically validate the results of the acoustic boundary-layer model by performing full linearized compressible Navier–Stokes (FLNS) simulations, in which non-slip and isothermal boundary conditions (including the wooden wall at $ΓR$⁠) are assigned at all solid walls. The FLNS simulations are performed in COMSOL Multiphysics using the “thermoviscous acoustics interface” in the acoustics module. Figure 17 verifies that the acoustic boundary-layer model approximates the visco-thermal losses in the setup very accurately.