Steady-state Wave Field Research Articles

Band relaxation triggered by modulational instability in topological photonic lattices

Modulational instability in topological photonic lattices enables the selective population of energy bands and generation of steady-state wavefields with well-defined topological invariants. This provides a way to measure bulk topological invariants, which determine the number of robust edge modes appearing at the lattice edges via the bulk-edge correspondence. Here we study numerically the process of wave thermalization arising from modulational instability in topological bands. We apply a grand canonical approach to compute the effective temperature β and chemical potential μ of the steady-state wavefields. The steady-state wavefields exhibit a strong wavevector k -dependence of β and μ throughout the Brillouin zone, suggesting the existence of a long-lived pre-thermal phase and the absence of thermalization for the moderate propagation times accessible using topological photonic lattices.

On two approaches to the third-order solution of surface gravity waves

A third-order approximate solution for surface gravity waves in the finite water depth is studied in the context of potential flow theory. This solution corresponds to the steady-state part of a multidirectional irregular wave field and provides explicit expressions for the surface elevation, free-surface velocity potential, and velocity potential. The amplitude dispersion relation is also provided. Two approaches are used to derive the third-order analytical solution, resulting in two types of approximate solutions: the perturbation solution and the Hamiltonian solution. The perturbation solution is obtained by a classical perturbation technique in which the time variable is expanded in multiscale to eliminate secular terms. The Hamiltonian solution is derived from the canonical transformation in the Hamiltonian theory of water waves. By comparing the two types of solutions, it is found that they are completely equivalent for the first- and second-order solutions and the nonlinear dispersion, but for the third-order part only the sum–sum terms are the same. Due to the canonical transformation that could completely separate the dynamic and bound harmonics, the Hamiltonian solutions break through the difficulty that the perturbation theory breaks down due to singularities in the transfer functions when the quartet resonance criterion is satisfied. Furthermore, it is also found that some time-averaged quantities based on the Hamiltonian solution, such as mean potential energy and mean kinetic energy, are equal to those in the initial state in which the sea surface is assumed to be a Gaussian random process. This is because there are associated conserved quantities in the Hamiltonian form. All of these show that the Hamiltonian solution is more reasonable and accurate to describe the third-order steady-state wave field. Finally, based on the Hamiltonian solution, some statistics are given such as the volume flux, skewness, excess kurtosis, and non-uniqueness of the induced mean flow and mean surface.

Imaging delamination in composite laminates using perturbation to steady-state wavefields

Delamination in composite laminates, though barely visible, can jeopardize integrity and safety of the composite structures. Interaction of guided waves with delamination in full wavefields have been widely used to graphically characterize the delamination. Recently, steady-state wavefields (SWs) in place of transient wavefields have been used for imaging delamination. However, interaction with delamination can be largely obscured by steady-state waves and hardly observed. Addressing this problem, this study proposes a novel approach for imaging delamination using perturbation to SWs. In particular, a new concept of delamination-caused perturbation to a SW is proposed. Compared to scattered waves from delamination, it is determined by delamination only and naturally sensitive to delamination. Delamination-caused perturbation to a SW is concentrated within the delamination region and rapidly attenuates at undamaged locations. On the basis of this new concept, a delamination index (DI) is established for imaging delamination, by which the delamination can be graphically characterized. The capability of the approach is experimentally validated on composite laminates with delamination, whose SWs are acquired through laser scanning. The experimental results validate that delamination imaging can be achieved by DIs, by which the occurrence, locations, and sizes of the delamination can be graphically characterized.

2D-wavelet wavenumber filtering for structural damage detection using full steady-state wavefield laser scanning

This paper presents the use of wavelet-based wavenumber filtering that can be used for damage detection with full steady-state wavefield laser scanning. A piezoelectric transducer provides a single-frequency continuous excitation while a scanning laser doppler vibrometer (SLDV) conducts the full steady-state wavefield measurement. To detect damage, a 2D wavelet transform with various scale components is applied to track any changes in wavenumber and a mapping process based on local wavenumber filtering is carried out. To demonstrate the performance of the proposed technique, several experiments are conducted on composite and aluminum structures, which contain various types of hidden damage, including Teflon inserts and machined thinning. The results demonstrate that this technique performs better than Fourier based wavenumber techniques in terms of damage detection and visualization capability for smaller size defects in a structure. In addition, by providing additional signal processing techniques along with the existing techniques, the proposed method aims to provide rich opportunities to selectively choose the damage detection algorithms for various types of structures and defects, thereby enhancing the damage detection capability of steady-state ultrasonic imaging techniques.

On the theoretical limitations in estimating thickness of a plate-like structure from a full-field single-tone response Lamb wave measurement

A persistent question in wavenumber analysis in the estimation of thickness from a steady-state wave field is identifying the theoretical sensitivity of the system. A well-known trade-off between spatial frequency/wavenumber resolution and thickness resolution exists. The current work presents a calculation of the Cramer-Rao Lower bound (CRLB), specifically as applied to thickness estimates, for a 2-dimensional multi-mode waveform in Gaussian noise. Cases of near-field and far-field excitation are considered, and transducer position with respect to the scan area is also varied. Additionally, we consider the CRLB in a plate with multiple sources, simulated as sources placed on the boundary of the plate. We conclude by presenting the CRLB values in terms of frequency for various thicknesses, and by presenting optimal excitation frequencies for a nominal thickness, based on the CRLB.

Damage detection on composite structures with standing wave excitation and wavenumber analysis

This paper describes the use of wavenumber filtering for damage detection with a signal-frequency standing wave excitation on composite structures. Using a single, fixed frequency excitation from a mounted piezoelectric transducer, the full steady-state wavefield could be obtained using a Laser Doppler Vibrometer with a mirror-tilting device. After completing the scanning, a wavenumber filtering is applied to determine dominant wavenumber components of the measured wavefield, which could be used for indicative of structural damage. Mapping processes based on local wavenumber filtering is then carried out for damaged area visualization. Also introduced are the comparison of two methods for damage identification and visualization: the local wavenumber mapping and acoustic wavenumber spectroscopy. To demonstrate the proposed techniques, several experiments are performed on composite structures with different types of damage, including debonding and delamination on composite plates. The results demonstrate that the techniques are very effective in localizing damage with the potential for the quick inspection of a variety of composite structural components.

Spontaneous Surface Generation and Interior Amplification of Internal Waves in a Regional-Scale Ocean Model

AbstractRecent theories, models, and observations have suggested the presence of significant spontaneous internal wave generation at density fronts near the ocean surface. Spontaneous generation is the emission of waves by unbalanced, large Rossby number flows in the absence of direct forcing. Here, spontaneous generation is investigated in a zonally reentrant channel model using parameter values typical of the Southern Ocean. The model is carefully equilibrated to obtain a steady-state wave field for which a closed energy budget is formulated. There are two main results: First, waves are spontaneously generated at sharp fronts in the top 50 m of the model. The magnitude of the energy flux to the wave field at these fronts is comparable to that from other mechanisms of wave generation. Second, the surface-generated wave field is amplified in the model interior through interaction with horizontal density gradients within the main zonal current. The magnitude of the mean-to-wave conversion in the model interior is comparable to recent observational estimates and is the dominant source of wave energy in the model, exceeding the initial spontaneous generation. This second result suggests that internal amplification of the wave field may contribute to the ocean’s internal wave energy budget at a rate commensurate with known generation mechanisms.

A steady-state solver and stability calculator for nonlinear internal wave flows

A steady solver and stability calculator is presented for the problem of nonlinear internal gravity waves forced by topography. Steady-state solutions are obtained using Newton’s method, as applied to a finite-difference discretization in terrain-following coordinates. The iteration is initialized using a boundary-inflation scheme, in which the nonlinearity of the flow is gradually increased over the first few Newton steps. The resulting method is shown to be robust over the full range of nonhydrostatic and rotating parameter space. Examples are given for both nonhydrostatic and rotating flows, as well as flows with realistic upstream shear and static stability profiles. With a modest extension, the solver also allows for a linear stability analysis of the steady-state wave fields. Unstable modes are computed using a shifted-inverse method, combined with a parameter-space search over a set of realistic target values. An example is given showing resonant instability in a nonhydrostatic mountain wave.

Two-step method for evolving nonlinear acoustic systems to a steady state

A two-step method for evolving two-dimensional nonlinear acoustic systems with flow to a periodic steady state is presented. In the first step of the method, the full nonlinear system governing acoustic disturbances is integrated numerically starting with arbitrary initial conditions using the explicit predictor-corrector method developed by MacCormack. In the second step, the Fourier Time Transform of the computed field is calculated to determine its frequency components. The transient wave field is then filtered from the spectrum and the inverse transform taken to provide an approximation to the steady-state wave field. This approximate field provides a new initial condition for subsequent iterations on the method. The method is tested on a benchmark acoustic problem for which exact steady-state solutions are known and on a nonlinear problem for which a steady-state solution has not been given before. Excellent agreement with the benchmark acoustic solution was obtained within four iterations for planar and nonplanar sources. Convergence to a steady state for the nonlinear problem occurred in six iterations. The two-step method eliminates the need to develop nonreflecting boundary conditions in order to obtain periodic steady-state solutions of two-dimension al acoustic problems and is easily extended to any number of spatial dimensions and to other hyperbolic systems. The procedure is shown to be numerically stable and may provide the only alternative for obtaining steady-state solutions to problems for which nonreflecting boundary conditions are not known or are physically incorrect.

Nonlinear theory of longitudinal oscillations of current plasma in a semiconductor with superlattice

It is shown that the behavior of the amplitude of the electric field has a band character, and is determined by its initial value and by the number of the resonance. i. A nonquadratic structure of the dispersion law for electrons in a semiconductor with superlattice leads to a considerable nonlinearity of the condution current, which manifests itself even in relatively weak electric fields. When the superlattice is acted upon simultaneously by electric field with frequency ~ and by a constant electric field Eo, the conduction current behaves in a resonant fashion [1-3]. A consequence of the nonparabolic structure of the energy spectrum of the superlattice is the possibility of resonance at the multiple frequencies nw = ~, where ~ = eEod (d is the period of the superlattice, and h = i), and n is an integer. For the occurrence of nonlinear resonances with n > i, the amplitude of the field must not be too small, so that the conduction current is principally nonlinear. It is known from the linear theory of the Stark resonance with n = 1 [4] that, for ~, the wave is unstable. It is clear that in a small neighborhood of the nonlinear resonance~n~, the corresponding term in the expression for the conduction current determines negative absorption, i.e., amplification of the wave. However, the contribution of the total absorption comes also from a large number of nonresonant terms. Their finite number with n no (in principle, an infinite number) describe positive absorption. It is clear that a steady-state wave field is established as a result of the competition between these groups of terms. In the present work, we take into account consistently the resonant and nonresonant terms in the expression for the conduction current, in the case of small deviation from the Stark resonance. The obtained results are used to find a condition for growth of the longitudinal oscillations, and to determine the steady-state values of the field. 2. It is assumed that electrons exist only in the lower miniband, and the electric fields do not cause transitions to the upper minibands. We consider homogeneous oscillations of the electric field. We assume that the constant (Eo) and variable (E) electric fields are directed along the axis of the superlattice. The starting equation is the nonlinear equation