Simulation Of Wave Propagation Research Articles

Hexagonal diffractive optical elements.

Diffractive optical elements (DOEs) have widespread applications in optics, ranging from point spread function engineering to holographic display. Conventionally, DOE design relies on Cartesian simulation grids, resulting in square features in the final design. Unfortunately, Cartesian grids provide an anisotropic sampling of the plane, and the resulting square features can be challenging to fabricate with high fidelity using methods such as photolithography. To address these limitations, we explore the use of hexagonal grids as a new grid structure for DOE design and fabrication. In this study, we demonstrate wave propagation simulation using an efficient hexagonal coordinate system and compare simulation accuracy with the standard Cartesian sampling scheme. Additionally, we have implemented algorithms for the inverse DOE design. The resulting hexagonal DOEs, encoded with wavefront information for holograms, are fabricated and experimentally compared to their Cartesian counterparts. Our findings indicate that employing hexagonal grids enhances holographic imaging quality. The exploration of new grid structures holds significant potential for advancing optical technology across various domains, including imaging, microscopy, photography, lighting, and virtual reality.

Frequency-dependent Q simulation and viscoacoustic reverse time migration based on the fractional Zener model

Seismic attenuation is a basic physical property of the earth, which significantly affects the characteristics of seismic wavefields. Accurately simulating wave propagation in the earth is essential to image subsurface structures. Some prevailing methods (e.g., the standard linear solid and fractional Laplacian equation) to describe seismic wave propagation in attenuating media are mainly based on the constant- Q model (CQM), which is valid at room temperature and pressure. However, laboratory measurements suggest that the quality factor Q is a function of frequencies in some regions. To simulate the frequency-dependent Q effect, we derive a viscoacoustic wave equation from the stress-strain relationship of the fractional Zener model (FZM) with variable fractional orders. During the implementation, we separate the real and imaginary parts of the modulus and introduce a low-rank decomposition method to solve the FZM equation. Because the amplitude dissipation and phase dispersion are decoupled, we establish a compensated reverse time migration ( Q-RTM) algorithm to mitigate adverse effects caused by seismic attenuation and improve the quality of seismic migration in frequency-dependent attenuating media. A two-layer model and the BP gas chimney model are used to perform Q-RTM tests. A low-pass filter with a Tukey window function is applied to suppress numerical instability during the compensation. Numerical results demonstrate that our FZM Q-RTM approach can produce high-resolution images with corrected reflector positions and amplitudes. Because the CQM equation ignores the frequency dependence of Q, it may lead to overcompensation in Q-RTM.

Numerical simulations of breaking wave propagation through the vegetation on a slope based on a drag coefficient prediction model

Coastal vegetation plays a key role in wave attenuation on sloping beaches. However, accurately predicting the vegetation-related drag coefficient (CD) under breaking waves on slopes remains a challenge. In this study, the Reynolds-averaged Navier–Stokes equations were modified by incorporating a vegetation resistance term to consider the vegetation-induced momentum loss. The stabilized k-ω shear stress transport turbulence model was improved by introducing the vegetation-induced turbulence generation and dissipation terms. The numerical model was validated by two laboratory experiments. Based on experimental wave height data, the CD was determined via a calibration method. The numerical results were found to be highly sensitive to the CD. The calibrated CD achieved excellent agreement with experimental data (RMSE < 0.2). In addition, the calibrated CD showed a high correlation (R2 = 0.85) with the Iribarren number ξ. Finally, a CD prediction formula expressed by the combination of Reynolds number and ξ was proposed via regression analysis, and its performance was evaluated by two additional cases. The limitations and applicability of the CD prediction formula were also discussed. The proposed numerical model and CD prediction formula have the potential to be useful for the numerical simulation of wave interactions with vegetation on slopes.

Задача о колебаниях в цепи связанных линейных осцилляторов с демпфирующей и антидемпфирующей границами

The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries. An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied. The problem is solved in Schrodinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

The Curie effect incorporation in the monodomain equation describing the action potential dynamics in cardiac tissue

In their in-depth study on cardiac tissue modeling, Clayton and Panfilov [1] present several monodomain or bidomain approaches for the mathematical description of the cardiac tissue action potential dynamics. For simulation of wave propagation in the cardiac tissue, the monodomain descriptions which use integer order derivatives reproduce many of the phenomena that are observed experimentally and are thus an appropriate analysis tool. The main objection concerning the monodomain approaches is that the electrical circuit capacitor, appearing in these descriptions, is considered ideal (the space between the capacitor plates is vacuum) and the Curie effect is ignored. The Curie effect consists of the fact that in case of a dielectric material, if at a moment of time a constant external voltage is applied, due to the capacitance of the capacitor and the properties of the dielectric, a supplementary electrical current is produced, besides the ohmic current. This supplementary contribution cannot be neglected in some cases. In this paper, the Curie effect, describing the action potential dynamics in cardiac tissue, assumed isotropic, is incorporated in the monodomain equation. The novelty is that this approach does not use fractional order derivatives and the obtained mathematical description with these equations is objective.

SPH Simulation of the Interaction between Freak Waves and Bottom-Fixed Structures

In this paper, the Smoothed Particle Hydrodynamics (SPH) method is used in a C# environment to simulate the interaction between freak waves and bottom-fixed structures by establishing a fluid dynamics model. Paraview software 5.10.1 was used to analyze and visualize the simulation results. In order to simulate wave propagation accurately, the reliability of the model was verified by comparing experimental and simulated data. A two-dimensional numerical wave flume was established based on the SPH method, a conservative Riemann solver was introduced, a repulsive boundary condition was adopted, and a slope was used to eliminate wave reflection. Bottom-fixed structures of different heights and lengths, as well as different wave conditions, were selected to numerically simulate the interaction between freak waves and bottom-fixed structures. The results show that the height of bottom-fixed structures and wave conditions have a significant effect on hindering the propagation of rogue waves, while the length has little effect on the propagation of deformed waves. When the amplitude of the wave remains constant, both the period andthe duration of the deformed wave are longer. This research is of certain significance for the prediction of freak waves in marine engineering and the application and promotion of SPH methods.

Realistic simulation of GPR for landmine and IED detection including antenna models, soil dispersion and heterogeneity

AbstractGround‐penetrating radar is an effective tool for detecting landmines and improvised explosive devices, but its performance is strongly influenced by subsurface properties as well as the characteristics of the target. To complement or replace labour‐intensive experiments on test sites, cost‐efficient electromagnetic wave propagation simulations using the finite‐difference time‐domain method are being increasingly used. However, to obtain realistic synthetic data, accurate modelling of signal alteration caused by dispersion, scattering from soil material, target contrast, shape, and inner setup, as well as the coupling effects of the antenna to the ground is required. In this study, we present a detailed three‐dimensional model of a shielded ground‐penetrating radar antenna applied to various scenarios containing metallic and non‐metallic targets buried in different soils. The frequency‐dependent intrinsic material properties of soil samples were measured with the coaxial transmission‐line technique, while a discrete random media was used to implement the heterogeneity of a gravel based on its grain‐size distribution. Our simulations show very good agreement with experimental validation data collected under controlled conditions. We accurately reproduce the amplitude, frequency, and phase of target signals, the subsurface background noise, antenna crosstalk and associated interference with target signals, and the effect of antenna elevation. The approach allows for a systematic investigation of the effects of soil, target, and sensor properties on detection performance, providing insight into novel and complex ground‐penetrating radar scenarios and the potential for a wide range of simulation possibilities for demining with ground‐penetrating radar. These investigations have the potential to improve the safety and effectiveness of landmine and improvised explosive device detection in the future, such as building a database for training deminers or developing automatic signal pattern recognition algorithms.

A perfectly matched layer for the absorption of electromagnetic waves using differential forms in three-dimensional domains

As a significant contribution, we present a novel formulation of the complex frequency-shifted perfectly matched layer (CFS-PML) for differential forms, introducing the innovative concept of a directional incidence matrix. This paper explores the application of differential forms in simulating wave propagation problems in three-dimensional domains. To confine the unbounded domain, we utilize a discretization technique based on the CFS-PML. Additionally, we provide an illustrative example that involves a system comprising a GPR (Ground Penetrating Radar) horn antenna, air, lossy soil, and a buried sphere.

Accurate simulations of pure-viscoacoustic wave propagation in tilted transversely isotropic media

Forward modeling of seismic wave propagation is crucial for the realization of reverse time migration (RTM) and full waveform inversion (FWI) in attenuating transversely isotropic media. To describe the attenuation and anisotropy properties of subsurface media, the pure-viscoacoustic anisotropic wave equations are established for wavefield simulations, because they can provide clear and stable wavefields. However, due to the use of several approximations in deriving the wave equation and the introduction of a fractional Laplacian approximation in solving the derived equation, the wavefields simulated by the previous pure-viscoacoustic tilted transversely isotropic (TTI) wave equations has low accuracy. To accurately simulate wavefields in media with velocity anisotropy and attenuation anisotropy, we first derive a new pure-viscoacoustic TTI wave equation from the exact complex-valued dispersion formula in viscoelastic vertical transversely isotropic (VTI) media. Then, we present the hybrid finite-difference and low-rank decomposition (HFDLRD) method to accurately solve our proposed pure-viscoacoustic TTI wave equation. Theoretical analysis and numerical examples suggest that our pure-viscoacoustic TTI wave equation has higher accuracy than previous pure-viscoacoustic TTI wave equations in describing qP-wave kinematic and attenuation characteristics. Additionally, the numerical experiment in a simple two-layer model shows that the HFDLRD technique outperforms the hybrid finite-difference and pseudo-spectral (HFDPS) method in terms of accuracy of wavefield modeling.

Data-driven algorithm based on the scaled boundary finite element method and deep learning for the identification of multiple cracks in massive structures

Structural defect identification is a vital aspect of structural health monitoring used to assess the safety of engineering structures. However, quantitatively determining the dimensions of structural defects is often difficult. Therefore, this study presents an innovative data-driven algorithm that combines the scaled boundary finite element method (SBFEM) and a deep learning framework based on a dilated causal convolutional neural network (CNN) to identify crack-like defects in large-scale structures. The SBFEM is used to simulate different crack-like defects. Mesh generation is significantly simplified by a simple procedure that requires only changing the scale centre at the crack tip and the positions of the nodes at the crack opening. This minimises remeshing and enables simple generation of sufficient data to train the neural network. In addition, an absorbing boundary model based on Rayleigh damping is used to avoid computing the entire model when simulating wave propagation in massive structures. To ensure that sequential data remain ordered and to obtain a large receptive field without increasing the complexity of the neural network, a dilated causal CNN is employed in the deep learning framework. Therefore, more historical information is captured, and the complex mapping relationship between the echo signal and the crack information is efficiently learnt. The proposed model can accurately identify the number, location, and depth of cracks in massive structures. Moreover, it is robust to noise, which is demonstrated via numerical examples. Therefore, the proposed algorithm provides valuable insight into the detection and diagnosis of structural defects, which can ultimately improve the safety of engineering structures.

A surrogate model for finite element simulation of elastic wave propagation in composite materials

Deep learning surrogate models can be employed in solid mechanics to forecast the behavior of structures subjected to various loading conditions, substantially decreasing the computational costs associated with simulations. In this letter, we have utilized convolutional neural networks and Fourier transform to predict the elastic wave output from composite bars. The microstructures of the bar are utilized as inputs to the deep learning model, while the output is the elastic wave response. The convolutional neural network learns to identify crucial input composite features and utilizes this information to predict the output elastic waves. Finally, the mean squared error of the predicted output signals is compared to the actual output signals, which was used to evaluate the model. The outcomes of this study demonstrate that the deep learning model can precisely and swiftly predict the output elastic waves of the composites, thus serving as a surrogate model for time-consuming finite element simulations.

Implicit coupling methods for nonlinear interactions between a large‐deformable hyperelastic solid and a viscous acoustic fluid of infinite extent

AbstractThis paper addresses the challenges in studying the interaction between high‐intensity sound waves and large‐deformable hyperelastic solids, which are characterized by nonlinearities of the hyperelastic material, the finite‐amplitude acoustic wave, and the large‐deformable fluid–solid interface. An implicit coupling method is proposed for predicting nonlinear structural‐acoustic responses of the large‐deformable hyperelastic solid submerged in a compressible viscous fluid of infinite extent. An arbitrary Lagrangian–Eulerian (ALE) formulation based on an unsplit complex‐frequency‐shifted perfectly matched layer method is developed for long‐time simulation of the nonlinear acoustic wave propagation without exhibiting long‐time instabilities. The solid and acoustic fluid domains are discretized using the finite element method, and two different options of staggered implicit coupling procedures for nonlinear structural‐acoustic interactions are developed. Theoretical formulations for stability analysis of the implicit methods are provided. The accuracy, robustness, and convergence properties of the proposed methods are evaluated by a benchmark problem, that is, a hyperelastic rod interacting with finite‐amplitude acoustic waves. The numerical results substantiate that the present methods are able to provide long‐time steady‐state solutions for a nonlinear coupled hyperelastic solid and viscous acoustic fluid system without numerical constraints of small time step sizes and long‐time instabilities. The methods are applied to investigate nonlinear dynamic behaviors of coupled hyperelastic elliptical ring and acoustic fluid systems. Physical insights into 2:1 and 4:2:1 internal resonances of the hyperelastic elliptical ring and period‐doubling bifurcations of the structural and acoustic responses of the system are provided.

Geophysical Research on an Open Pit Mine for Geotechnical Planning and Future Land Reclamation: A Case Study from NW Macedonia, Greece

In open pit mining areas, knowledge of geotechnical conditions (e.g., overburden thickness, background slope, and fault locations) ensures geotechnical safety during exploitation as well as reclamation planning. The Greek Public Power Corporation initiated a research program after stability issues emerged on the southern side of the Mavropigi open pit mine in NW Macedonia. Geotechnical wells revealed steeply dipping bedrock and thin tectonic contact, indicating the need for the detailed imaging of the subsurface for future stability measures. For this purpose, a geophysical investigation aimed to extract information mostly for the dip of the interface between schist bedrock and overlaying Neogene sediments and/or limestones. Based on the high contrast of electrical properties between schists and limestones, as well as the differences in acoustic impedance and formation thickness, the seismic reflection and electrical resistivity tomography (ERT) methods were selected. The suitability of the seismic reflection for its application in this area was checked by generating synthetic seismic data, which resulted from the simulation of seismic wave propagation for geological models of the area. The acquisition parameters were determined after the noise test. Field seismic data processing produced a depth-migrated section, which revealed the existence of a fault. The use of dipole–dipole and gradient arrays, in 2D and 3D electrical resistivity measurements, ensured both the lateral and vertical mapping of schist bedrock and detected limestone bodies within the overburden. Also, the tectonic contact zone between limestone and schist formations was properly imaged. The comparison between geoelectrical and seismic sections indicated that the seismic reflection method provided a more accurate estimate of fault inclination. Finally, the geophysical survey enriched the geotechnical models necessary for sustainable mining (e.g., rational exploitation, the optimization of productivity, and zero accidents) including the planning of future reclamation.

Simulation of wave propagation with obstacles: Time invariance operator applied to interference and diffraction

We present here a computational numerical operator, and we name it as Time Invariance Operator (TIO). This operator can add obstacles to the domain of the differential equation that describes a physical phenomenon. After the TIO acts, the wave equation recognizes the introduced points as non-interacting zones without affecting the rest of the domain. Computational physics has been consolidated as an important field of study, especially when connected with the fundamentals of physics. In many cases, simulations are conducted considering the ideal case of a wave in an infinite domain and open space without considering obstacles, barriers, or other aspects of the real world. The results presented in this paper allow us to infer that the TIO is the easiest way to apply the physical domain to wave propagation simulations and successfully recreate wave interaction experiments through computer simulations. Our motivation is to perform wave simulations that interact with obstacles, barriers, single slits, and double slits. We aim to investigate the results obtained in images to determine if the methodology we used to introduce realistic physical characteristics was successful in presenting the expected phenomenology. The simplicity of the TIO’s action in creating locally time-invariant regions over the domain makes it suitable not only for waves but also for equations with transient terms. Heat transfer, mass transfer, computational fluid dynamics, and other time evolution equations can take some benefit from the operator presented in this paper. The TIO ensures local conservation that mimics interaction regions or ensures free space characteristics if it is the case like a 2D tensor of local conservation. The principal result from this paper is the validation of the TIO to impose realistic conditions with minimal modifications over a running code of wave equation simulation originally in free space. The TIO is innovative because it imposes dynamic conditions that mimic realistic interacting zones.

High performance calculation of nonlinear wave propagation on GPU for focused ultrasound treatment planning

High intensity focused ultrasound (HIFU) is very promising new technology that has many therapeutic application, and among them are the treatment of cancer in different organs without major side effects. Some of the limitations of HIFU are the long treatment time and incomplete ablation. High performance computing can help in this regard. Simulation of nonlinear wave propagation in three dimensional geometry is very time consuming process especially when shocks are presented. In the current work, we are going to present three dimensional parallel solver on GPU. Some examples for the simulation in a patient specific geometry will be also presented. The presented results can be used for the optimization of the treatment. For the validation of the results, ex-vivo and in-vivo experiments have been performed. It has been shown that large volume can be ablated within a short period of time.

TFM imaging simulations of defects near or in the weld bevel: modelling approach and customized finite element tools

In safety-critical industries, including the nuclear sector, ultrasonic testing (UT) of welded components is an area of continuous improvement. We have collaborated on developing advanced UT simulation tools to address challenges in monitoring and imaging defects near or within a weld bevel. These tools, integrated into the CIVA software platform, use hybrid approaches combining ray models and finite elements (FE) to accurately simulate wave propagation and diffraction phenomena in both 2D and 3D inspection configurations. While computation time is longer compared to ray-based approaches, the results serve as a reference for improving reliability and validating simulation tools. A comprehensive parameterization of the weld is proposed, utilizing knowledge on anisotropy, attenuation properties, and variations within the bevel, to achieve reliable UT results. This paper reviews the new features developed, including the integration of the MINA model for the weld description and improvements in bevel geometry. The automated coupling methodology and meshing procedures, which do not require any specific numerical expertise to be used, are discussed. As an application, we present TFM imaging simulations, comparing results from the standard paraxial ray-based approach with the new module utilizing the FE computational core.

A review of misfit functions for adjoint full waveform inversion in seismology

SUMMARY In seismological full waveform inversion, the choice of misfit functions plays a critical role in quantifying the discrepancy between observed and synthetic data, affecting convergence rate and also final results. We revisit and compare six commonly used misfit functions, including cross-correlation time-shift (CC), least-square waveform difference (L2), multitaper time-shift (MT), exponentiated phase shift (EP), time–frequency phase shift (TF) and zero-lag cross-correlation coefficient (CCC), with respect to their definitions, adjoint sources and misfit kernels for velocity perturbations. Synthetic tests are performed for several canonical models. First, we simulated wave propagation in a model with a single rectangular anomaly with sharp boundaries and a smoothed variant of that model. We analysed the resulting misfit kernels first for the P-wave phase, which is highly distorted in the sharp model due to strong heterogeneities, and mostly experiences traveltime perturbations in the smooth model. Second, we considered a model where a laterally limited region is subject to layered anomalies (low velocity in the middle crust and high velocity in the lower crust) and determine misfit kernels for S and surface waves in this model. Based on these two simplified seismological scenarios, we further perform iterative test inversions using different misfit functions. Combining the features of misfit kernels and synthetic inversion results, we find that CCC, L2 and EP are the most effective at identifying the sharpness of velocity anomalies from the direct body waves and their scattered phases. Consequently, inversion based on those misfit measures yielded the best recovery in the inversion test. For surface and S waves from crustal sources, TF appears to be the most effective in constraining the heterogeneous structure in the crust but needs more iterations for convergence than other misfit functions.

Spectral-infinite-element simulations of seismic wave propagation in self-gravitating, rotating 3-D Earth models

SUMMARY Although observation of gravity perturbations induced by earthquakes is possible, simulation of seismic wave propagation in a self-gravitating, rotating Earth model with 3-D heterogeneity is challenging due to the numerical complexities associated with the unbounded Poisson/Laplace equation that governs gravity perturbations. Therefore, gravity perturbations are generally omitted, and only the background gravity is taken into account using the so-called Cowling approximation. However, gravity perturbations may be significant for large earthquakes (Mw ≥ 6.0) and long-period responses. In this study, we develop a time-domain solver based on the spectral-infinite-element approach, which combines the spectral element method inside the Earth domain with a mapped-infinite-element method in the infinite space outside. This combination allows us to solve the complete, coupled momentum-gravitational equations in a fully discretized domain while accommodating complex 3-D Earth models. We compute displacement and gravity perturbations considering various Earth models, including Preliminary Reference Earth Model and S40RTS and conduct comprehensive benchmarks of our method against the spherical harmonics normal-mode approach and the direct radial integration method. Our 3-D simulations accommodate topography, bathymetry, rotation, ellipticity and oceans. Results show that our technique is accurate and stable for long simulations. Our method provides a new scope for incorporating earthquake-induced gravity perturbations into source and adjoint tomographic inversions.

Effect of intermittent joint distribution on the mechanical and acoustic behavior of rock masses

The mechanical characteristics and acoustic behavior of rock masses are greatly influenced by stochastic joints. In this study, numerical models of rock masses incorporating intermittent joints with different numbers and dip angles were produced using the finite element method (FEM) with the intrinsic cohesive zone model (ICZM). Then, the uniaxial compressive and wave propagation simulations were performed. The results indicate that the joint number and dip angle can affect the mechanical and acoustic properties of the models. The uniaxial compressive strength (UCS) and wave velocity of rock masses decrease monotonically as the joint number increases. However, the wave velocity grows monotonically as the joint dip angle increases. When the joint dip angle is 45°–60°, the UCS of the rock mass is lower than that of other dip angles. The wave velocity parallel to the joints is greater than that perpendicular to the joints. When the dip angle of joints remains unchanged, the UCS and wave velocity are positively related. When the joint dip angle increases, the variation amplitude of the UCS regarding the wave velocity increases. To reveal the effect of the joint distribution on the velocity, a theoretical model was also proposed. According to the theoretical wave velocity, the change in wave velocity of models with various joint numbers and dip angles was consistent with the simulation results. Furthermore, a theoretical indicator (i.e. fabric tensor) was adopted to analyze the variation of the wave velocity and UCS.

3D Wave Propagation Simulations of Mw 6.5+ Earthquakes on the Tacoma Fault, Washington State, Considering the Effects of Topography, a Geotechnical Gradient, and a Fault Damage Zone

ABSTRACT We simulate shaking in Tacoma, Washington, and surrounding areas from Mw 6.5 and 7.0 earthquakes on the Tacoma fault. Ground motions are directly modeled up to 2.5 Hz using kinematic, finite-fault sources; a 3D seismic velocity model considering regional geology; and a model mesh with 30 m sampling at the ground surface. In addition, we explore how adjustments to the seismic velocity model affect predicted shaking over a range of periods. These adjustments include the addition of a region-specific geotechnical gradient, surface topography, and a fault damage zone. We find that the simulated shaking tends to be near estimates from empirical ground-motion models (GMMs). However, long-period (T = 5.0 s) shaking within the Tacoma basin is typically underpredicted by the GMMs. The fit between simulated and GMM-derived short-period (T = 0.5 s) shaking is significantly improved with the addition of the geotechnical gradient. From comparing different Mw 6.5 earthquake scenarios, we also find that the response of the Tacoma basin is sensitive to the azimuth of incoming seismic waves. In adding surface topography to the simulation, we find that average ground motion is similar to that produced from the nontopography model. However, shaking is often amplified at topographic highs and deamplified at topographic lows, and the wavefield undergoes extensive scattering. Adding a fault damage zone has the effect of amplifying short-period shaking adjacent to the fault, while reducing far-field shaking. Intermediate-period shaking is amplified within the Tacoma basin, likely due to enhanced surface-wave generation attributable to the fault damage zone waveguide. When applied in the same model, the topography and fault damage zone adjustments often enhance or reduce the effects of one another, adding further complexity to the wavefield. These results emphasize the importance of improving near-surface velocity model resolution as waveform simulations progress toward higher frequencies.