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The paper outlines analytically and exemplifies numerically the basic aspects, characterizing dispersion spectra of the shear horizontal (SH) waves in transversely periodic layers and half-spaces with a monoclinic functionally graded unit cell. On introducing the background, the ’quasi-orthorhombic’ formulation is pointed out. Further analytical consideration bypasses explicit intricacy of the wave solutions in continuously varying media and relies only on a few basic traits of the governing equation of SH motion. An elementary reasoning pinpoints the key features of the Floquet eigenmodes and their link to the traction-free boundary conditions in question. This simple grounds suffices to generalize the remarkable property, previously restricted to the orthorhombic piecewise homogeneous periodic stacks, which implies that the SH dispersion spectrum for a unit cell, assumed free of traction at the faces, is embedded into the spectrum for the finite periodic structure of these unit cells and contains the locus of surface-wave solutions for the semi-infinite periodic structure. The conclusion is valid for an arbitrary continuous and/or discrete transverse periodic inhomogeneity. Numerical results, presented for the case of continuously inhomogeneous unit cell, are based on the Peano series of multiple integrals.

The impedance method provides a convenient framework for analysing the velocity spectrum of surface acoustic waves, propagating in anisotropic-coated substrates within the subsonic range where there is no energy leakage into the substrate. The dispersion equation determines subsonic wave velocities as zeros of eigenvalues of the aggregate impedance which are real monotonic functions of velocity at any fixed wavenumber. The impedance method lends insight into the properties of a coated substrate spectrum by elucidating its connection to the individual acoustic parameters of the substrate and the coating layer materials. By using this approach, certain bounds and basic trends for the dispersion of velocity branches in the coated substrates are revealed. The low-frequency and high-frequency dispersion is explored in detail. The criterion is examined for either speeding or slowing trend of the low-frequency linear dispersion of the velocity branch evolving from the Rayleigh velocity of the substrate. Specific behaviour of this branch due to the higher-order dispersion in substrates coated by a relatively light or dense, “slow” or “fast” layer is interpreted. The impedance method facilitates a similar analysis for various layered structures, for instance, substrate with multilayered coating, a plate coated on both sides, and a layer embedded between two substrates.

Basic patterns of the velocity versus wavenumber dispersion of the surface waves in solids coated by a relatively light or dense, “slow” or “fast” layer are discussed in the general case of an arbitrary anisotropy of substrate and coating materials. The onset of the subsonic wave branch, characterized by either a speeding or a slowing trend, is examined. Competitive tendencies, which pertain to the low-frequency dispersion in the case of dense “fast” layer, are revealed.

Propagation of sagittally polarized plane acoustic waves is considered in an orthorhombic medium with a periodic system of N+1 infinite planar cuts maintaining sliding contact (ideal cracks). The reflection and transmission rates are derived by the propagator-matrix method. Two essentially different types of stop bands exist, in which the imaginary part of the Bloch vector either remains finite or reaches infinity. The latter corresponds to the transmission cut-off, which may come about specifically at the sliding-contact interface. Coupling of the Bragg phenomenon with the cutting-off effect produces quite specific resonant features of reflection and transmission. Especially sharp filtering properties of the spectra come about at a small deviation Δθ from such angles of incidence, which provide total transmission (anti-reflection) independent of frequency, namely, at nearly normal incidence of the fast mode, and at angles of incidence of the slow mode close to a certain critical value. At ∣Δθ∣⪡1, the spectrum of transmission (without mode conversion) represents a nearly periodic group of abrupt dips to zero and a modulated group of secondary drops increasing with growing (Δθ)2N2, whereas the general spectral background is close to unity. In turn, the reflection spectrum at ∣Δθ∣⪡1 contains sharp principal peaks with modulated heights, reaching nearly unit height, and the secondary peaks against almost zero background. Changes in the spectra shape on varying the angle of incidence θ become drastic near the specific threshold value of θ, which corresponds to the mutual transformation of the ordinary stop bands and cutting-off bands. After the cross-over, the transmission spectrum contains significantly wide step-wise dips, within which the rate stays very close to zero if N2⪢1.

Geometrical-acoustics approach interprets vibration modes localized at the edge of wedges as the quasi-plane flexural waves propagating in a plate of variable thickness. This approach is combined with the dispersion relation for flexural wave in a thin anisotropic fluid-loaded plate to analytically determine the subsonic velocities c of the localized modes in anisotropic immersed wedges. The transcendent equation in c is established for an arbitrarily anisotropic wedge material and a general case of the wedge–fluid coupling. An approximate explicit solution for c is obtained in the cases when the parameter of the wedge-fluid coupling θn/r is either small or large (here θ is the apex angle, n is the modal order, and r is the ratio of the fluid density and the wedge density). In both cases, the ratio of the wedge-mode velocities c/c0in the immersed and free wedge is a corresponding function of the coupling parameter θn/r. Provided that the wedge–fluid coupling is sufficiently pronounced, the ratio c/c0in the presence of anisotropy acquires the scaling factor, which depends appropriately on elastic coefficients of the wedge and turns to unity in the isotropic limit.

Propagation of flexural localized vibration modes along edges of anisotropic wedges is considered in the framework of the geometrical-acoustics approach. Its application allows for straightforward evaluation of the wedge-mode velocities in the general case of arbitrary elastic anisotropy. The velocities depend on the wedge apex angle and on the mode number in the same way as in the isotropic case, but there appears to be additional dependence on elastic coefficients. The velocities in tetragonal wedges (with the midplane orthogonal to the four-fold axis) and in "weakly" monoclinic wedges are explicitly calculated and analyzed. Bounds of the wedge-wave velocity variation in tetragonal materials are established.

The specific feature of the interface, which maintains sliding contact between elastic media, is that it can be impervious to the wave field existing in one of the adjoined materials. As a result, reflection–transmission of plane acoustic waves at the sliding-contact interface may enjoy the cutting-off effect, which implies that neither bulk, nor inhomogeneous modes are being transmitted at particular angles of incidence. The necessary and sufficient criteria for this phenomenon are obtained for a binary structure, constituted by two elastic half-spaces in sliding contact, and for a sandwich structure with sliding-contact interfaces between the enclosed layer and the substrates. In the generic case of unrestricted anisotropy (triclinic materials), the criterion for cutting-off in a binary structure involves acoustic parameters of solely that of the half-spaces, which contains the incident mode, and proves to be independent of an adjacent medium. The frequency-dispersive criterion for the absence of transmission through a triclinic layer in the sliding-contact sandwich structure is independent of substrates. By appeal to the Stroh formalism, the cutting-off conditions in a binary and a sandwich structure are further elaborated under the assumption that one of the half-spaces, or a layer, is orthorhombic, and its two symmetry planes are parallel, respectively, to the plane of incidence and to the sliding-contact interface with arbitrary adjacent media. It is shown that the transmission cut-off in a binary structure is necessarily accompanied by the absence of mode conversion at reflection, but the reverse is not true. The angles of incidence which give rise to these effects are determined in terms of elastic coefficients. Transmission cut-off through an orthorhombic layer comes about at an arbitrary angle of incidence, related to guided-modes range in the layer, for the corresponding aperiodic infinite set of the frequency values. Relations for the coefficients of reflection and transmission at the sliding-contact interface between two orthorhombic half-spaces are obtained in concise form, expressed solely via normal components of the partial Stroh-normalized traction amplitudes. Provided that the adjoined orthorhombic half-spaces in sliding contact are identical, the same value of wave-vector tangential projection, which stipulates transmission cut-off at the incidence of, say, the fast mode, entails total transmission at the incidence of the slow mode.

Oblique incidence of a shear horizontal elastic wave on a periodic structure of identical piezomagnetic layers, separated by thin superconductive coatings, is considered. The Bragg resonance of reflection arising due to the s state on the interfaces is analyzed. It is shown that for a sufficiently large number of layers the reflection coefficient R may change from zero to R ≈1 for the n-s transition.

Conditions are derived for the existence of focusing cusps in ballistic phonon intensity patterns for propagation directions in crystal symmetry planes. Line caustics are known to be associated with lines of vanishing Gaussian curvature (parabolic lines) on the acoustic slowness surface, while cusps are associated specifically with points where the direction of vanishing principal curvature is parallel to the parabolic line. A parabolic line meets a crystal symmetry plane sigma at a right angle, and so it is the vanishing of the slowness-surface curvature transverse to sigma that conditions the existence of a cusp. A relation for the transverse curvature is derived and analyzed. It is shown that in an arbitrary symmetry plane sigma there may be up to four pairs of inversion-equivalent cuspidal points for SH (out-of-plane polarized) waves, and up to eight pairs of cuspidal points associated with the in-plane polarized (usually quasi-transverse) waves. In tetragonal crystals, the symmetry planes containing the four-fold axis can have at most two pairs of cusps for the SH waves and up to six pairs of cusps for the in-plane waves. In cubic crystals, the face symmetry planes sigma cannot have cuspidal points for SH waves, as is known, while four pairs of cusps for in-plane waves exist in sigma if and only if the outer-most slowness sheet has a concave region embracing the four-fold axis. The points of vanishing transverse curvature on the slowness surface in symmetry planes of tetragonal and cubic media are identified by concise relations, facilitating their explicit analysis.

A detailed analysis is presented of the shape of the acoustic slowness surface of anisotropic elastic solids near axes of rotational symmetry. Particular emphasis is given to the consequences of phase-speed degeneracy along three- and fourfold symmetry axes, where the curvature of the slowness surface does not have a unique value but instead depends on the direction of approach. All the geometrically distinct patterns of curvature variation in the vicinity of three- and fourfold symmetry axes are identified, and the algebraic criteria for their occurrence are established. The results are of relevance to many problems concerned with acoustic-wave propagation and ballistic phonon transport in anisotropic solids. \textcopyright{} 1996 The American Physical Society.