Since World War II, there have been many efforts worldwide to design hull coatings for naval and underwater defense applications in order to reduce either noise radiated from the hull (discretion) or acoustic reflection from the hull (stealth). This paper presents a non-exhaustive review of concepts that have emerged since the iconic Alberich coating and highlights new trends and promising candidates, such as metamaterials, to improve the discretion and stealth performance of underwater defense systems. The need for acoustic materials for naval and underwater applications is first explained, along with some specific design requirements for acoustic coatings. Two coatings technologies, called micro-inclusion coatings and macro-inclusion coatings, are then successively reviewed. New trends in materials design for naval and underwater defense applications are also presented in the following section. This paper concludes by introducing the need for acoustic materials in the frame of new threats to the naval industries as well as other non-naval areas, opening new possibilities for their design.

We herein investigate the scattering of orthogonal counterpropagating waves and one-way propagating bulk waves in discrete acoustic superlattices subjected to a scattering potential applied to one of the superlattice unit cells. We demonstrate theoretically that the orthogonality of counterpropagating modes does not provide robust protection against backscattering. By contrast, the one-way propagating modes do satisfy a no-reflection condition, i.e., they exhibit immunity to backscattering, for a wide range of applied scattering potentials, which represent defects and disorder.

We used the transfer matrix method to investigate the conditions supporting the existence of directional bulk waves in a two-dimensional (2D) phononic crystal. The 2D crystal was a square lattice of unit cells composed of rectangular subunits constituted of two different isotropic continuous media. We established the conditions on the geometry of the phononic crystal and its constitutive media for the emergence of waves, which, for the same handedness, exhibited a non-zero amplitude in one direction within the crystal’s 2D Brillouin zone and zero amplitude in the opposite direction. Due to time-reversal symmetry, the crystal supported propagation in the reverse direction for the opposite handedness. These features may enable robust directional propagation of bulk acoustic waves and topological acoustic technology.

A nonreciprocal system composed of a one-dimensional piezoelectric phononic crystal whose periodic electrical conditions are modulated in time is presented. One-way longitudinal wave propagation is studied experimentally and compared to finite element temporal simulations. The modulation is performed by prescribing grounded or floating potential conditions on a periodic set of electrodes through external circuits. This approach makes it possible to consider a wide range of modulation speeds, and the large number of unit cells of the phononic crystal allows us to characterize experimentally the full dispersion curves of the system. This permits to observe the presence of directional bandgaps and to follow the shift in frequencies of these bandgaps as a function of the modulation speed. The experiments show the linear evolution of the central position of the bandgaps with the increase in the modulation speed, as well as their progressive closure, over a wide range of frequencies. Experiments are also used to estimate the evolution of bandgaps in a dispersive system, a problem discussed in several theoretical works but never observed experimentally. This work may constitute the foundation for experimental analysis of Floquet acoustic metamaterials, accelerated-modulation space-time metamaterials, or acoustic analog of the event horizon.

The Plane Wave Expansion (PWE) method allows the calculation of dispersion curves, i.e., the relation linking the frequency to the wave number for any propagating mode of periodic structures made of elastic materials such as phononic crystals. The method is relatively easy to implement numerically but presents some limitations. After recalling some fundamental aspects of crystallography that are necessary to the study of periodic structures, the PWE method described in detail for the case of bulk phononic crystals, i.e., structures of infinite extent, and its advantages and drawbacks are discussed. It is also shown that the method can be used for calculating the band structure of phononic crystals of finite thickness and for analysing the evanescent waves within the phononic band gaps.

Propagation of longitudinal acoustic waves in a one-dimensional piezoelectric structure with space-time modulated electrical boundary conditions is investigated. An analytical model allowing the calculation of eigenmodes for spatially continuous shifts of electrical boundary conditions is compared with finite difference time domain simulation results for a discrete set of time-varying spatially fixed conditions. Both models predict that such a structure behaves as a nonreciprocal device exhibiting unidirectional propagation properties in some frequency ranges. The modulus and direction of the modulation speed vector strongly affect this nonreciprocal behavior. Moreover, other nonlinear phenomena such as frequency up and down conversions, wave packet distortion, and parasitic echoes occurring in such systems are investigated in detail. The importance of these phenomena is discussed in the context of nonlinear acoustic wave-based components for radio-communication systems.

A piezoelectric plate, poled along its thickness and supporting on its top and bottom surfaces a periodic grating of electrodes, is considered. An analytical model allowing band structure calculation is derived for the first symmetrical mode propagating along the length of the plate. Analytical results show that an electrical Bragg (EB) bandgap can be observed for this mode, depending on the electrical boundary conditions applied on the electrodes. This “EB bandgap” is associated with a discontinuity of the electric field between two successive unit cells. These results are validated with the numerical simulations based on the finite element method. Analytical and numerical results prove that the EB bandgap is highly tunable and can be optimized by changing the crystallographic orientation of the material. A simple tunable filter exploiting this bandgap is designed and fabricated. Experimental results of electrical impedance and electrical potential at the output together with a scanning laser vibrometer analysis are presented, which confirm the theoretical predictions.

We introduce a formalism that enables the calculation of elastic wave functions supported by parallel arrays of coupled one-dimensional elastic waveguides. These wave functions are expressed as tensor products of a spinor part associated with directional degrees of freedom and an orbital angular momentum (OAM) part associated with the phase of the coupled waveguides. We demonstrate that one can construct wave functions as a superposition of these elastic waves, which cannot be written as a tensor product of a spinor part and an OAM part. These elastic wave functions are not separable in the tensor product Hilbert space of directional and OAM subspaces. We show that we can construct maximally nonseparable states that are similar to Bell states.

Non-conventional topology of elastic waves arises from breaking symmetry of phononic structures either intrinsically through internal resonances or extrinsically via application of external stimuli. We develop a spacetime representation based on twistor theory of an intrinsic topological elastic structure composed of a harmonic chain attached to a rigid substrate. Elastic waves in this structure obey the Klein–Gordon and Dirac equations and possesses spinorial character. We demonstrate the mapping between straight line trajectories of these elastic waves in spacetime and the twistor complex space. The twistor representation of these Dirac phonons is related to their topological and fermion-like properties. The second topological phononic structure is an extrinsic structure composed of a one-dimensional elastic medium subjected to a moving superlattice. We report an analogy between the elastic behavior of this time-dependent superlattice, the scalar quantum field theory and general relativity of two types of exotic particle excitations, namely temporal Dirac phonons and temporal ghost (tachyonic) phonons. These phonons live on separate sides of a two-dimensional frequency space and are delimited by ghost lines reminiscent of the conventional light cone. Both phonon types exhibit spinorial amplitudes that can be measured by mapping the particle behavior to the band structure of elastic waves.

A phononic crystal constituted by a one-dimensional piezoelectric material with a periodic distribution of electrodes submitted to space and time-dependent electrical boundary conditions is considered in this work. Interaction of an incident elastic pulse propagating with such phononic crystal is studied using a specific Finite Difference Time Domain model. Simulations are conducted in the case of periodic grounded electrodes “moving” at constant subsonic or supersonic speed. Various nonlinear phenomena resulting from this interaction are observed: Brillouin-like acoustic scattering, non-reciprocity of transmission at fundamental frequency, parametric amplification of the incident wave. The dispersion curve of the wave propagating in the phononic crystal with space-time modulated electrical boundary conditions is also deduced from simulation results.

The properties of a one-dimensional phononic crystal made of identical piezoelectric elements separated by thin metallic electrodes connected to the ground are studied theoretically for cases where the locations of the electrical connections change as a function of time with a specific speed. This spatio-temporal modulation of the electrical boundary conditions results in significant non-linear effects that are evidenced numerically. The interaction between an incident harmonic longitudinal wave and the time-dependent phononic crystal is shown to lead to frequency splitting analogous to Brillouin scattering. Moreover, the boundaries of the Bragg bandgaps are strongly affected, and for some specific modulation speed, one-way wave propagation can be achieved.

Spatiotemporal modulations of the elastic properties of materials can be used to break time and parity symmetry of elastic waves. We show that the form of the elastic band structure depends not only on the spatial and temporal periodicity of a spatiotemporal modulation but also on its shape through its Fourier components. We demonstrate that hybridization gaps open from interactions between the Bloch modes of the periodic medium in absence of the temporal variation of the modulation and the combined sinusoidal components of the Fourier decomposition of the periodic modulation.

Periodic spatio-temporal modulations (STM) of the elastic properties of materials are used to break time and parity symmetry of elastic waves. The shape of the STM is shown to affect band structure asymmetry, independent of its period.Several approaches offer pathways to access the deliberate design of elastic media with broken symmetry, including intrinsic and extrinsic methods. Symmetry in intrinsic systems is broken without addition of energy from the outside. In contrast, symmetry can be broken extrinsically by moving fluids1, 2, gyroscopic inclusions3, or directed and externally driven spatio-temporal modulations of the properties of the medium4, 5.

This work focuses on elastic wave propagation in phononic crystals constituted by homogeneous piezoelectric materials with periodic distributions of electrodes. In such periodic structures, dispersion curves are strongly modified by the variations of the external electrical impedances connected to the electrodes. The case of elastic pulses interacting with such piezoelectric phononic crystals with time dependent electrical boundary conditions is considered. Simulations are performed using a specific Finite Difference Time Domain model developed for the one-dimensional case. Two configurations of time-dependent periodic electrical boundary conditions on the electrodes are studied: i) switch from floating to grounded electrodes during a finite time window; ii) grounded electrodes moving at constant speed. For each case, non linear effects in pulse transmission through a phononic crystal slab are analysed.

Phononic crystals made of piezoelectric composites with 1-3 connectivity are studied theoretically and experimentally. It is shown that they present Bragg band gaps that depend on the periodic electrical boundary conditions. These structures have improved properties compared to phononic crystals composed of bulk piezoelectric elements, especially the existence of larger band gaps and the fact that they do not require severe constraints on their aspect ratios. Experimental results present an overall agreement with the theoretical predictions and clearly show that the pass bands and stop bands of the device under study are easily tunable by only changing the electrical boundary conditions applied on each piezocomposite layer.

We have used multiple-time-scales perturbation theory as well as the numerical methods of molecular dynamics and spectral energy density (SED) to investigate the phonon band structure of a two-chain model with second-order anharmonic interactions. We show that when one chain is linear and the other is nonlinear, the two-chain model exhibits a nonlinear resonance near a critical wave number due to mode self-interaction. The nonlinear resonance enables wave number-dependent interband energy transfer. We have also shown that there exist nonlinear modes within the spectral gap separating the lower and upper branches of the phonon band structure. These modes result from three phonon interactions between a phonon belonging to the nonlinear branch and two phonons lying on the lower branch. This phenomenon offers a mechanism for phonon splitting.