Theoretical requirements for broadband perfect absorption of acoustic waves by ultra-thin elastic meta-films

We derive and numerically demonstrate that perfect absorption of elastic waves can be achieved in two types of ultra-thin elastic meta-films: one requires a large value of almost pure imaginary effective mass density and a free space boundary, while the other requires a small value of almost pure imaginary effective modulus and a hard wall boundary. When the pure imaginary density or modulus exhibits certain frequency dispersions, the perfect absorption effect becomes broadband, even in the low frequency regime. Through a model analysis, we find that such almost pure imaginary effective mass density with required dispersion for perfect absorption can be achieved by elastic metamaterials with large damping. Our work provides a feasible approach to realize broadband perfect absorption of elastic waves in ultra-thin films.

In this work, we analyze the mechanism and theoretical requirements to achieve broadband perfect absorption for elastic waves by using an ultra-thin elastic meta-film. Based on transfer matrix theory, we find out two types of ultra-thin films with such possibility. One requires one a large value of almost pure imaginary effective mass density and a free space boundary, while the other requires a small value of almost pure imaginary effective modulus and a hard wall boundary. In the former case, the displacement is almost a constant across the ultra-thin film, while the stress tends to zero abruptly. In the latter case, the situation is the opposite with the stress being almost a constant and the displacement tending to zero abruptly. We further show that when the almost pure imaginary effective mass density is inversely proportional to frequency, or the almost pure imaginary effective modulus is proportional to frequency, the perfect absorption effect becomes broadband. Through a simple model analysis, we demonstrate that elastic metamaterials with large damping can be designed to realize such effective media with almost pure imaginary parameters with the required frequency dispersions in certain frequency regimes, therefore providing a feasible approach for broadband absorption of elastic waves.
We consider elastic waves with angular frequency ω propagating along the z direction, which are incident on an ultra-thin film embedded in a background. Here, we assume the background and the film are both composed of isotropic solids. Therefore, when the incidence is in the normal direction, the transverse and longitudinal waves are uncoupled. For simplicity, in the following we only consider longitudinal waves of normal incidence. Similar conclusions can be obtained for transverse waves.
If we define r u (r τ ) and t u (t τ ) as the transmission and reflection coefficients through the ultra-thin elastic meta-film, with respect to the displacement (stress), respectively. Then, the transmission and reflection may be classified into four cases. In case 1, both the stress τ and displacement u are almost constants across the film. In this case, the transmission is almost unity and the reflection is almost zero. And there is no absorption. In case 2, the displacement u is almost a constant across the film, while the stress τ varies abruptly across the film. In this case, we have In case 3, the stress τ is almost a constant across the film, while the displacement u varies abruptly across the film. In this case, we have In case 4, both the stress τ and displacement u vary abruptly across the film. In this case, we have 1 + r τ ≠ t τ and 1 + r u ≠ t u . Actually, case 4 indicates that the wavelength inside the film is generally comparable to the thickness of the film, therefore resulting in a non-negligible phase change across the film. In this work, we will focus on cases 2 and 3. As we shall show later, such cases exhibit interesting possibilities of achieving broadband perfect absorption, while case 4, though also capable of achieving perfect absorption, is usually limited to a narrow bandwidth due to the resonance effect.
In the following, we first analytically derive the conditions for the elastic waves to be perfectly absorbed based on a transfer matrix approach 42,43 . We denote the mass density, Lamé's first and second parameters of the ultra-thin film (background medium) to be ρ(ρ 0 ), λ(λ 0 ) and μ(μ 0 ), respectively.
We define a transfer matrix as, Here k p is the wave number of longitudinal wave, d is the thickness of the ultra-thin film. And δ p (= k p d) is the phase change across the film. Thus, the reflection and transmission coefficients can be derived as,  where M p,mn is the element in m-th row and n-th column of the transfer matrix M p . k p,0 2 is the wave number in the background medium. Here, we assume the phase change δ p across the film is negligible, i.e.
Now, we investigate the requirement of the parameters of the ultra-thin film and background in the cases 2 and 3 mentioned before. By substituting the reflection and transmission coefficients in Eq. (7) into Eqs (1) and (2), we obtain, for cases 2 and 3, respectively. Since the ultra-thin film has ωd ≪ 1, thus traditional solids have 2ω 2 ρd ≈ 0 and k d 2 . This indicates that an ultra-thin film made of traditional solids will exhibit almost constant displacement and stress across the film, leading to case 1 with no absorption. In order to realize cases 2 and 3 with absorption, from Eqs (8) and (9), we find that either the effective mass density of the film must be unusually large (case 2), or the effective modulus must be unusually small (case 3).
However, in cases 2 and 3, when the environment is symmetric (i.e., the media in the incident and transmitted regions are the same), from Eqs (1) and (2), we can easily find that the ultra-thin film has a maximal absorption rate of 50% with − r u ≈ t u ≈ 0.5 or − r τ ≈ t τ ≈ 0.5 44 .
In order to obtain perfect absorption, we must break the symmetry. Here, we employ a free space boundary or a hard wall boundary attached to the ultra-thin elastic meta-film, as illustrated in Fig. 1(a,b). Free space boundary enforces zero stress, while hard wall boundary enforces zero displacement. Therefore, free space boundary is associated with case 2, in which the displacement is almost a constant while the stress can decrease sharply to zero across the film. Hard wall boundary is associated with case 3, in which the stress is almost a constant while the displacement can decrease sharply to zero across the film. When the free space boundary is applied, the total reflection coefficient of the whole system is R u t t  44 . This implies that when the ultra-thin elastic meta-film satisfy t u = t τ = 2/3 and r u = − r τ = − 1/3, perfect absorption can be achieved. On the other hand, when the hard wall boundary is applied, perfect absorption can be obtained when the ultra-thin elastic meta-film satisfies t u = t τ = 2/3 and r u = − r τ = 1/3. Thus, from Eqs (7)-(9), the effective parameters of the ultra-thin film for perfect absorption can be derived as, for case 2 with a free space boundary, and case 3 with a hard wall boundary, respectively. From Eqs (10) and (11), we can see that when the imaginary effective mass density is inversely proportional to frequency, or the effective modulus is proportional to frequency, the perfect absorption condition can be satisfied in a broad range of frequencies, leading to broadband perfect absorption. As we shall prove later in model analysis, this type of dispersion is possible in damping elastic metamaterials. Two schematic graphs are shown in Fig. 1(a,b) to describe the physical picture of the two types of ultra-thin films, respectively.
Another interesting fact that is worth noting is that Eqs (10) only requires a particular value of imaginary effective mass density for the case of constant displacement, while the effective modulus of the film can vary within a relatively large range, as long as Eqs (1) and (8) are satisfied. Similarly, Eq. (11) only requires a particular value of imaginary effective modulus for the case of constant stress. Any effective mass density is fine as long as Eqs (2) and (9) are satisfied. This property allows us to focus on only one parameter instead of both of them, which greatly simplifies the design process.
Likewise, we can also derive the requirement of effective parameters for transverse waves as for the cases with almost constant displacement and shear stress, respectively. From Eqs (10)-(13), one may also find that generally the longitudinal and transverse waves cannot be perfectly absorbed simultaneously. In order to achieve perfect absorption, we require the effective mass density of the film to be anisotropic for case 2, or the effective modulus of the film to satisfy In order to verify the analytical results, we perform numerical simulations based on finite element software, COMSOL Multi-physics, as shown in Fig. 2. The background material in the incident region is selected as epoxy with λ 0 = 4.428 × 10 9 Pa, μ 0 = 1.590 × 10 9 Pa and ρ 0 = 1180 kg/m 3 . The incident waves from left are longitudinal waves with a working frequency of 500 Hz. Therefore, the wavelength in epoxy First, we consider the u-constant case. From Eq. (10), we obtain the required large imaginary mass density as ρ = i9538 kg/m 3 . Since there is no strict requirement on the modulus, we choose the same values of epoxy, i.e., λ = 4.428 × 10 9 Pa and μ = 1.590 × 10 9 Pa. In Fig. 2(a,b), we show the distributions of real part of displacement Re(u z ) (color), normalized amplitudes of stress |τ zz |/|τ zz,in | (red solid lines) and displacement |u z |/|u z,in | (blue solid lines). u z,in and τ zz,in are, respectively, the normal displacement and normal stress of incident waves. Figure 2(a) demonstrates the transmission of longitudinal waves through the ultra-thin film embedded in epoxy, while Fig. 2(b) demonstrates the perfect absorption of Figure 2. Simulations of perfect absorption. Snapshots of the real part of displacement Re(u z ) (color), and normalized amplitudes of stress |τ zz |/|τ zz,in | (red solid lines) and displacement |u z |/|u z,in | (blue solid lines) for cases of (a) an ultra-thin film of imaginary mass density in epoxy, (b) with free space on the right side of the film. The parameters of the ultra-thin film are λ = 4.428 × 10 9 Pa, μ = 1.590 × 10 9 Pa, ρ = i9538 kg/m 3 and d = 0.1 m. Snapshots of real part of stress Re(τ zz ) (color), and normalized amplitudes of stress |τ zz |/|τ zz,in | (red solid lines) and displacement |u z |/|u z,in | (blue solid lines) for cases of (c) an ultra-thin film of imaginary bulk modulus in epoxy, (d) with hard wall on the right side of the film. The parameters of the ultra-thin film are λ = μ = − i3.138 × 10 8 Pa, ρ = 1180 kg/m 3 and d = 0.1 m. (e) Dependence of absorptance on the incident angle for the u-constant case (green lines) and τ-constant case (magenta lines) with the same material parameters as those in (b) and (d), respectively. The incident waves are longitudinal waves with working frequency 500 Hz.
wave energy when a free space boundary is attached to the right side of the film. From the normalized amplitudes in Fig. 2(a), it is observed that the transmission coefficient is about 2/3. The displacement is almost a constant across the film while the stress experiences an abrupt change due to large mass density. In Fig. 2(b), it is observed that when a free space boundary is attached, almost all the incident wave energy is absorbed. There are almost no reflected waves, as can be deducted from the nonexistence of variance in the normalized amplitudes in the incident region in Fig. 2(b).
Since there is no strict requirement on the mass density, we choose the same value of epoxy as ρ = 1180 kg/ m 3 . In Fig. 2(c,d), the distributions of Re(u z ), |τ zz |/|τ zz,in |, and |u z |/|u z,in | are presented. Figure 2(c) shows the transmission of longitudinal waves through the ultra-thin film embedded in epoxy, while Fig. 2(d) demonstrates the perfect absorption of wave energy when a hard wall boundary is attached to the right side of the film. From the normalized amplitudes in Fig. 2(c), it is observed that the transmission coefficient is also about 2/3. However, unlike the case in Fig. 2(a), the stress is almost a constant across the film, while the displacement experiences an abrupt change due to the small modulus. In Fig. 2(d), it is observed that when a hard wall boundary is attached, almost all the incident wave energy is absorbed. There are almost no reflected waves, which can be deducted from the nonexistence of variance in the normalized amplitudes in the incident region in Fig. 2(d).
Moreover, we calculate the incident angle-dependent absorptance in Fig. 2(e) for the u-constant case (green lines) and τ-constant case (magenta lines) with the same material parameters as those in Fig. 2(b,d), respectively. It is seen that large absorption can be obtained in a wide range of incident angle. Therefore, the above numerical simulations coincide excellently with our analytical results. Although we only verify the longitudinal waves, the perfect absorption of transverse waves can be easily confirmed in a similar process.
Previously we have demonstrated that an ultra-thin film can with large imaginary mass density and a free space boundary, or with small imaginary modulus and a hard wall boundary can achieve perfect absorption of elastic waves. When the imaginary mass density is inversely proportional to frequency, or the imaginary modulus is proportional to frequency, such perfect absorption effect can be broadband. However, how to realize such imaginary parameters remains an unresolved issue. It is known that positive imaginary value of mass density and negative imaginary value of modulus correspond to absorption 29 . However, in most natural materials, the absorption is relatively small, rendering the parameters having relatively larger real parts than imaginary parts. In the following, we will propose a model of damping elastic metamaterials exhibiting effective mass density proportional to i/ω, therefore enabling the ability of broadband perfect absorption of elastic waves.
As illustrated by the inset of Fig. 3(a), we propose a simple one-dimensional mass-spring-mass model composed of a background mass M I embedded with an inner mass M II , with large damping induced by the frictional losses between the two masses. We assume that the frictional losses are proportional to the velocity. Thus, according to Newton's second law, we have, large contrast masses and small γ. Especially, our analysis only applies to the linear regime with small displacements. However, there are also some advantages for elastic waves. For electromagnetic waves, the perfect magnetic conductor boundary is required, which is inherently narrow band [46][47][48][49] . However, for elastic waves, both free space and hard wall boundaries are naturally broadband, which makes the realization of broadband absorption of acoustic and elastic waves with an ultra-thin film easier than that of electromagnetic waves.
For conclusions, we have theoretically proved and numerically demonstrated the absorption of elastic waves in ultra-thin films with either imaginary large mass density and a free space boundary, or imaginary small modulus and a hard wall boundary. Broadband perfect absorption can be achieved when the frequency dispersions of the imaginary mass density or modulus can be inversely proportional to or proportional to the frequency in a certain frequency regime. We demonstrate that elastic metamaterials with large damping provides a feasible approach to realize the imaginary mass density with suitable dispersions for broadband absorption. Therefore, ultra-thin films composed of such metamaterials can in principle achieve broadband perfect absorption of elastic waves.