Wave Propagation Process Research Articles

Strong resonance response with ultrahigh quality factor in grating-multilayer systems based on quasi-bound states in the continuum

Optical bound states in the continuum (BICs) exist in many photonic crystals and periodic structures with a strong resonance and ultrahigh Q factor. Such phenomena can be used in the designs of narrowband transmission filters, lasers, and sensors. In this paper, we consider the energy bands of a complex structure consisting of a grating and a multilayer substructure to obtain the position of the BIC in the structure. Hence, the higher Q factor can be obtained in the grating-multilayer structure than can be realized in the simple grating geometry. We analyze the wave propagation process in the complex structure and the change in the Q value via the use of transmission matrix theory. In addition, the reflectance spectrum is found to exhibit a series of asymmetric line-shapes with different values of the asymmetry parameter, δ, due to the interference between the two channels. One of these channels is the broadband channel, induced by the Fabry–Perot resonance, and the other channel is the narrowband channel, induced by guided mode resonance. Quasi-BICs are seen to transform into BICs as the value of δ is decreased gradually to zero. Therefore, a large Goos–Hänchen shift can be achieved as a result of the high Q factor and quasi-BIC. This work designs a complex structure with ultrahigh Q factor and strong resonance properties, which has significant implications for exploring the phenomenon of BICs.

Hybrid HVDC system fault transient analysis considering traveling wave propagation and converter control response

Accurate fault analysis is essential for protection and control design of hybrid HVDC transmission system. This paper proposes a dc fault transient analysis method, which considers both the traveling wave propagation process and converter control response. Firstly, the multiple reflected and refracted traveling waves at line terminals as well as fault point are calculated by building propagation coefficient matrices. Secondly, the transfer function between the converter equivalent source and the traveling wave transients is built, then the fault current and voltage are calculated by compensating multiple traveling waves with corresponding control response. Compared with existing methods, the proposed method can calculate long-period traveling waves in a simplified way, and implement more accurate analysis for the converter response caused by the basic control strategy and the fault control strategy. An LCC-MMC HVDC system is built based on PSCAD/EMTDC, and the simulation cases are carried out to verify the correctness and superiorities of the proposed method. Finally, the impact of control response on the performance of traveling-wave-based directional protection is evaluated by the proposed fault analysis method.

Dynamics of wave generation and dissipation processes during cold wave events in the Bohai Sea

Wave dynamics were investigated by applying the wind-wave model Simulating Waves Nearshore to the semi-enclosed Bohai Sea during cold wave events. Wind quality was examined by comparing three wind data sources with buoy observations, and then numerical experiments were conducted to investigate the impacts of model physics settings on simulated waves. After model calibration, the wave dynamics were examined from the aspects of response to various wind conditions (speed, fetch, duration), wave dominant component (wind wave or swell), as well as the dissipation processes during a pair of cold wave events (e.g., northwesterly in 2014 and northeasterly in 2015) with the consideration of spatial differences. The maximum significant wave heights during two cold wave events are similar (e.g., 3.5 m in the central Bohai Basin), which attributes to the impacts of limited fetch difference and insufficient high wind duration. Wind wave is the dominant component, which is reflected in the wave spectra (e.g., wave age less than 1.2). Spatial regions of wave dissipations controlled by whitecapping, bottom friction, and depth-induced breaking during cold wave events are quantified. Whitecapping dominates the wave energy dissipation in deep-water areas, while the bottom friction dissipation controls the major dissipation process as waves propagate towards the coast (e.g., at 5–10 m water depths). Depth-induced dissipation only occurs over a narrow strip along the southern coastal area but is significantly beyond the others. Depth-induced dissipation decreases suddenly when reaches a flat bottom, and it decreases with the increase of the width of the bottom slope.

Dynamic Response Characteristics of Roadway Surrounding Rock and the Support System and Rock Burst Prevention Technology for Coal Mines

Anchor cables (bolts) act as the main support system and play an important role in improving the rock burst resistance and stability of the roadway surrounding the rock. In this study, the dynamic response characteristics of the roadway surrounding the rock and the support system under different shock intensities were investigated. The following findings were obtained. The stress wave propagation process under dynamic shock was divided into a stress vibration initiation stage, a stress fluctuation stage, and a stress adjustment stage. In the stress vibration initiation stage, the surface mass of the roadway surrounding the rock started to vibrate, and the pretension of the anchor cables (bolts) was reduced; in the stress fluctuation stage, the failure of the roadway surrounding the rock intensified, and the anchor cables (bolts) were damaged to some extent; and in the stress adjustment stage, the roadway deformation of the surrounding rock and the axial forces of the anchor cables (bolts) tended to stabilize. As the dynamic shock intensity increased, the vibration velocity, displacement increment, and acceleration amplitude of the mass of the roadway surrounding the rock increased exponentially. The critical shock energy of the roadway surrounding the rock was 105 J, above which the damage to the rock was aggravated. The larger the pretension of the anchor cables (bolts) was and the higher the dynamic shock intensity was, the more severe the damage to the anchor cables (bolts) was. Given the dynamic response characteristics of the roadway surrounding the rock and support elements under shock, a full anchor cable yielding support technology is proposed to effectively control the stability of the roadway surrounding the rock under dynamic shock, providing a reference for the construction of the support systems for preventing rock bursts in similar roadways.

Investigation of dynamic impact responses of layered polymer-graphene nanocomposite films using coarse-grained molecular dynamics simulations

Polymer nanocomposite films have recently shown superior energy dissipation capability by using the micro-projectile impact testing method. However, how stress waves interact with nanointerfaces and the underlying deformation mechanisms have remained largely elusive. This paper investigates the detailed stress wave propagation process and dynamic failure mechanisms of layered poly(methyl methacrylate) (PMMA)-graphene nanocomposite films during piston impact through coarse-grained molecular dynamics simulations. The spatiotemporal contours of stress and local density clearly demonstrate shock front, reflected wave, and release wave. By plotting shock front velocity (Us) against piston velocity (Up), we find that the linear Hugoniot Us−Up relationship generally observed for bulk polymer systems also applies to the layered nanocomposite system. When the piston reaches a critical velocity, PMMA crazing can emerge at the location where the major reflected wave and release wave meet. We show that the activation of PMMA crazing significantly enhances the energy dissipation ratio of the nanocomposite films, defined as the ratio between the dissipated energy through irreversible deformation and the input energy. The ratio maximizes at the critical Up when the PMMA crazing emerges and then decreases as Up further increases. We also find that a critical PMMA-graphene interfacial strength is required to activate PMMA crazing instead of interfacial separation. Additionally, layer thickness affects the amounts of input energy and dissipated energy for nanocomposite films under impact. This study provides important insights into the detailed dynamic deformation mechanisms and how nanointerfaces/nanostructures affect the energy dissipation capability of layered nanocomposite films.

Fractal Analysis for Wave Propagation in Combustion–Explosion Fracturing Shale Reservoir

The in-situ combustion–explosion fracturing technology in shale reservoirs can promote continuous fracture expansion with a radial detonation wave first converging into a shock wave and then decaying into an elastic wave. The transformation scale of the shale reservoir is determined by the range of wave propagation during combustion–explosion. As wave propagation paths are usually tortuous and fractal, the previous integer wave models are not competent to describe the wave propagation and estimate the impact range of the combustion–explosion fracturing process. This study develops two fractional wave propagation models and seeks analytical solutions. Firstly, a novel fractional wave model of rotation angle is proposed to describe the process of detonation waves converting into shock waves in a bifurcated structure. The radial displacement gradient of the detonation wave is represented by the internal expansion and rotation deformation of the shale. Secondly, another fractional wave propagation model of radial displacement is proposed to show the process of a shock wave decaying into an elastic wave. Thirdly, the proposed models are analytically solved through the fractional variable separation method and variational iteration method, respectively. Analytical solutions for rotation angle and radial displacement with fractal time and space are obtained. Finally, the impacts of the branching parameter of the detonation wave converge bifurcation system, aggregation order of detonation compression wave, and different types of explosives on the rotation angle of the shock wave are investigated. The propagation mechanism of the primary wave (P-wave) with time and space is analyzed. The analytical solutions can well describe the wave propagation process in fractured shales. The proposed fractional wave propagation models can promote the research of wave propagation in the combustion–explosion fracturing process of shale reservoirs.

Investigation on propagation mechanism of leakage acoustic waves in gas-liquid stratified flow

Propagation speed of acoustic waves is of vital importance in pipeline leakage detection using acoustic method. However, the wave front shape and thermodynamic process of leakage acoustic waves in gas-liquid stratified flow remain unclear, which restricts the proper calculation of sound speed in this flow pattern. This paper presents a simulation study using CFD software Fluent to investigate the propagation process of leakage acoustic waves in stratified flow. The leakage acoustic waves are generated in different fluid phases and small time-step simulation is implemented to study the pressure behavior during the wave propagation. The results show that leakage acoustic waves propagate as single plane wave in gas-liquid stratified flow and the propagation process is nearly adiabatic. The value of polytropic exponent when calculating the sound speed in gas phase should be 1.4. Further, experimental study is conducted to verify the simulation results. Influencing factors including pressure and void fraction are considered. The present results contribute to the proper selection of sound speed calculation model in stratified flow, which can improve the leakage detection accuracy in gas-liquid stratified flow pipelines.

Stochastic Semi-Physical Model for Nonstationary Spatially Variable Ground Motions in an Engineering Site

Spatially variable ground motions are crucial for the seismic analysis and design of extended and multi-support structures. In this study, a new random nonstationary spatially variable ground motion model based on the propagation process of seismic waves is proposed. In particular, the amplitude and phase transfer functions of seismic waves at the local site are studied herein. The probability density function of the corresponding parameters is given. Two simple examples are used to show the generation of random ground motions. Compared with traditional methods, our proposed model not only has a clear physical background but also shows good practicability. The model should be for those researchers who want to use nonstationary spatially variable ground motions to study the seismic response of lifeline systems or building portfolios.

Research on the Influence of Impedance-Layer Changes on the Protective Properties of Wave-Impedance Materials under Hypervelocity Impact

The hypervelocity impact of space debris causes damage or destruction to spacecraft. The continuous damage caused by space debris creates greater requirements for protective materials. Wave-impedance gradient-protection material is a new type of space-debris-protection material with high kinetic-energy dissipation. However, the relationship between the distribution characteristics of the impedance layer and the protective performance is still unclear. This study provides guidance for the design of high-performance wave-impedance gradient materials by establishing the quantitative relationship between impedance-layer distribution characteristics and protective performance. Based on the one-dimensional shock-wave theory, this paper analyzes the propagation process of shock waves in wave-impedance gradient materials, establishes a transmission model of shock waves with changes in impedance layers, and quantitatively studies the influence of the change in wave impedance on the impact pressure, internal-energy conversion, and projectile- breaking characteristics by means of a numerical simulation. The results show that, when the surface and back of the wave-impedance gradient material are titanium alloy and nylon, respectively, the total transmission coefficient increases from 0.206 to a maximum of 0.339 with the continuous change in the gradient. The reduction amplitude of the shock wave with time under the three working conditions is Ti-Al-Mg-Ny > Ti-Al-Ny > Ti-Ny. The relationship of the transformed internal energy is Ti Al-Mg-Ny > Ti-Al-Ny > Ti-Ny, and the projectile breaking area is Ti-Al-Mg-Ny > Ti-Al-Ny > Ti-Ny. The analysis shows that the continuous change in wave impedance is beneficial to reduce the attenuation of the amplitude of the shock wave in the hypervelocity projectile, to keep the stress amplitude of the shock wave at a higher level, and to improve the internal-energy conversion and impact-breaking degree of the projectile, thereby enhancing the protection performance of the wave-impedance gradient material.

Design and Implementation of an Ultrasonic Flowmeter Based on the Cross-Correlation Method.

Ultrasonic flowmeters play an important role in industrial production, aerospace and other fields. In this paper, a high-precision ultrasonic flowmeter based on the cross-correlation method is designed, and the commercial finite element software COMSOL Multiphysics 5.6 is used to simulate the propagation process of ultrasonic waves during flow measurement, and the implementation process of the cross-correlation algorithm is simulated by Python language. The flowmeter adopts the cross-correlation algorithm to improve the measurement accuracy of ultrasonic time of flight and adopts the method of combining FPGA and an embedded microprocessor to improve operation efficiency. In order to verify the performance of the flowmeter, we tested the flowmeter on the National Institute of Metrology and the self-built test platform, using the still water dragging method, the dynamic volume method and the field comparison method, respectively. The results show that the flowmeter has the ability to test the flow under the condition of high flow velocity (26 m/s) and a pipe diameter in the range of DN6~DN1600, that the absolute value of the relative indication error does not exceed 0.815% and that the repeatability does not exceed 0.150%. The designed ultrasonic flowmeter has high measurement accuracy, good repeatability, strong stability and a wide application range.

A Numerical Study of Wave Propagation and Cracking Processes in Rock-Like Material under Seismic Loading Based on the Bonded-Particle Model Approach

An earthquake is usually followed by a considerable number of aftershocks that play a significant role in earthquake-induced landslides. During the aftershock, the cracking process in rocks becomes more complex because of the formation of faults. In order to investigate the effects of seismic loading on the cracking processes in a specimen containing a single flaw, a numerical approach based on the bonded-particle model (BPM) was adopted to study the seismic loading applied in two orthogonal directions. The results reveal that no transmission and reflection phenomena were observable in the small specimens (76 mm × 152 mm) because they were considerably smaller than the wavelength of the P-wave. Furthermore, under seismic loading, the induced crack was solely tensile in nature. Repeated axial seismic loading did not induce crack propagation after the first axial seismic loading. Cracks began to propagate only when the seismic loading direction was changed from axial to lateral, and then back to axial, ultimately resulting in the failure of the specimen.

An h-adaptive local discontinuous Galerkin method for second order wave equation: Applications for the underwater explosion shock hydrodynamics

The underwater explosion (UNDEX) phenomenon involves hydrodynamic behaviors with a wide range of spatial scales, from macroscopic cavitation regions to refined areas with fluid-structure interactions around the shock wavefront. Correct description of the refined shock wavefront is important for determining the shock wave position and pressure peak. In this work, the local discontinuous Galerkin (LDG) method for the two-dimensional wave equation, which allows for the local triangular meshes refinement and merging during the shock wave propagation process, is presented to model the UNDEX shock hydrodynamics. This article is an extension of our previous work, where we discretize the UNDEX total wave formulation in the LDG framework and simulate the UNDEX shock wave and cavitation phenomenon. In order to save computational cost (including CPU time and computational storage) and capture the shock wavefront more accurately, the LDG method is extended to be in combination with the mesh adaption technique. The troubled-cell indicator, depending on the local jump of the pressure, is adopted to identify elements which require to be refined or coarsened. Typical numerical examples are tested to assess the capability of the present h-adaptive LDG method for capturing the UNDEX shock wave. The numerical results clearly demonstrate that the mesh adaption technique enables us to dynamically increase the mesh resolution in close proximity to the flow discontinuities with a large pressure jump. The local increase of the mesh resolution results in a substantial increase in the accuracy of the pressure peak. In terms of the computational efficiency, the h-adaptive LDG method can save nearly half of the calculation time in comparison to the non-adaptive LDG method on uniform meshes, whose mesh resolution is the same as that of the maximum level of refinement of the h-adaptive LDG method.

Comparative Study on Blast Damage Features of Reinforced Concrete Slabs with Polyurethane Sacrificial Cladding Based on Different Numerical Simulation Methods.

The defense effects of sacrificial cladding have been extensively studied in the field of blast resistance. As a polymer material with a cellular structure, polyurethane also has the potential to act as sacrificial cladding due to its good mechanical properties. The purpose of this study is to compare and select a numerical simulation method that is suitable for exploring the blast damage mitigation effect of polyurethane sacrificial cladding on reinforced concrete slabs. To this end, three numerical models were developed using the Fully Coupled Eulerian–Lagrangian (CEL) method, the Arbitrary Lagrangian–Eulerian (ALE) coupling method, and the Smoothed Particle Hydrodynamics and Finite Element Method (SPH–FEM) coupling method, respectively. These three numerical models were used to investigate the damage features of reinforced concrete slabs with polyurethane sacrificial cladding (PU–RCS) under contact explosions. A field test was also carried out to provide a comparison for numerical simulation results. Moreover, the advantages and disadvantages of the three simulation results and the applicability of the three coupled models were discussed. The results show that compared with the CEL model and the ALE coupling model, the SPH–FEM coupling model can better simulate the damage features of PU–RCS, such as the cracks on the bottom surface of the RC slab and the large deformation failure state of polyurethane sacrificial cladding, while the CEL model and the ALE coupling model can simulate the propagation process of shock waves and have a lower computational cost. In conclusion, the SPH–FEM coupling method is the most applicable method for exploring the blast damage features of PU–RCS in this study.

Experimental and Numerical Studies on Bond Quality of Fully Grouted Rockbolt under Confining Pressure and Pull-Out Load

In mining engineering, the in situ stress changes with the stress induced by the surrounding mining activities. It positively or negatively affects the propagation of ultrasonic guided waves in rockbolts. Therefore, the effect of in situ stress in rockbolt support was determined by applying confining pressure and pull-out load in a laboratory test and using ultrasonic guided waves to test the rockbolt. Furthermore, the propagation law of ultrasonic guided waves and bond quality of the rockbolt under the interaction of the pull-out load and confining pressure were studied. Numerical simulations were performed to deduce the guided wave propagation process in grouted systems, and the influencing mechanism of the pull-out load and confining pressure on the guided wave propagation was discussed. The laboratory test and numerical simulation results show that the confining pressure weakens the guided wave propagation without pull-out load. Under the same pull-out load, the guided wave propagation gradually strengthens with increasing confining pressure. A larger confining pressure weakens the weakening effect of the pull-out load and suppresses the discreteness of the guided wave propagation. Under the same confining pressure, the guided waves did not diffract well into the cement mortar and concrete with increasing pull-out load, the confining pressure restricted the radial vibration of the guided waves, and the guided wave propagation law weakened. Thus, the pull-out load plays a weakening role in the propagation law of ultrasonic guided waves.

Propagation path tracing of hydraulically created fractures based on microseismic monitoring

To evaluate the effect of hydraulic fracturing in a low-permeability reservoir, a propagation path tracing method for hydraulically created fractures was established based on microseismic monitoring data. First, the numerical simulation of the wave propagation process, grid search, and error-weight coefficient method was combined to locate the microseismic source. Then, the moment tensor inversion method was used to determine the tensile angle and source mechanism of hydraulically created fractures. Next, the tensile angle was used as the weight-index to determine the size of the mixed-source mechanism fracture combined with the shear and tensile source size quantization model. Finally, the spatial topological relationship between fractures was determined by comprehensively considering the spatial location and radius of the fractures, to realize the propagation path tracing of hydraulically created fractures. These tracking results can be used as one of the bases for the evaluation of the hydraulic fracturing effect.

A Time-Domain Artificial Boundary for Near-Field Wave Problem of Fluid Saturated Porous Media

The near-field wave problem of the saturated soil involves the energy radiation effect of the truncated infinite media. A viscous spring boundary is proposed for the fluid-saturated porous media. Based on the process of wave propagation under internal point source, the stress and flow velocity boundaries are constructed by reasonable assumptions of outgoing waves and Green’s function, respectively. Without the permeability assumption, the proposed boundary avoids the low accuracy caused by the assumption of zero permeability that is widely used in the existing methods. The boundary simultaneously has a simple form, clear physical meaning, and less computational cost due to its local character. Meanwhile, a completely explicit integration algorithm considering the damping is constructed to solve the finite element equations of saturated porous media with the proposed boundary. The accuracy and high computational efficiency of the wave numerical method are verified in the examples.

Study on Propagation Characteristics of Ground Penetrating Radar Wave in Dikes and Dams with Polymer Grouting Repair Using Finite-Difference Time-Domain with Perfectly Matched Layer Boundary Condition

Non-aqueous reactive polymer grouting technology has been widely used in the repair engineering of dikes and dams with shallow diseases. By using the finite difference time-domain method and perfectly matched layer boundary conditions, the calculation model of dikes and dams with shallow diseases such as water-filled cave, air-filled cave and incompact area is established. The propagation process of electromagnetic waves of ground-penetrating radar in dikes and dams with shallow diseases using polymer grouting repair is simulated, and the forward simulation profiles and single-channel waveforms are obtained. The propagation characteristics such as waveform amplitude, waveform shape, transmission time, and reflection time are compared and analyzed. The results show that the forward simulation profiles of dikes and dams with water-filled caves before and after polymer grouting repair present two clusters of hyperbolas, but three clusters of hyperbolas with different amplitudes were observed at 50% repair. The amplitude of the hyperbola and the single-channel reflected waves before repair and 50% repair of the cave and incompact area are greater than those of 100% repair. The propagation characteristics of ground-penetrating radar can effectively explain the degree of polymer grouting repair for dikes and dams with shallow diseases, and provide a theoretical basis for using the ground-penetrating radar to evaluate the effect of polymer grouting technology to repair dikes and dams with shallow diseases.

Loading issues in coupled acoustic-structural analysis of dynamic responses of dams in underwater explosion events: A gravity-dam case study

The coupled acoustic-structural analysis is an efficient and practical approach to underwater explosion (UNDEX) resistance evaluation of various underwater structures, including dams. It features its simple manner of implementing UNDEX shock loadings on structures in a direct way without in detail considering the detonation physics and shock wave propagation processes. The loading implementation is thus believed to be the most critical step in determining the reliability of coupled acoustic-structural analyses of structures in UNDEX events. The present paper aims to clarify two important loading issues frequently confronting the coupled acoustic-structural analysis of UNDEX responses of dams, i.e. the reservoir water thickness and the rise time in a typical UNDEX loading pulse. The ABAQUS code is used for the present numerical implementation. The centrifuge-based physical UNDEX test on a gravity dam performed by Wang (2020) is numerically simulated with the coupled acoustic-structural analysis strategy. The simulation results are found to be in reasonable agreement with the corresponding test ones under the planar-wave assumption for the shock loading implementation. The water thickness is found to have little effect on the simulation results. This validates the reliability of the present simulation strategy and indicates the feasibility of using a relatively thin water domain with the planar-wave assumption in UNDEX analyses of gravity dams. However, it is indicated that the predicted response of the dam is significantly stronger than the measured one when the spherical-wave assumption is made with the total wave formulation. This is due to the insufficient water thickness modeled that leads to wave reflection at the fluid boundaries. After that, a parameter analysis is carried out concerning the rise time period in the loading pulse. The comparison results show that the effect of the rising duration on the UNDEX response of the dam is limited. It is thus suggested that a rise time period of around 30–50 μs should be reasonable in most UNDEX resistance evaluation scenarios for general dams. The current findings should be of interest to practitioners who are engaged in the UNDEX resistance evaluation of dams.

Quantitative evaluation in classification and amplitude of near-field landslide generated wave induced by granular debris

Landslides falling into water bodies can generate impulsive waves. The process of wave generation induced by far subaerial granular landslide has been investigated through series physical experiments. The slide Froude number has been quantified considered the parameters of slide mass and water such as grain size, water depth and impact angle. The effect of impact angle, grain size and water depth on the variation of generated wave amplitude is 0.11%, 1.52% and 2.89% respectively when every 1% change in the value of the parameter. On the three governing parameters the water depth is the most sensitive to the wave amplitude, secondly the grain size and final for the impact angle. The intruding pattern of granular debris, the landslide generated wave types and maximum wave amplitude have been systematically discussed. The degree of impact interaction between the slides and water under different conditions is characterized by the value of the slide Froude number, and the empirical relationships linking the slide Froude number with the produced wave amplitude and wave classifications have been conducted based on experimental data, which is useful to illustrate the triggering mechanisms and dynamic process of impact landslide waves on high debris avalanches.

Impact damping performance and mechanisms of 3D-printed density gradient cylindrical shell chains

Granular chain structures may improve the impact attenuation ability and the tunability of the wave propagation. In this paper, we present theoretical, numerical and experimental studies on the response of 3D-printed density gradient cylindrical shell chains under striker impact. A simplified continuum-based model considering the wave dispersion due to transverse inertia is developed to describe the complicated wave propagation process in graded chains. An approximate solution is derived by using the Laplace transform method. An analytical solution is obtained when the transverse inertia effect is ignored. The one-dimensional wave propagation model can well represent the experimental/numerical results of the force history, velocity distributions, and impact attenuation performances. It is found that the positive/negative density gradient chains with linear density distributions can strongly attenuate the impact impulse. As the linear density distribution parameter increases, the transmitted peak force first increases and then decreases, and it reaches the maximum value when the chain is uniform. The transmitted peak force decreases with the increase of the transverse inertia parameter. The impact attenuation mechanism is explained by analyzing the wave impedance mismatch, and it is found that the relationship between the normalized transmitted peak force and the linear density distribution parameter γ of the chains can be characterized by a concave downward even function when −2 < γ < 2. These understandings can be applied to guide the structural design of density gradient cylindrical shell chains to efficiently and precisely manipulate wave propagation.