On the surface Bloch waves in truncated periodic media: scalar-wave primer

TL;DR

The paper addresses boundary-layer waves along rational-slope cuts of periodic media, introducing a quadratic eigenvalue problem (QEP) on an effective unit cell to capture evanescent Bloch states. A reduced-order model (ROM) assembles these evanescent QEP eigenstates to construct surface Bloch waves that satisfy homogeneous Neumann boundaries, enabling rapid exploration of surface undulations on dispersion and skin depth. Key contributions include a rigorous QEP formulation for surface waves, a ROM framework with robust convergence properties, and quantified power flow/skin-depth measures to assess energetic relevance, all demonstrated on 2D scalar phononic-like media. The approach generalizes to other boundary conditions and governing equations, offering a design pathway to manipulate surface waves via surface-cut engineering with potential impact on wave regulation in metamaterials.

Abstract

Much like their counterparts in homogenous elastic solids, waves in periodic media can be broadly classified into Floquet-Bloch body waves, and evanescent surface waves. Our goal is to elucidate the latter boundary layers, termed surface Bloch (SB) waves, affiliated with rational surface cuts and homogeneous Neumann data. To this end we adopt a two-dimensional (2D) scalar wave equation with periodic coefficients (describing anticline shear waves in phonoic crystals) as a test bed and develop a unit cell-of-periodicity-based, reduced order model of the SB waves that is capable of describing both their dispersion, waveforms, and ``skin depth''. The centerpiece of our analysis is a quadratic eigenvalue problem (QEP) for the effective unit cell of periodicity -- deriving from a geometric interplay between the mother Bravais lattice and orientation of the surface cut -- that seeks the complex wavenumber normal to the cut plane given (i) the excitation frequency and (ii) wavenumber in the direction of the cut plane. In this way the sought boundary layer is derived via superposition of the evanescent QEP eigenstates, whose relative amplitudes are obtained by imposing the homogeneous boundary condition. With the QEP eigenspectrum at hand, evaluation of an SB wave -- in terms of both dispersion characteristics and evanescent waveforms -- entails only a low-dimensional eigenvalue problem. This feature caters for rapid exploration of the effect of (periodic) surface undulations, and so enables manipulation of the SB waves via optimal design of the surface cut. Our analysis also includes an account for the power flow and ``skin depth'' of a surface Bloch wave, both of which are critical for the energetic relevance of boundary layers.

Figures (10)

• Figure 1: Direction of propagation, $\boldsymbol{e}\in\mathbb{R}^2$, of a surface Bloch wave featuring a "1:3" rational slope ($q^1\!=3,\,q^2\!=1$) relative to lattice $\boldsymbol{R}$. The new lattice basis and unit cell catering for the surface wave analysis are $(\tilde{\boldsymbol{a}}_1,\tilde{\boldsymbol{a}}_2)$ and $\tilde{Y}$, respectively.
• Figure 2: Schematics of an undulated cut ($P=3$) at depth $\mathfrak{d}\!=\!-\tfrac{1}{2}$ with relevant features and notations. In the example, the surface cut $\mathcal{S}$ is made at "1:2" rational slope ($q^1\!=2,\space q^2\!=1$) relative to an orthogonal lattice $\boldsymbol{R}$. In the context of \ref{['evan1']} and \ref{['s3']}, the surface traction on the ligaments extending "above" the cut plane $\mathcal{S}$ is computed via the periodicity of quadratic eigenfunctions $\tilde{\phi}_n$.
• Figure 3: Unit prism of periodicity, $\Pi_{\space\mathcal{S}_{\space\delta}}$, of the semi-infinite domain $\Omega_{\mathcal{S}_\delta}$ and re-tailored unit cell of periodicity $\tilde{Y}_{\space\delta}$ (with $|\tilde{Y}_{\space\delta}|=|\tilde{Y}|$) catering for the surface undulation $\mathcal{S}_{\delta}$.
• Figure 4: Dispersion of surface Bloch waves: (a) schematics of an evanescent Bloch wave \ref{['evan1']} propagating in direction $\boldsymbol{e}\space=\space\boldsymbol{e}_1$ along the free surface $\mathcal{S}\space=\space\{\boldsymbol{x}\!: \xi_2\space=\space\mathfrak{d}\}$ of a periodic half-space for some $\mathfrak{d}\!\in\![-0.5,0.5)$; (b) example QEP solution, fundamental mode: $\phi_1(\boldsymbol{x})$ and $\phi_1(\boldsymbol{x})\space e^{{\boldsymbol{k}}\cdot\boldsymbol{x}}$ (real parts) which illustrate the genesis of SB waves; (c) the first two branches of the surface-wave dispersion diagram for $\mathfrak{d}\space=\space -0.5$: bullets -- spectral data $k_e^\star(\omega)$ computed via QEP \ref{['QEP-surf2']} and condition \ref{['cond1']}, and solid lines -- dispersion of (standard) Floquet-Bloch waves \ref{['floquet']} propagating in direction $\boldsymbol{e}_1$ through an unbounded periodic medium. In the last panel, example (scaled) distributions of $C_{\boldsymbol{\Uppsi}} (k_e)$ for $\omega\space\in\space\{0.6, 1.3,2\}$ each feature a sharp peak with an $O(10^3)$ magnitude, locating a point on the SB dispersion diagram. By contrast, $C_{\boldsymbol{\Uppsi}} (k_e)$ for $\omega\!=\!1.7$ (inside the surface-wave band gap, shaded area) is monotonic with an $O(10)$ maximum value.
• Figure 5: Effect of the depth of cut $\mathfrak{d}$ on: (a) surface wavenumber $k_e^\star$, and (b) skin depth $\mathfrak{s}$ due to \ref{['skindepth2b']} with $\vartheta = 0.1$.

Theorems & Definitions (12)

• Remark 1
• Remark 2
• Lemma 1
• Lemma 2
• Theorem 1
• Definition 1
• Remark 3
• Remark 4
• Remark 5
• Definition 2