Modeling elastic wave mode conversion within zero-phase-difference ultrathin anisotropic medium

We investigate a novel mechanism for elastic wave mode conversion in structured media composed of periodic resonant scatterers. Traditional models typically rely on phase accumulation between longitudinal and transverse wave components as they propagate through an anisotropic layer. This limits their effectiveness in the low-frequency or thin-layer regime. In contrast, we propose a new model based on oblique resonance within a homogenized anisotropic block, which generates oblique displacements and introduces tangential motion at the interface. It is carried out under the limiting condition where the scatterer thickness approaches zero. Previous studies did not consider this zero-thickness limit. If one directly substitutes d = 0 into their formulations, the result becomes a trivial solution.This fundamentally differs from existing literature that derive methods for finite-thickness structures. We derive the boundary conditions and establish a coupled system of equations to describe the transmission and reflection behavior. Effective parameters, including mass density and impedance, are extracted from field quantities within the computational region and are shown to be angle-dependent. The conversion rates predicted by our model show excellent agreement with simulations, confirming both the physical assumptions and the analytical formulation. The proposed approach provides a new pathway for low-frequency, compact, and efficient control of elastic wave modes.

Introduction

The manipulation of elastic waves, particularly the mode conversion between longitudinal (P) and transverse (S) waves, plays a crucial role in applications such as acoustic metamaterials, ultrasonic imaging, and vibration isolation (Liu et al., 2000; Zhao et al., 2007; Wu et al., 2011; Zhu et al., 2014; Ma et al., 2016; Li et al., 2017; Zheng et al., 2019; Lee et al., 2022). Traditionally, elastic wave mode conversion occurs only under oblique incidence due to symmetry constraints at solid interfaces, which prohibits conversion at normal incidence. This limitation significantly restricts the design and functionality of compact and low-frequency elastic wave devices. Recent advances in elastic metamaterials have enabled mode conversion at normal incidence by exploiting anisotropic distributions of mass density and elastic moduli. These metamaterials—designed with engineered microstructures—act as homogeneous anisotropic media, permitting complete P-to-S and S-to-P conversion under specific conditions. However, such designs often require material thickness comparable to the wavelength, posing challenges in low-frequency regimes where wavelengths are inherently large. To overcome this, recent studies have proposed locally resonant structures capable of inducing efficient mode conversion at frequencies two orders of magnitude lower than those of traditional designs, with subwavelength-thickness devices. While promising, these mechanisms are still underexplored and lack a comprehensive theoretical model.

In this work, we develop a theoretical framework to analyze elastic wave propagation and mode conversion in ultrathin anisotropic media inspired by locally resonant metamaterials. By modeling asymmetric resonant units as zero-thickness anisotropic layers and applying effective medium theory, we derive analytical expressions for reflection and transmission coefficients. Our model not only matches numerical simulations but also reveals key physical insights into the reversible nature of mode conversion and the influence of resonator geometry. These findings provide theoretical guidance for designing compact, efficient elastic wave devices for low-frequency environments, demonstrating broad application prospects. Specifically, the breakthrough zero-thickness resonance mechanism proposed in this study enables elastic wave conversion devices to effectively overcome the limitations of traditional structures in low-frequency and thin-layer applications, paving the way for innovations across multiple core technology fields: In industrial non-destructive testing, such devices can achieve precise identification of ultra-thin structures and hidden defects; In biomedical applications, they offer new pathways for developing multimodal ultrasound imaging and targeted therapy technologies; In high-precision equipment, they provide effective solutions for vibration mode filtering and isolation; Additionally, this technology lays theoretical foundations for mode diversity multiplexing in future chip-level phonon communication systems. Furthermore, in seismic protection, this mechanism provides innovative strategies for designing novel metamaterials based on wave mode conversion, expanding potential approaches for vibration damping technologies. In this work, we overcome the aforementioned limitations by proposing a model that remains effective even in the limit of zero-thickness layers, enabling mode conversion under conditions where traditional approaches fail. In the present study, we only analyzed the simplest case—vertical incidence without considering loss—for clarity of theoretical derivation. In future work, we plan to extend the model to more general cases including oblique incidence and complex parameters.

Consider a row of anisotropic resonant units embedded in epoxy matrix, as shown in Figure 1a. The unit is composed of a steel cylinder coated with elliptical soft rubber embedded in an epoxy matrix. The used materials parameters are: ${\rho }_{\mathrm{s}}=7900\text{kg}/{\mathrm{m}}^3$ , $M_{\mathrm{s}}=266\text{GPa}$ , and $G_{\mathrm{s}}=81\text{GPa}$ for steel; ${\rho }_{\mathrm{r}}=1300\text{kg}/{\mathrm{m}}^3$ , $M_{\mathrm{r}}=0.69\text{MPa}$ , and $G_{\mathrm{r}}=0.04\text{MPa}$ for rubber; ${\rho }_{\mathrm{e}}=1180\text{kg}/{\mathrm{m}}^3$ , $M_{\mathrm{e}}=7.6\text{GPa}$ , and $G_{\mathrm{e}}=1.6\text{GPa}$ for epoxy, where $\rho$ is the mass density, and $M$ and $G$ are longitudinal and shear modulus, respectively. The radius of the steel cylinder is $r=0.2p$ , where $p$ is the period constant. The semimajor and semiminor axis of the elliptical rubber are $r_a=0.38p$ and $r_b=0.24p$ , respectively. The elliptical rubber is rotated with an angle $\theta =0.31\pi$ . The scatterer exhibits two resonance frequencies (Qing et al., 2022). We let longitudinal plane waves normally incident from the left side. At the resonance frequency, the strong oblique resonance of the scatterer induces significant oblique displacement, which in turn leads to mode conversion (Yu et al., 2023). At the angle $\theta =0.31\pi$ , the conversion rate reaches its maximum value of 1/4.

Figure 1

Figure 1. (a) Schematic of the system studied. A row of periodically arranged scatterers is embedded in a solid matrix. (b) Model illustration corresponding to the system in (a).

We constructed a model based on two underlying physical concepts. First, the scatter unit used in this study, as steel cylinders coating with elliptical rubber layers, can be modeled as an anisotropic homogeneous material. When the frequency of the incident wave is near to the resonant frequency, the scatterer obliquely resonates along its major axis, which is also along the θ-direction. This oblique vibration brings mode conversion (Climente et al., 2014; Kweun et al., 2017; Kim et al., 2018; Chai et al., 2023; Chen et al., 2024; Liu and Peng, 2024; Zhao et al., 2024). Secondly, the transmitted wave on the right can be viewed as the interference between the wave that penetrates the scatterer and the wave that bypasses it (Qing et al., 2022; Zhang et al., 2025).

Based on these physical pictures, we construct a wave propagation model, as illustrated in Figure 1b. The scatterer, together with the surrounding portion of the matrix, is modeled as an effective homogeneous blocks with effective mass density ${\rho }_{\text{eff}}$ , effective longitudinal modulus $M_{\text{eff}}$ and effective shear modulus ${\mathrm{G}}_{\text{eff}}$ . As the elastic wave propagates through them, both transmission and reflection occur, accompanied by mode conversion. Consequently, the transmitted and reflected waves contain both longitudinal and transverse components. The incident wave is represented by $u_{l+}\left(x\le 0\right)=A_{l+}{e}^{i\left(k_{l1}x-\omega t\right)}$ , with the subscript l (or t) referring to the longitudinal (or transverse) mode and “+” (or “-”) indicating propagation along the positive (or negative) x-axis. Here $u$ , $k$ , and $\omega$ are displacement, wave vector, and angular frequency, respectively. Similarly, the reflected longitudinal wave, reflected transverse wave, transmitted longitudinal wave, and transmitted transverse wave are represented by $u_{l-}\left(x\le 0\right)=A_{l-}{e}^{i\left(-k_{l1}x-\omega t\right)}$ , $u_{t-}\left(x\le 0\right)=A_{t-}{e}^{i\left(-k_{t1}x-\omega t\right)}$ , $u_{l+}\left(x\ge d\right)=D_{l+}{e}^{i\left(k_{l1}x-\omega t\right)}$ , and $u_{t+}\left(x\ge d\right)=D_{t+}{e}^{i\left(k_{t1}x-\omega t\right)}$ , respectively (Zhao et al., 2022).

Within the intermediate layer (with thickness d), there are two propagation channels: one through the effective blocks, and the other through the narrow gaps between the blocks. We assume that the elastic wave within the block takes the form of an “oblique wave”, whose vibration direction forms an angle θ with respect to the propagation direction. The forward- and backward-propagating oblique waves are described by $u_{+}\left(0\le x\le d\right)=C_{+}{e}^{i\left(k_{o2}x-\omega t\right)}$ and $u_{-}\left(0\le x\le d\right)=C_{-}{e}^{i\left(-k_{o2}x-\omega t\right)}$ , respectively. It is characterized by propagation along the x-direction, while the displacement occurs along the θ-direction. The oblique wave provides both horizontal and vertical displacement components, denoted by $u_x=u\cos \theta$ and $u_y=u\sin \theta$ along the x- and y-direction, respectively. The vertical component gives rise to transverse wave contributions in both the transmitted and reflected fields. A portion of the incident wave remains unconverted and propagates through the narrow gaps. This unconverted component remains as longitudinal wave and can be described by $u_l\left(0\le x\le d\right)=B_l{e}^{i\left(k_{l1}x-\omega t\right)}$ and $u_t\left(0\le x\le d\right)=B_t{e}^{i\left(-k_{l1}x-\omega t\right)}$ .

Accordingly, the boundary conditions between the intermediate layer input surface and output surface can be established to derive (Equations 1–4). The detailed analysis is presented below. At the boundary $x=0$ , the continuity of the normal displacement gives

$A_{l+}+A_{l-}=B_l+B_t+\cos \theta C_{+}+\cos \theta C_{-}.(1)$

The continuity of tangential displacement gives

$A_{t-}=\sin \theta C_{+}+\sin \theta C_{-}.(2)$

The normal and tangential stress are ${\tau }_{xx}=M\mathrm{\partial }{u}_l/\mathrm{\partial }x$ and ${\tau }_{xy}=G\mathrm{\partial }{u}_t/\mathrm{\partial }x$ , respectively. The continuity of normal stress gives

$Z_{l1}\left(A_{l+}-A_{l-}\right)=Z_{l2}\left(B_l-B_t\right)+Z_{l2}^{\prime }\left(C_{+}-C_{-}\right)\cos \theta .(3)$

Here $Z_{l1}$ and $Z_{l2}$ are acoustic impedances for longitudinal waves in left background and slits between blocks, and $Z_{l2}^{\prime }=\sqrt{{\rho }_{x,eff}{M}_{eff}}$ is the effective longitudinal impedances for the blocks. The continuity of tangential stress gives

$-Z_{t1}{A}_{t-}=Z_{t2}^{\prime }\left(C_{+}-C_{-}\right)\sin \theta .(4)$

Here is $Z_{t1}=\sqrt{G\rho }$ and $Z_{t2}^{\prime }=\sqrt{{\rho }_{y,eff}{G}_{eff}}$ are acoustic impedances for transverse waves. At the boundary $x=d$ , the corresponding continuity boundary conditions give equations

$B_l{e}^{i{k}_{l1}d}+B_t{e}^{-i{k}_{l1}d}+C_{+}{e}^{i{k}_{2}d}\cos \theta +C_{-}{e}^{-i{k}_{2}d}\cos \theta =D_{l+}{e}^{i{k}_{l1}d},(5)$

$C_{+}{e}^{i{k}_{2}d}\sin \theta +C_{-}{e}^{-i{k}_{2}d}\sin \theta =D_{t+}{e}^{i{k}_{t1}d},(6)$

$Z_{l1}\left(B_l{e}^{i{k}_{l2}d}-B_t{e}^{-i{k}_{l2}d}\right)+Z_{l2}\left(C_{+}{e}^{i{k}_{2}d}-C_{-}{e}^{-i{k}_{2}d}\right)\cos \theta =Z_{l1}{D}_{l+}{e}^{i{k}_{l1}d},(7)$

and

$Z_{t2}\left(C_{+}{e}^{i{k}_{2}d}-C_{-}{e}^{-i{k}_{2}d}\right)\sin \theta =Z_{t1}{D}_{t+}{e}^{i{k}_{t1}d}.(8)$

In summary, the boundary conditions yield a system of Equation 9.

$\left\{\begin{array}{l}{A}_{l+}+A_{l-}=B_l+B_t+\cos \theta C_{+}+\cos \theta C_{-}\\ A_{t+}+A_{t-}=\sin \theta C_{+}+\sin \theta C_{-}\\ Z_{l1}\left(A_{l+}-A_{l-}\right)=Z_{l2}\left(B_l+B_t\right)+\cos \theta Z_{l2}^{\prime }\left(C_{+}-C_{-}\right)\\ Z_{t1}\left(A_{t+}-A_{t-}\right)=+\sin \theta Z_{t2}^{\prime }\left(C_{+}-C_{-}\right)\\ B_{\mathrm{l}}{e}^{i{k}_{l2}d}+{\mathrm{B}}_t{e}^{-i{k}_{l2}d}+\cos \theta C_{+}{e}^{i{k}_{o2}d}+\cos \theta C_{-}{e}^{-i{k}_{o2}d}=D_{l+}{e}^{i{k}_{l1}d}\\ \sin \theta C_{+}{e}^{i{k}_{o2}d}+\sin \theta C_{-}{e}^{-i{k}_{o2}d}=D_{t+}{e}^{i{k}_{t1}d}\\ Z_{l2}\left(B_l{e}^{i{k}_{l2}d}-B_t{e}^{-i{k}_{l2}d}\right)+\cos \theta Z_{l2}^{\prime }\left({\mathrm{C}}_{+}{e}^{i{k}_{o2}d}-C_{-}{e}^{-i{k}_{o2}d}\right)=Z_{l1}{D}_{l+}{e}^{i{\mathrm{k}}_{\mathrm{l}1}\mathrm{d}}\\ \sin \theta Z_{t2}^{\prime }\left({\mathrm{C}}_{+}{e}^{i{k}_{o2}d}-C_{-}{e}^{-i{k}_{o2}d}\right)=Z_{t1}{D}_{t+}{e}^{i{k}_{t1}d}\end{array}\right.(9)$

The wave energy is defined as the energy fluxes along the horizontal direction (x-axis). The energy of the incident wave is $E_I=-v_x^{*}·{\tau }_{xx}$ , where $v$ and $\tau$ denote local velocity (* is conjugate) and stress. Energy conservation gives $\left|E_{D,l+}/E_I\right|+\left|E_{D,t+}/E_I\right|+\left|E_{A,l-}/E_I\right|+\left|E_{A,t-}/E_I\right|=1$ , with $E_{D,l+}$ , $E_{D,t+}$ , $E_{A,l-}$ , and $E_{A,t-}$ are wave energy for transmitted longitudinal wave, transmitted transverse wave, reflected longitudinal wave, and reflected transverse wave, respectively. For the epoxy matrix, we have $v=\mathrm{\partial }u/\mathrm{\partial }t$ and $\tau =M\mathrm{\partial }u/\mathrm{\partial }x$ , which subsequently lead to $\left|E_{A,t-}/E_I\right|=\left|c_{t,e}{A}_{t-}/c_{l,e}{A}_{l+}\right|$ . Accordingly, the principle of energy conservation leads to Equation 10

$\left|\frac{D_{l+}}{A_{l+}}\right|+\left|\frac{c_{t,e}{D}_{t+}}{c_{l,e}{A}_{l+}}\right|+\left|\frac{A_{l-}}{A_{l+}}\right|+\left|\frac{c_{t,e}{A}_{t-}}{c_{l,e}{A}_{l+}}\right|=1.(10)$

Which also should be considered.

We focus on the mode conversion that occurs at the resonant frequency. The conversion ratio is defined by term $E_{D,t+}/E_{l+}=\left|c_{t,e}{D}_{t+}/c_{l,e}{A}_{l+}\right|$ . The analytical expression is given by:

${\mathrm{D}}_{\mathrm{t}+}=\frac{2{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\cos \mathrm{\theta }\left|{\mathrm{A}}_{\mathrm{l}+}\right|}{{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\cos \mathrm{\theta }+{\mathrm{Z}}_{\mathrm{t}1}}\sqrt{\frac{1-\frac{{\left|{\mathrm{Z}}_{\mathrm{l}1}-{\mathrm{Z}}_{\mathrm{l}2}\right|}^2+4{{\mathrm{Z}}_{\mathrm{l}1}}^2{\cos }^2\left({\mathrm{k}}_{\mathrm{l}2}\mathrm{d}\right)}{{\left|{\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right|}^2}}{\mathrm{\alpha }}}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{t}1}\mathrm{d}}(11)$

with $\mathrm{\alpha }={\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }{\mathrm{Z}}_{\mathrm{t}1}\cot \mathrm{\theta }}{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right)}\frac{\mathrm{p}}{2\mathrm{x}+2\mathrm{y}}+\left(-\frac{\mathrm{q}\cot \mathrm{\theta }}{2\mathrm{x}+2\mathrm{y}}+\frac{\mathrm{q}\left({\mathrm{Z}}_{\mathrm{l}2}^{\prime }-{\mathrm{Z}}_{\mathrm{t}1}\right)\cot \mathrm{\theta }}{\mathrm{n}\left({\mathrm{Z}}_{\mathrm{l}2}^{\prime }+{\mathrm{Z}}_{\mathrm{t}1}\right)}\right)\right|}^2+\frac{{\mathrm{c}}_{\mathrm{t}}}{{\mathrm{c}}_{\mathrm{l}}}{\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }-{\mathrm{Z}}_{\mathrm{t}1}}{\mathrm{n}\left({\mathrm{Z}}_{\mathrm{l}2}^{\prime }+{\mathrm{Z}}_{\mathrm{t}1}\right)}\right|}^2+{\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }{\mathrm{Z}}_{\mathrm{t}1}\cot \mathrm{\theta }}{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right)}\right|}^2+\frac{{\mathrm{c}}_{\mathrm{t}}}{{\mathrm{c}}_{\mathrm{l}}}\mathrm{m}={\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}},\mathrm{n}={\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}},\mathrm{x}={\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{l}2}\mathrm{d}},\mathrm{y}={\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{l}2}\mathrm{d}},\mathrm{p}={\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{l}1}\mathrm{d}},\mathrm{q}={\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{t}1}\mathrm{d}}.$ When $\theta =0$ , the Equation 11 reduces to

${\mathrm{D}}_{\mathrm{t}+}=\frac{2{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\left|{\mathrm{A}}_{\mathrm{l}+}\right|}{{\mathrm{Z}}_{\mathrm{t}2}^{\prime }+{\mathrm{Z}}_{\mathrm{t}1}}\sqrt{\frac{1-\frac{{\left|{\mathrm{Z}}_{\mathrm{l}1}-{\mathrm{Z}}_{\mathrm{l}2}\right|}^2+4{{\mathrm{Z}}_{\mathrm{l}1}}^2{\cos }^2\left({\mathrm{k}}_{\mathrm{l}2}\mathrm{d}\right)}{{\left|{\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right|}^2}}{\mathrm{\alpha }}}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{t}1}\mathrm{d}},$

It can be directly shown to be zero. Similarly, when $\theta =\pi /2$ , the Equation 11 also can be directly shown to be zero. This reflects a physically consistent conclusion: no mode conversion occurs when the system is perfectly symmetric. For other angles, a certain amount of mode conversion generally occurs, and the exact value is determined by the parameters in Equation 11.

In this work, we particularly consider the low-frequency limit in which the slab becomes sufficiently thin compared to the wavelength, a condition referred to as zero-phase-difference. The term zero-phase-difference refers to the condition in which the phase difference between the incident and exit surfaces of the scatterer is zero. Physically, this means that the thickness of the scatterer is neglected in terms of phase variation, corresponding to the low-frequency limit where the wavelength is much larger than the scatterer’s thickness. In the limit $d\to 0$ , we have terms with $e^{ikd}\to 1$ . Thus, (Equations 5–8) reduce to Equations 12–15

$B_{l+}+B_{l-}+C_{+}\cos \theta +C_{-}\cos \theta =D_{l+},(12)$

$C_{+}\sin \theta +C_{-}\sin \theta =D_{t+},(13)$

$Z_{l1}\left(B_{l+}-B_{l-}\right)+Z_{l2}\left(C_{+}{e}^{i{k}_{2}d}-C_{-}{e}^{-i{k}_{2}d}\right)\cos \theta =Z_{l1}{D}_{l+},(14)$

and

$Z_{t2}\left(C_{+}{e}^{i{k}_{2}d}-C_{-}{e}^{-i{k}_{2}d}\right)\sin \theta =Z_{t1}{D}_{t+}.(15)$

The system of Equation 12 reduces to

$\left\{\begin{array}{l}{\mathrm{A}}_{\mathrm{l}+}+{\mathrm{A}}_{\mathrm{l}-}={\mathrm{B}}_{\mathrm{l}}+{\mathrm{B}}_{\mathrm{t}}+\cos \mathrm{\theta }{\mathrm{C}}_{+}+\cos \mathrm{\theta }{\mathrm{C}}_{-}\\ {\mathrm{A}}_{\mathrm{t}+}+{\mathrm{A}}_{\mathrm{t}-}=\sin \mathrm{\theta }{\mathrm{C}}_{+}+\sin \mathrm{\theta }{\mathrm{C}}_{-}\\ {\mathrm{Z}}_{\mathrm{l}1}\left({\mathrm{A}}_{\mathrm{l}+}-{\mathrm{A}}_{\mathrm{l}-}\right)={\mathrm{Z}}_{\mathrm{l}2}\left({\mathrm{B}}_{\mathrm{l}}+{\mathrm{B}}_{\mathrm{t}}\right)+\cos \mathrm{\theta }{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{C}}_{+}-{\mathrm{C}}_{-}\right)\\ {\mathrm{Z}}_{\mathrm{t}1}\left({\mathrm{A}}_{\mathrm{t}+}-{\mathrm{A}}_{\mathrm{t}-}\right)=+\sin \mathrm{\theta }{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\left({\mathrm{C}}_{+}-{\mathrm{C}}_{-}\right)\\ {\mathrm{B}}_{\mathrm{l}}+{\mathrm{B}}_{\mathrm{t}}+\cos \mathrm{\theta }{\mathrm{C}}_{+}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}+\cos \mathrm{\theta }{\mathrm{C}}_{-}{\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}={\mathrm{D}}_{\mathrm{l}+}\\ \sin \mathrm{\theta }{\mathrm{C}}_{+}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}+\sin \mathrm{\theta }{\mathrm{C}}_{-}{\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}={\mathrm{D}}_{\mathrm{t}+}\\ \cos \mathrm{\theta }{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{C}}_{+}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}-{\mathrm{C}}_{-}{\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}\right)={\mathrm{Z}}_{\mathrm{l}1}{\mathrm{D}}_{\mathrm{l}+}\\ \sin \mathrm{\theta }{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\left({\mathrm{C}}_{+}{\mathrm{e}}^{\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}-{\mathrm{C}}_{-}{\mathrm{e}}^{-\mathrm{i}{\mathrm{k}}_{\mathrm{o}2}\mathrm{d}}\right)={\mathrm{Z}}_{\mathrm{t}1}{\mathrm{D}}_{\mathrm{t}+}\end{array}\right.$

And the solution of conversion ratio Equation 13 reduces to

${\mathrm{D}}_{\mathrm{t}+}=\frac{2{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\cos \mathrm{\theta }\left|{\mathrm{A}}_{\mathrm{l}+}\right|}{{\mathrm{Z}}_{\mathrm{t}2}^{\prime }\cos \mathrm{\theta }+{\mathrm{Z}}_{\mathrm{t}1}}\sqrt{\frac{1-\frac{{\left|{\mathrm{Z}}_{\mathrm{l}1}-{\mathrm{Z}}_{\mathrm{l}2}\right|}^2+4{{\mathrm{Z}}_{\mathrm{l}1}}^2{\cos }^2\left({\mathrm{k}}_{\mathrm{l}2}\mathrm{d}\right)}{{\left|{\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right|}^2}}{\mathrm{\alpha }}}$

with $\mathrm{\alpha }={\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }{\mathrm{Z}}_{\mathrm{t}1}\cot \mathrm{\theta }}{4{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right)}+\left(-\frac{\cot \mathrm{\theta }}{4}+\frac{\left({\mathrm{Z}}_{\mathrm{l}2}^{\prime }-{\mathrm{Z}}_{\mathrm{t}1}\right)\cot \mathrm{\theta }}{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }+{\mathrm{Z}}_{\mathrm{t}1}}\right)\right|}^2+\frac{{\mathrm{c}}_{\mathrm{t}}}{{\mathrm{c}}_{\mathrm{l}}}{\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }-{\mathrm{Z}}_{\mathrm{t}1}}{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }+{\mathrm{Z}}_{\mathrm{t}1}}\right|}^2+{\left|\frac{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }{\mathrm{Z}}_{\mathrm{t}1}\cot \mathrm{\theta }}{{\mathrm{Z}}_{\mathrm{l}2}^{\prime }\left({\mathrm{Z}}_{\mathrm{l}1}+{\mathrm{Z}}_{\mathrm{l}2}\right)}\right|}^2+\frac{{\mathrm{c}}_{\mathrm{t}}}{{\mathrm{c}}_{\mathrm{l}}}$ It can be seen that, due to the simplification $d\to 0$ , the phase difference $e^{ikd}$ between the two interfaces of the intermediate layer no longer plays a significant role. As a result, the conversion rate becomes highly dependent on the effective parameters of the scatterer at different angles. In the following, we describe the procedure employed to extract the effective parameters and present the resulting values.

The region over which the effective parameters are calculated is shown in the Figure 2 by the dashed box. It is a rectangular domain of length $a=\sqrt{{r_a}^2{\sin }^2\theta +{r_b}^2{\cos }^2\theta }+0.01$ along the x-direction and width $a=\sqrt{{r_a}^2{\cos }^2\theta +{r_b}^2{\sin }^2\theta }+0.01$ along the y-direction. This domain includes the entire scatterer and a portion of the surrounding matrix. The effective mass density ${\rho }_{x,eff}=\text{force}/\text{acceleration}/\text{volume}$ . The force is defined as the difference between the normal stresses on the right and left boundaries of the dashed box. In principle, the shear stresses on the top and bottom boundaries should also be considered. However, since these boundaries are located very close to the periodic boundaries, the shear stresses at the top and bottom are nearly identical, and their difference can be approximated as zero. The acceleration is the integral of acceleration over the matrix region within the selected domain. The acceleration inside the scatterer is not included in the calculation, in accordance with the assumptions of Milton’s model (Climente et al., 2014; Kweun et al., 2017; Kim et al., 2018). The volume, equaling $a$ time $b$ , denotes the volume of the selected region. It is noticed that the scatterer exhibits strong anisotropy in mass density. In the present calculation, we evaluate the effective mass density along the x-direction, which is relevant for computing the effective longitudinal impedance. To determine the effective shear modulus, however, the mass density along the y-direction is required. In this work, we assume that the effective mass density along the y-direction at angle theta is equivalent to the effective mass density along the x-direction at angle 90-theta. The variation of effective mass densities ${\rho }_{xeff}$ and ${\rho }_{yeff}$ with respect to angle is shown in the Figures 3a,b.

Figure 2

Figure 2. Schematic of effective parameter calculation. The structure within the dashed box region is treated as an equivalent homogeneous medium (yellow block).

Figure 3

Figure 3. Variation of material properties as a function of angle. (a) Density (ρ_x ); (b) Density (ρ_y ); (c) Elastic coefficient (C11); (d) Elastic coefficient (C44).

The calculation of the effective modulus follows the approach described in Ref (Liu et al., 2000; Zheng et al., 2019; Qing et al., 2022; Kweun et al., 2017; Yang et al., 2021). The effective normal stress $t_{eff}$ is obtained from the normal stress on the left and right boundaries of the region, while the effective normal strain $e_{eff}$ is determined from the corresponding boundary displacements. The effective longitudinal modulus $M_{eff}=t_{eff}/e_{eff}$ (Zhao et al., 2024) is then calculated from the ratio of the effective stress to the effective strain. Similarly, the effective shear modulus $G_{eff}=t_{xy}/e_{xy}$ (Zhang et al., 2025) can be calculated from the effective shear stress $t_{xy}$ and effective shear strain $e_{xy}$ evaluated on the boundaries of the region. The calculation results are shown in the Figures 3c,d.

We now proceed to calculate the acoustic impedance of different materials. More specifically, we are interested in the ratio of the structure’s acoustic impedance to that of the background matrix. The longitudinal and shear impedances for matrix are $Z_{l1}=\sqrt{{\rho }_x/M}$ and $Z_{t1}=\sqrt{{\rho }_y/G}$ respectively. When an acoustic wave propagates from a wide waveguide into a narrower one, an impedance mismatch arises due to the unbalanced load. This mismatch depends on the ratio of the cross-sectional areas of the two waveguides. Therefore, we approximate the impedance ratio as $Z_{l2}/Z_{l1}=d/\left(d-b\right)$ . The value of $Z_{l2}^{\prime }$ and $Z_{t2}^{\prime }$ are calculated by ${Z_{l2}}^{\prime }=\sqrt{{\rho }_x/\left(M*C_{11}\right)}$ and ${Z_{t2}}^{\prime }=\sqrt{{\rho }_y/\left(G*C_{44}\right)}$ . The results are shown in Figure 3.

With all the impedance parameters obtained, we are now able to compute the mode conversion rate. The results are shown in the Figure 4. The red line represents the result calculated from our model, while the black line corresponds to the simulation result obtained using COMSOL. The agreement between the two confirms the validity of our model. Inprevious studies on mode conversion, the underlying mechanism typically relies on elastic waves propagating through a finite region of anisotropic medium. In such models, mode conversion is achieved by exploiting the propagation difference between longitudinal and transverse waves within the anisotropic layer. This process requires a certain propagation distance to accumulate sufficient phase difference. A representative example is the FP-type resonance mechanism described in Ref (Liu et al., 2000; Kweun et al., 2017; Yang et al., 2021). However, when the thickness of the anisotropic layer becomes vanishingly small, such as in the low-frequency regime, this mechanism inevitably breaks down. Indeed, as shown in Refs (Qing et al., 2022; Kweun et al., 2017; Kim et al., 2018; Chai et al., 2023; Zhao et al., 2022; Fahy et al., 2007; Wang et al., 2016; Cao et al., 2018; Yang et al., 2018; Tian and Yu, 2019; Xu et al., 2019; Ruan and Liang, 2021; Dong et al., 2022; Tian et al., 2022; Jiang et al., 2023), if we impose the limit $d\to 0$ , it will lead to a trivial solution with zero conversion rate. In contrast, the present work introduces a fundamentally different mechanism that remains valid even in the thin-layer limit. The local resonance mechanism used in our study is the same as that in Refs (Qing et al., 2022; Kweun et al., 2017; Kim et al., 2018; Chai et al., 2023; Zhao et al., 2022; Fahy et al., 2007; Wang et al., 2016; Cao et al., 2018; Yang et al., 2018; Tian and Yu, 2019; Xu et al., 2019; Ruan and Liang, 2021; Dong et al., 2022; Tian et al., 2022; Jiang et al., 2023). The main difference lies in the oblique arrangement of the elliptical coating in our tri-component structure, which breaks the structural symmetry. The advantage of this design is that the symmetry breaking enables elastic wave mode conversion between longitudinal and transverse waves at extremely low frequencies. Moreover, previous works only reported numerical simulations without providing a theoretical explanation. In this work, we for the first time propose a theoretical model for such low-frequency mode conversion and quantitatively compare it with the simulation results. Our model enables nontrivial mode conversion when $d\to 0$ , by leveraging a new physical process: oblique resonance leading to oblique displacement. We use the oblique resonance to describe the condition where the vibration direction at resonance is neither purely vertical nor horizontal, but oblique. The key innovation lies in the introduction of a special block that supports purely oblique displacement. This block produces a tangential displacement component at the interface, which drives the conversion between wave modes. This mechanism is particularly suited for low-frequency applications where traditional anisotropic-layer models fail.

Figure 4

Figure 4. Conversion rate as a function of angle. The black line shows the result calculated from the model (calculated by Equation 11), while the red line corresponds to the simulation results (calculated by COMSOL).

We acknowledge that the narrow bandwidth is a well-known challenge inherent to the local resonance mechanism, for which no effective solution has been identified to date. Our discussion further addresses the correlation between conversion efficiency and bandwidth, highlighting this as a fundamental limitation in current designs.

Based on the established model for elastic wave mode conversion in zero-phase-difference ultrathin anisotropic media, we employed COMSOL software to calculate the variation in conversion rate with frequency. The results are shown in the Figure 5. As the frequency increases, the mode conversion rate exhibits a trend of first increasing and then decreasing. Near 9.74 Hz, the conversion rate reaches its maximum value of 25%. Once frequency deviates from this value, the conversion rate drops sharply, approaching essentially zero at 9.5 Hz.

Figure 5

Figure 5. The conversion efficiency as a function of the excitation frequency.

In conclusion, we presented a new theoretical model for elastic wave mode conversion in periodic structures, introducing a mechanism based on oblique resonance and oblique displacement. Unlike conventional models that require wave propagation through a finite anisotropic layer, our approach remains valid in the limit of zero thickness. The essential mechanism is the generation of tangential displacement by a block supporting only oblique motion, which drives the mode conversion. A full set of governing equations was derived under general boundary conditions, and effective parameters were obtained directly from the stress and displacement fields. The model accurately predicts conversion rates that are in excellent agreement with numerical simulations. These results demonstrate the effectiveness of the model and suggest its potential application in the design of compact elastic metamaterials operating at low frequencies.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Author contributions

XB: Writing – original draft, Writing – review and editing. YG: Writing – review and editing.

Funding

The author(s) declare that no financial support was received for the research and/or publication of this article.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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The author(s) declare that no Generative AI was used in the creation of this manuscript.

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