Wave Propagation Research Articles

A study of the nonlinear behaviour of the irregular Beni Haroun cable-stayed bridge to progressive seismic failure under SVGM

The progressive failure of the nonlinear irregular Cable-Stayed Bridges (CSBs) under Spatial Variability of Ground Motions (SVGM) counting seismic wave propagation, incoherence effect and site-response effect is investigated. A spectral representation method is used in order to develop a computer code. This code is used to simulate SVGM time histories to be compatible with an overall coherency function and a target response spectrum, A three-dimensional Finite Element Model (FEM) of an asymmetric bridge: the Beni Haroun cable-stayed bridge (CSB) (North Eastern Algeria) is developed in order to include the effects of the main components of SVGM on response parameters in terms of variations of absolute maximum vertical displacements and flexural moments a long the deck and on height of short and tall pylons of study CSB, in particular, nonlinear bridge response quantities are examined in terms of rotational ductility demands at the ends of the short and tall piers of the bridge under all components of SVGM. The numerical results are comforted with those obtained assuming uniform seismic excitations. The analysis results reveal, among others, that the bridge responses are very sensitive to the SVGM, with site-response effects being more critical for bridge response than those of incoherence components and wave propagation. As a general trend, it is showed that the assumption of uniform excitation results in much lower ductility demands on bridges than incoherence and site SVGM components and that the latter should be considered for the seismic response assessments and progressive seismic failure studies of irregular cable-stayed bridges.

Three-Dimensional Analysis of Vocal Fold Oscillations: Correlating Superior and Medial Surface Dynamics Using Ex Vivo Human Hemilarynges.

The primary acoustic signal of the voice is generated by the complex oscillation of the vocal folds (VFs), whereby physicians can barely examine the medial VF surface due to its anatomical inaccessibility. In this study, we investigated possibilities to infer medial surface dynamics by analyzing correlations in the oscillatory behavior of the superior and medial VF surfaces of four human hemilarynges, each in 24 different combinations of flow rate, VF adduction, and elongation. The two surfaces were recorded synchronously during sustained phonation using two high-speed camera setups and were subsequently 3D-reconstructed. The 3D surface parameters of mean and maximum velocities and displacements and general phonation parameters were calculated. The VF oscillations were also analyzed using empirical eigenfunctions (EEFs) and mucosal wave propagation, calculated from medial surface trajectories. Strong linear correlations were found between the 3D parameters of the superior and medial VF surfaces, ranging from 0.8 to 0.95. The linear regressions showed similar values for the maximum velocities at all hemilarynges (0.69-0.9), indicating the most promising parameter for predicting the medial surface. Since excessive VF velocities are suspected to cause phono-trauma and VF polyps, this parameter could provide added value to laryngeal diagnostics in the future.

Experimental and numerical analysis of the geometry of a laboratory-scale overtopping wave energy converter using constructal design

An experimental study is conducted to investigate the influence of the design of a laboratory-scale onshore overtopping device over the mass of water accumulated in its reservoir, and generate a database to validate computational models. The geometric evaluation is based on the constructal design, being investigated one degree of freedom (ratio between the height and length of the ramp, H1/L1) for various depths of free surface of the water (h1). Four geometric constraints delimit the ramps configurations. The exhaustive search is applied to obtain the ramp geometry that maximizes the amount of water that enters the device reservoir, represented by the water level accumulated in the reservoir (hR). For h1 = 0.392 m cases, a numerical model based on the finite volume method is also used for validation purposes. Results indicated that the lowest H1/L1 ratios of the ramp led to the best performances for all studied magnitudes of h1, which agreed with previous findings of literature for similar wave conditions. The proposed numerical model was validated for predicting wave propagation, instantaneous amount of water discharge, and prediction of the effect of H1/L1 over hR, with a maximum relative error (RE) of 3.92 %.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Convergence and divergence of storm waves induced by multi-scale currents: Observations and coupled wave-current modeling

Current effects on waves (CEW) are among the most intricate physical processes in wave evolution. In this study, we used a coupled wave-tide-circulation model for the Northwest Atlantic to investigate current effects on storm waves during Hurricane Igor in 2010. Validated with extensive buoy and altimeter data, the inclusion of CEW in the model significantly improves the accuracy in simulating significant wave heights (Hs) by up to 21.3% for a wave buoy. Storm waves experience significant temporal and spatial modulation by multi-scale currents. Storm-driven currents have the most pronounced impact to the right of the storm track, which typically align with wave propagation and reduce Hs by up to 12.1%. The subsequent near-inertial oscillations induce temporal fluctuations of wave convergence and divergence at near-inertial frequencies, which also occurs in regions with strong tidal currents but at tidal frequencies. Furthermore, storm waves are modulated by the Gulf Stream, Labrador Current and associated mesoscale eddies. Overall, these multi-scales yield strong effects on storm waves (Hs > 3.0 m), significantly modulating Hs (−25.2%–+55.4%) and mean wave periods (−14.9%–+15.7%). The mean wave energy power shows more significant modulation by multi-scale currents, reflecting the combined effects of changing wave states and current-induced transport of wave energy. CEW are governed by the interactive dynamic and kinematic effects. The relative wind effect is the primary mechanism for lower storm waves by reducing energy input to waves and influences downstream wave states. Among kinematic effects, current-induced wave refraction typically plays a dominant role in redistributing wave energy. This study systematically quantified the modulation of storm waves by multi-scale currents and revealed the underlying mechanisms, providing a comprehensive understanding of extreme wave states under coupled ocean dynamics.

Real-time phase-resolved ocean wave prediction in directional wave fields: Second-order Lagrangian wave models

The Improved Choppy Wave Model (ICWM) was established to achieve a second-order Lagrangian expansion of surface waves. The second-order Lagrangian nonlinear interaction terms were found to be negligible and were therefore discarded. However, these interactions in directional wave fields remained unexplored. ICWM was successfully used, especially for floating offshore wind turbines, but it was noted that its accuracy in predicting directional sea states needed improvement. Hence, we formulated the Complementary ICWM (CICWM) with the nonlinear terms for free surface elevation in directional waves. Detailed formulations for data assimilation and wave propagation were provided, enabling real-time wave forecasting for directional seas using a simplified assimilation method. Comparing the model performances against tank-scale experiments showed that CICWM enhances the surface elevation description in directional seas. Ultimately, it was confirmed that the second-order Lagrangian model can reduce the prediction error of the linear wave model by 90% in directional seas for the experimental setups and sea states investigated in this study.

Dynamics of Love-type wave propagation in composite transversely isotropic porous structures

The present study aims to analyze the propagation behavior of Love-type wave in a composite transversely isotropic porous structure. The structure comprises an inhomogeneous sandy porous layer lying between a non-homogeneous magneto-poroelastic layer and a heterogeneous fractured porous half-space. Analytic solutions of the field equations of the respective media involve the application of the variable separable method and Wentzel-Kramers-Brillouin (WKB) asymptotic approach for the conversion of partial differential equations into ordinary differential equations. Through careful imposition of boundary conditions and subsequent elimination of arbitrary constants, we derive a complex dispersion relation governing the propagation of Love-type waves. This dispersion equation yields both the phase velocity curve, corresponding to the real expression, and the damping velocity curve, derived from the imaginary expression. To represent our findings, we conduct extensive calculations and graphical simulations illustrating the influence of various material parameters such as heterogeneity, porosity, volume fraction of fractures, sandiness, magnet-oelastic coupling, angle at which wave crosses the magnetic field, and layer thickness on the dispersive nature of Love-type waves using MATHEMATICA software. Furthermore, we conduct case-specific analyses, revealing instances where the dispersion equation simplifies to the standard Love wave equation, thereby validating our mathematical framework. Our findings underscore the significant influence of the aforementioned material parameters on the phase and damping velocities of Love-type wave. This interdisciplinary investigation into different porous media opens new avenues for future research and has significant implications in various disciplines, ranging from engineering and geophysics to environmental science and beyond.

Solid face sheets enable lattice metamaterials to withstand high-amplitude impulsive loading without yielding

Owing to their ability to provide tunable mechanical responses, lattice materials are frequently studied to elucidate their response to static and dynamic loads. However, these roles are typically in opposition: static loads must be supported sufficiently far away from the onset of buckling or yielding, whereas dynamic loads are typically ameliorated by crushing of the lattice, which provides excellent energy-absorption due to the large plastic deformation accompanying densification. In contrast, this work considers the octet truss as an exemplar topology, in a structural role where it must simultaneously support static loads while enduring high-amplitude impulsive loads. This study focuses on the ability to withstand impulsive loads without yielding, an essential prerequisite to enduring dual loading. Computational studies using the ALE3D hydrocode were performed to examine the response of the octet truss under a short temporal width impulse shape associated with laser-driven shocks. A key finding was that covering the lattice with a solid face sheet and treating this face sheet thickness as a design variable allows the Taylor-like pulse to be attenuated prior to entering the weaker lattice, at the cost of added mass up front. Experimental validation was accomplished by laser-driven shock testing, using octet trusses printed out of Ti-5Al-5V-5Mo-3Cr. The results show that for a given quantity of mass, the attenuation is maximized when as much mass as possible is moved into the face sheet, leaving a more slender lattice structure. The effect of placing mass in the face sheet rather than lattice beams dominates the effect of relative density, to the point where a low-mass structure with most of the mass concentrated in the face sheet can outperform a high-mass structure with most of the mass in the lattice. By further understanding the propagation of short pulse width waves within under-dense structures, this study expand the domain of applicability of such structures, including lattice materials, to challenging dual-loading regimes spanning decades of strain rates.

Physical analysis of high refractive index metamaterial-based radiation aggregation engineering of planar dipole antenna for gain enhancement of mm-wave applications

A metamaterial-incorporated high-gain and broadband dipole antenna is proposed for 5G mm-wave applications. The designed high refractive index metamaterial (HRIM) properties are presented in detail with supporting results. The proposed dipole antenna achieved broadband features, and the antenna gain increased significantly by incorporating HRIM in the electromagnetic (EM) wave propagation path. Besides, the EM wave aggregation engineering of utilizing the HRIM on the dipole antenna is analyzed using the electric field, magnetic field, and power flow distributions. Both conventional and HRIM-based dipole antenna (HRIMDA) were fabricated on thin (0.254 mm) Rogers RT5880 substrate material with a low dielectric constant of 2.2, where the noticeable gain enhancement by HRIM is observed. The measured results show the operational bandwidth (BW) from 23 to 38 GHz frequency, and the highest gain of 9.5 dBi is accomplished at 35 GHz frequency. Finally, a four-element MIMO configuration is numerically and experimentally analyzed where the isolation of the two conjugated MIMO elements is < − 20 dB. The MIMO parameters like envelop correlation coefficient (ECC) < 1 × 10–3 and diversity gain (DG) value of > 9.9 were also achieved. Hence, the proposed HRIM base dipole antenna is a potential candidate for 5G mm-wave applications.

Electromagnetic field produced by radiation source submerged in non-homogeneous seawater

Electromagnetic (EM) waves are one of the important carriers for information exchange in seawater. The EM parameters of seawater are important factors that affect the performance of EM wave propagation. In the real marine environment, non-constant EM parameters make it impossible to consider seawater as a homogeneous media in the model. However, there are few targeted analyses in the existing studies. In this paper, an N-layered media model was established to investigate the influence of non-constant EM parameters on EM wave propagation in seawater. A direct global matrix method is proposed to solve the EM fields, whose result accuracy does not decrease with the increasing number of layers. The necessity and correctness of the model and methods are verified through numerical simulations and sea experiments. Combined with the existing marine environment database, the propagation characteristics of EM waves in different EM parameter profiles of seawater were analyzed through numerical experiments. The results indicated that the non-homogeneous seawater has a great impact on the intensity, phase, change rate, and spatial distribution of EM waves. Furthermore, the influence is not only reflected in the underwater EM field but also in the air and increases with frequency.

Инструментальные наблюдения и статистический анализ вертикальных профилей температуры в юго-западной части залива Петра Великого Японского моря

The study is based on data from four moorings stretched out in the north-northwest – south-southeast direction across the isobaths in the southwestern Peter the Great Bay, the Sea of Japan, on 3–14 October, 2022. For this purpose, timeseries of vertical profiles of temperature and its gradient were expanded into empirical orthogonal functions; timescales were estimated using wavelet transform. In all cases, the leading mode captures vertical stratification variability related to the thermocline shoaling or deepening, manifesting itself by concerted oscillations of temperature and its vertical gradient, which occurred on the near-inertial, semi-diurnal and two-diurnal timescales. The near-inertial oscillations were the most intense during most of the time, while the semi-diurnal ones were the most intense on the fourth and fifth days. From the joint wavelet spectra it was found that temperature anomalies moved towards the shallower area and the speed was estimated as equal to 0.44–0.55 and 0.95–1.11 m/s for the near-inertial and semi-diurnal timescales, respectively. The speed on the two-diurnal timescale differs by an order of magnitude between the pairs of two deeper and two shallower moorings, equaling to 1.17 and 0.15 m/s, respectively. The results were interpreted as related to the near-inertial and semi-diurnal internal wave propagation from the continental slope towards the inner shelf and two-diurnal timescale was speculated to be a difference between them caused by non-linearity.

Low-affinity ligands of the epidermal growth factor receptor are long-range signal transmitters during collective cell migration of epithelial cells.

Epidermal growth factor receptor ligands (EGFRLs) consist of seven proteins. In stark contrast to the amassed knowledge concerning the epidermal growth factor receptors themselves, the extracellular dynamics of individual EGFRLs remain elusive. Here, employing fluorescent probes and a tool for triggering ectodomain shedding of EGFRLs, we show that EREG, a low-affinity EGFRL, exhibits the most rapid and efficient activation of EGFR in confluent epithelial cells and mouse epidermis. In Madin-Darby canine kidney (MDCK) renal epithelial cells, EGFR- and ERK-activation waves propagate during collective cell migration in an ADAM17 sheddase- and EGFRL-dependent manner. Upon induction of EGFRL shedding, radial ERK activation waves were observed in the surrounding receiver cells. Notably, the low-affinity ligands EREG and AREG mediated faster and broader ERK waves than the high-affinity ligands. The integrity of tight/adherens junctions was essential for the propagation of ERK activation, implying that the tight intercellular spaces prefer the low-affinity EGFRL to the high-affinity ligands for efficient signal transmission. To validate this observation in vivo , we generated EREG-deficient mice expressing the ERK biosensor and found that ERK wave propagation and cell migration were impaired during skin wound repair. In conclusion, we have quantitatively demonstrated the distinctions among EGFRLs in shedding, diffusion, and target cell activation in physiological contexts. Our findings underscore the pivotal role of low-affinity EGFRLs in rapid intercellular signal transmission.

Modeling shock attenuation in hydrogels via frequency-dependent acoustic drag

A new method for assimilating a frequency-dependent drag coefficient into time-domain acoustic simulations is presented. The method combines structural (wave propagation) simulations together with acoustic attenuation of the individual frequencies through a model for the frequency-dependent drag coefficient. An incident pressure pulse is obtained experimentally or from a preliminary finite element simulation. This pulse is then decomposed into its spectral components. The propagation of each frequency component is simulated separately with the appropriate drag coefficient. In the final stage, the nodal pressure for all single frequency simulations are summed to reconstruct the transmitted attenuated pressure pulse. This method is demonstrated using a previously calibrated spectral model of the attenuation of methyl cellulose hydrogel, but it can be used for any other damping material for which a frequency response function can be obtained.

Analysis of Lie symmetry, bifurcations with phase portraits, sensitivity and diverse W − M-shape soliton solutions for the (2 + 1)-dimensional evolution equation

This research investigates the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation, a key model in nonlinear wave dynamics, particularly for shallow water waves. By integrating the features of the KP and BBM equations, it offers a comprehensive framework for studying the propagation, stability, and interactions of long waves. Using Lie group analysis for symmetry reductions and bifurcation with phase portraits to explore dynamic behaviour, the study also applies chaos theory to examine system features. Additionally, the new extended direct algebraic method (NEDAM) is employed to derive novel soliton solutions, including dark, bright-dark, W-shape, singular, periodic, and mixed forms. Sensitivity analysis is performed, and 3D, 2D, contour, and density plots visually demonstrate the results. These findings show that NEDAM effectively extracts exact solitons, providing a basis for further studies in nonlinear wave theory and its applications in engineering and physics.

Effect of loading rate on local deformation of rock-like models with locked segment

To investigate the influence of loading rate on the local deformation and failure characteristics of rock-like models with a locked segment, this study fabricated them using quartz sand and barite powder as aggregates, and gypsum as cementing material. Uniaxial compression tests, combined with strain cube monitoring and the digital image correlation technique, were employed to analyze the mechanical properties, internal and external deformation characteristics, crack propagation, and failure modes at different loading rates. The results showed that as the loading rate increased, the strength of these models initially increased significantly and then slightly decreased, primarily due to the complex interaction between the longitudinal and transverse deformations of the models and the stress concentration intensity at the crack tips. The fluctuation in the principal strain axis's deflection angle revealed the complex process of stress wave propagation and particle adjustment within the models, with more pronounced fluctuations observed at higher loading rates. The stress intensity factors, KIand KII, at the crack tips first increased and then decreased with increasing loading rate, with KI consistently higher than KII. The strain concentration phenomenon in the rock-like models occurred earlier with increasing loading rates, and the corresponding crack closure stress exhibited a decreasing trend. Higher loading rates caused earlier crack initiation and faster propagation. As the loading rate increased, the crack paths shortened, and both damage density and severity increased significantly, indicating brittle failure. This study provides important experimental evidence and theoretical support for understanding the effects of loading rate in rock mass engineering.

Prediction of dynamic shear and maximum displacement of clamped reinforced concrete beams subjected to impact loading

This paper presents a novel approach for predicting the dynamic shear forces and the maximum displacement of clamped reinforced concrete (RC) beams subjected to impact loading. By integrating wave propagation effects, membrane actions and the time-dependent acceleration distribution into the analysis, the study presents an improved approach based on single-degree-of-freedom (SDOF) analysis and overcomes the limitations of conventional SDOF method. The proposed model is validated against experimental data and finite-element simulations, demonstrating its reliability and accuracy in predicting dynamic response. Based on the validated model, a series of design charts are generated facilitating quick predictions of the maximum shear force at support and the maximum displacement of RC beams under impact, offering practical tools for engineers to enhance the safety and resilience of RC beams against impact loading.

An Assessment of HF Radio Wave Propagation in Antarctica for a Radio Link Between McMurdo and South Pole Station

AbstractIn this work, we analyze data collected by an HF transmitter/receiver radio link, operating as an oblique ionosonde between the McMurdo Station (transmitter) and South Pole Station (receiver) at 4.1, 5.1, 6.0, 6.4, and 7.2 MHz between 28 February and 14 March 2019. To help contextualize the link's data we have performed numerical raytrace simulations to help understand the observations. By considering both the data and simulations, we have identified both single‐ and two‐hop E‐ and F‐region propagation modes in the data, where the multi‐hop modes were observed in the hours around sunrise and sunset in the 4.1 and 5.1 MHz channels. This is an unexpected result given the accepted wisdom that multi‐hop modes, which require a ground scatter component, cannot be supported in Antarctica because of the highly absorptive ice covering much of the continent. Our results show that multi‐hop propagation modes can be supported in the region under specific ionospheric conditions—around sunrise and sunset—if the mode's ground scatter component is collocated with the Transantarctic Mountains. The mountains are located along the great‐circle path between the link's transmitter and receiver. However, the combination of favorable ionospheric and ground scattering conditions makes the detection of the multi‐hop mode a rare occurrence in the data set analyzed here. These findings are critical to data analysis efforts of any current or future oblique ionosonde systems operating in Antarctica and other regions such as the Arctic.

Highly resolved peta-scale direct numerical simulations: Onset of Kelvin–Helmholtz Rayleigh–Taylor instability via pressure pulses

The study presents a comprehensive numerical investigation of the Kelvin–Helmholtz Rayleigh–Taylor Instability (KHRTI) onset using highly resolved peta-scale direct numerical simulations by solving the compressible Navier–Stokes equations (NSE). The numerical framework incorporates a three-dimensional (3D) cuboidal domain with differential heating applied to two air streams, fostering the development of the KHRTI. A novel numerical methodology with selective mesh refinement near critical regions is employed with the help of a non-uniform compact scheme to capture small-scale phenomena accurately. Analysis of pressure disturbances during early KHRTI stages reveal distinct wave propagation patterns influenced by Rayleigh–Taylor (RT) and Kelvin–Helmholtz (KH) mechanisms. Enstrophy dynamics are quantified through the compressible enstrophy transport equation (CETE), highlighting dominant contributions from viscous stresses during early receptivity stages. The study provides insights into KHRTI evolution, shedding light on shear-buoyancy-driven instabilities and their implications for transition to turbulence.

Study on the evolution characteristics of coal mass spalling during coal and gas outbursts

The process of coal and gas outbursts (called outbursts for short) are commonly accompanied by coal mass spalling and fragmentation, but the formation mechanism of spalling is still unclear. To clarify this mechanism, this research constructs multi-layered combined coal-rock mass model. Based on the stress wave propagation mechanism, it’s found that outbursts initially start from ‘weak layer’. Besides, there are two significant results by analyzing the process of unloading waves formed by sudden pressure relief of the excavation face. Firstly, in the axial direction, due to the internal impact of the coal mass during the unloading wave propagation, the load shock wave is formed. The tensile stress is formed on the surface of the coal mass, resulting in rapid damage to the coal mass and even multi-layered spalling. Secondly, in the radial plane, permanent standing surfaces will be formed when the unloading wave catches up with the plastic load wave. Under effect of the unloading wave, internal impact will occur a series of load shock wave, resulting in the formation of multi-layered spalling in the radial plane. In addition, the attenuation law of the load shock wave generated by the internal impact in multi-layered coal mass is also analyzed, results showed that the thickness of coal mass spalling formed far away from the free surface will increase. Finally, the three-dimensional dynamic damage path and the spindle shape damage area formation mechanism in coal mass under external dynamic load are analyzed, which provides a new idea for elucidating the outbursts mechanism.

The onset of parametric decay of lower hybrid waves during lower hybrid current drive experiments on Experimental Advanced Superconducting Tokamak

Abstract An ad hoc calculation is presented to interpret the growth of sideband waves during lower hybrid current drive (LHCD) experiments on Experimental Advanced Superconducting Tokamak (EAST). We adopt the solution of the non-linear dispersion relation for parametric instabilities to describe the growth rate of ion cyclotron quasi-modes (ICQMs) in the peripheral plasma, and use the WKB solution of the propagation of electrostatic waves in two dimensions to model the variation of the electric field in the scrape-off-layer (SOL). We then calculate the growth rates and amplification factors of the sideband waves associated with ICQM in the parallel and perpendicular coupling limits. The onset and substantial growth of the first down-shifted sideband of lower hybrid waves (LHWs) are associated with elevation of the plasma density in the outer SOL where the growth rate of ICQM in the parallel coupling limit increases sharply after the density is elevated. The peak frequencies associated with the first harmonic of ICQM obtained from calculating the amplification factors in the parallel coupling limit ( A para ) agree very well with the observed frequencies of the first down-shifted sideband of 2.45 GHz LHWs in EAST pulse 43 772 and 86 521. The consistency between the calculation and the observation indicates that the sideband waves are associated with parametric decay instability of ICQM excited in the outer SOL through parallel coupling. The amplitude of A para obtained from a single pass of LHWs through the SOL of EAST is, however, one order of magnitude smaller than unity, meaning that it remains elusive whether the substantial growth of sideband waves mainly occurs in front of the antenna or requires multiple passes of LHWs across the SOL region.