Shear Research Articles

Ultrasonic imaging of concrete-embedded corroded steel liners using linear and nonlinear evaluation

ABSTRACT This study explores the efficacy of horizontally polarised shear (SH) waves in ultrasonic inspection of concrete-embedded steel liner plates, focusing on corrosion detection. SH-waves are found to significantly improve the localisation and potential dimensional assessment of embedded plate-like structures compared to compressional (P) waves, as evidenced by SH C-scans. Anomalies corresponding to severe corrosion are discernible in SH C-scans, whereas they are minimally visible in P-wave scans. Nonlinear evaluation explores harmonic-based imaging, including second harmonic generation (SHG) and third harmonic generation. Nonlinear SH-wave evaluation accurately locates the most severely corroded embedded plate, whereas comparative P-wave evaluations fail. However, potential FFT sidelobe contamination in the results warrants consideration. These results suggest a promising use for nonlinear waveform distortion in SH-wave-based corrosion detection though conclusive evidence of the source of SHG is uncertain. Experiments utilised an automatic scanner system for consistent control. The concrete specimen investigated herein contained pre-corroded plates which hint at heightened nonlinear effects when corrosion occurs post-embedment due to corrosion-induced cracking. These findings encourage further investigation into SH-waves for nonlinear ultrasonic detection of concrete-embedded steel liner plates and the development of instrumentation tailored to such methods.

A flexible yield criterion for strength modeling from biaxial compression to biaxial tension

Accurate strength modeling from equi-biaxial tension (EBT) to equi-biaxial compression (EBC) is critical for the plastic behavior prediction covering the wide-range of stress triaxiality encountered in sheet metal forming. To date, however, few yield criteria are available that can precisely model the initial yield and hardening behavior under six typical stress states between EBC and EBT, simultaneously. Furthermore, there is still a lack of a unified yield criterion for accurate strength modeling across various stress state ranges. To address the issues, a theoretical framework for constructing yield criteria dependent on stress states is provided and a new analytically described isotropic yield criterion is presented in this study. The flexibility in terms of the yield locus and application range is thoroughly explored to make the new yield criterion general. Subsequently, the isotropic yield criterion is extended into an analytically described anisotropic-asymmetric yield criterion. Furthermore, the extended yield criterion is applied to capture the initial yield behavior of DP980, AA5754-O, and AZ31 sheets, and the strain hardening behavior of QP1180 sheets at various stress states ranging from EBC to EBT along different loading directions. The predicted results from the extended criterion agree well with the corresponding experimental findings. The applications demonstrate that the proposed anisotropic-asymmetric yield criterion can effectively model the initial yield and hardening behavior of HCP, BCC, and FCC metal sheets under EBT, EBC, uniaxial tension (UT), plane strain tension (PST), shear (SH), and uniaxial compression (UC) in an analytical way.

Application of iterative elastic reverse time migration to shear horizontal ultrasonic echo data obtained at a concrete step specimen

AbstractThe ultrasonic echo technique is broadly applied in non‐destructive testing (NDT) of concrete structures involving tasks such as measuring thickness, determining geometry and locating built‐in elements. To address the challenge of enhancing ultrasonic imaging for complex concrete constructions, we adapted a seismic imaging algorithm – reverse time migration (RTM) – for NDT in civil engineering. Unlike the traditionally applied synthetic aperture focusing technique (SAFT), RTM takes into account the full wavefield including primary and reflected arrivals as well as multiples. This capability enables RTM to effectively handle all wave phenomena, unlimited by changes in velocity and reflector inclinations. This paper concentrates on applying and evaluating a two‐dimensional elastic RTM algorithm that specifically addresses horizontally polarized shear (SH) waves only, as these are predominantly used in ultrasonic NDT of concrete structures. The elastic SH RTM algorithm was deployed for imaging real ultrasonic echo SH‐wave data obtained at a concrete specimen exhibiting a complex back wall geometry and containing four tendon ducts. As these features are frequently encountered in practical NDT scenarios, their precise imaging holds significant importance. By applying the elastic SH RTM algorithm, we successfully reproduced nearly all reflectors within the concrete specimen. In particular, we were capable of accurately reconstructing all vertically oriented reflectors as well as the circular cross sections of three tendon ducts, which was not achievable with traditional SAFT imaging. These findings demonstrate that elastic SH RTM holds the ability to considerably improve the imaging of complex concrete geometries, marking a crucial advancement for accurate, high‐quality ultrasonic NDT in civil engineering.

Analytical solution of Tunnel-Soil-Tunnel (TST) interaction under seismic excitation

ABSTRACT This research proposes the analytical formulations for the vertically aligned twin tunnels under the sinusoidal form of seismic excitation in the case of the damped free vibration. The analytical formulae are derived to calculate the interactive tunnel moment, tunnel shear force and soil pressure for the transverse horizontal shear (SH), vertical shear (SV) and body (P) waves. For the derivation of the analytical formulations, the twin tunnels are assumed as beam elements connected by the linear elastic springs. The tunnel-soil-tunnel (TST) interaction model is verified by the previous analytical solution and numerical analysis of the case studies. The ranges of the difference in results between the present research and the previous analytical solution are found to be (3.8 ~ 8.1) %, (6.73 ~ 10.95) % and (2.3 ~ 11.79) % in the case of the SH, SV and P waves, respectively. These differences may represent good agreement about verifications.

A study of Griffith crack in nonlocal infinite magneto-elastic media

The main objective of this article is to study how a finite crack interacts with horizontally polarized shear (SH) waves in a nonlocal isotropic medium when a magnetic field is present. The study aims to understand the effects of various parameters, including the magnetic field, non-locality, frequency of SH waves, on the behavior of the crack. Here, we are concerned with Griffith’s theory of a brittle fracture crack mechanism. The article starts by formulating the governing equations for the propagation of SH waves in the given medium. These equations are converted into dual integral equations using Fourier transformation. Subsequently, an Abel transformation is applied to obtain a Fredholm integral equation of the second kind. Daubechies wavelet family has been used to find an approximate solution for the integral equation. The article aims to determine specific physical quantities including stress intensity factor (SIF), shear stress, and crack energy related to the behavior of the crack. These parameters are essential in understanding how the crack responds to the external conditions. The application of Daubechies wavelets in this context is a relatively new and innovative approach. The study performed a comprehensive analysis of the effects of various factors on the crack behavior. The findings are presented in graphical form to help illustrate the conclusions. To assess the accuracy of the proposed method, the obtained results are compared with known results from the literature for local media. This is an important step in demonstrating the reliability and effectiveness of the proposed approach.

Convolutional neural network for identifying effective seismic force at a DRM layer for rapid reconstruction of SH ground motions

AbstractWe introduce a novel data‐informed convolutional neural network (CNN) approach that utilizes sparse ground motion measurements to accurately identify effective seismic forces in a truncated two‐dimensional (2D) domain. Namely, this paper presents the first prototype of a CNN capable of inferring domain reduction method (DRM) forces, equivalent to incident waves, across all nodes in the DRM layer. It achieves this from sparse measurement data in a multidimensional setting, even in the presence of incoherent incident waves. The method is applied to shear (SH) waves propagating into a domain truncated by a wave‐absorbing boundary condition (WABC). By effectively training the CNN using input‐layer features (surface sensor measurements) and output‐layer features (effective forces at a DRM layer), we achieve significant reductions in processing time compared to PDE‐constrained optimization methods. The numerical experiments demonstrate the method's effectiveness and robustness in identifying effective forces, equivalent to incoherent incident waves, at a DRM layer.

Application of Iterative Elastic SH Reverse Time Migration to Synthetic Ultrasonic Echo Data

The ultrasonic echo technique is widely used in non-destructive testing (NDT) of concrete objects for thickness measurements, geometry determinations and localization of built-in components. To improve ultrasonic imaging of complex concrete structures, we transferred a seismic imaging technique, the Reverse Time Migration (RTM), to NDT in civil engineering. RTM, in contrast to the conventionally used synthetic aperture focusing technique (SAFT) algorithms, considers all wavefield types and thus, can handle complex wave propagations in any direction with no limit on velocity variations and reflector dip. In this paper, we focused on the development, application and evaluation of a two-dimensional elastic RTM algorithm considering horizontally polarized shear (SH) waves only. We applied the elastic SH RTM routine to synthetic ultrasonic echo SH-wave data generated with a concrete model incorporating several steps and circular cavities. As these features can often be found in real-world NDT use cases, their imaging is extremely important. By using elastic SH RTM, we were able to clearly reproduce almost all reflectors inside the concrete model including the vertical step edges and the cross sections of the cavities. We were also capable to show that more features could be mapped compared to SAFT, and that imaging of complex reflectors could be sharpened compared to elastic P-SV (compressional-vertically polarized shear) RTM. Our promising results illustrate that elastic SH RTM has the potential to significantly enhance the reconstruction of challenging concrete structures, representing an important step forward for precise, high-quality ultrasonic NDT in civil engineering.

Vibration isolation of infinitely periodic pile barriers for anti-plane shear waves: An exact series solution

The application of periodic pile barriers for isolating ambient vibrations has emerged as a significant research topic, which is essentially the problem of wave scattering by periodically distributed structures. This study presents an analytical series solution for anti-plane shear (SH) wave scattering by infinitely periodic pile barriers using the wave function expansion method, incorporating the wave field characteristics of infinitely periodic distributed scatterers subjected to plane waves. From the perspective of periodic theory, the wave fields around each scatterer are completely identical, differing only by a phase shift (frequency domain) or a time difference (time domain). As a result, as long as the boundary condition of one scatterer is satisfied, boundary conditions of the remaining scatterers can be satisfied simultaneously. This novel approach allows the selection of an arbitrary single pile as a reference in the infinitely periodic model to be solved, yielding accurate results without truncation errors and significantly reducing computational and storage requirements. A FORTRAN code is developed to achieve numerical evaluation of the theoretical formulation, followed by verification of the solution's accuracy using two degenerate examples. Numerical examples are conducted, focusing on the influence of pile stiffness, pile spacing, and pile type on the vibration isolation. The results demonstrate that decreasing the spacing between piles can effectively increase the bandgap width and reduce the displacement response. The screening effectiveness of solid piles is better than that of hollow pipe piles and filled pipe piles. In vibration mitigation design for engineering projects, the larger the shear stiffness ratio of the pile-soil is not always better, an economical and reasonable pile stiffness should be selected.

Nonlocal effect on shear wave propagation in a fiber-reinforced poroelastic layered structure subjected to interfacial impulsive disturbance

The present study illuminates the influence of impulsive line source on Horizontally polarized shear (SH) waves propagating in a stratified structure consisting of two distinct nonlocal fiber-reinforced layers with voids over a foundation with nonlocal functionally graded fiber-reinforced substrate with voids. The governing equations for a nonlocal fiber-reinforced poroelastic medium are established. The proposed mathematical model incorporates the Green’s function technique and the Fourier transformation to yield the complex dispersion relation of the propagating wave. The study discloses the existence of two wave fronts of SH-waves propagating with different speeds through the layered structure. The second wave front arises as a result of existence of void pores, while the first wave front defines SH-wave propagation in a nonlocal fiber-reinforced layered structure. Each wave front possesses individual critical frequency beyond which they fail to propagate. Speeds of both wave fronts are influenced by the appearance of nonlocality parameter. Using MATHEMATICA, several graphs have been plotted to demonstrate the impact of key parameters on the dispersion and attenuation of both wave fronts. Some notable cases have been derived which validate the present mathematical model.

A new anisotropic-asymmetric yield criterion covering wider stress states in sheet metal forming

The stress state-dependent plastic anisotropy of sheet metals has drawn significant attention. Nevertheless, existing phenomenological plasticity models limitedly capture the distinctive yield strengths and plastic flow (r-values) under a wide range of stress states covering uniaxial tension (UT), equi-biaxial tension (EBT), uniaxial compression (UC), plane strain tension (PS), and pure shear (SH). This study aims to propose a new anisotropic-asymmetric plasticity model with enhanced flexibility based on the non-associated flow rule. The yield function is formulated based on the additive coupling of two stress potentials: coupling between a new anisotropic pressure-sensitive fourth-order polynomial function, and an isotropic stress invariant-based function. Then, all the anisotropic/asymmetric parameters of the yield function can be analytically identified from 7 yield stresses measured from UT tests along 0°, 45°, and 90° to the rolling direction (RD), UC tests along 0° and 90° to the RD, EBT test, and SH test along 45° to the RD. Within the scheme of no-associated flow rule, a new asymmetric fourth-order plastic potential function is proposed to analytically describe the plastic flow of sheet metals under UT, UC, EBT and SH. The additively coupled stress invariant-based term is introduced to adjust the yield stresses and plastic flow under biaxial tension including PS, which leads to an independent description of the yield stresses and plastic strain directions under SH and PS states. Various automotive sheet metals such as advanced high-strength steels, aluminum alloy, and magnesium alloy with strong asymmetry and anisotropy are investigated to verify the proposed yield criterion. The results demonstrate remarkable flexibility and accuracy of the proposed yield criterion against other existing models. Evolving yield surface and plastic potential of a quenching and partitioning steel are accurately captured by the proposed anisotropic-asymmetric yield criterion. Besides, the advantages of new model are further discussed in terms of the proof of yield surface convexity, identified parameters, and the regulation between anisotropy and isotropy in comparison with existing multiplicative coupling method.

Characteristic and processing method of SH-wave data generated by vibroseis source

Enhancing the energy of horizontally-polarized shear (SH) waves while suppressing compressional (P) waves is a crucial prerequisite for processing and interpreting SH-wave data obtained from urban engineering surveys. In the case of processing SH-wave data generated by a hammer-impact source, a widely used and effective technique is to calculate half the difference between the two datasets collected from opposite force directions at the same shot point. However, this method shows the opposite effect when processing SH-wave data generated by a shear wave vibroseis source. To address this issue, we begin with investigating the characteristic of related reflections in SH-wave data generated by the vibroseis source, and then propose an effective processing method that computes half the sum of the two vibroseis-generated datasets collected in the same manner. Synthetic test shows that the proposed method can effectively enhance the energy of SH-waves while suppressing that of P-waves. The field data application suggests that the proposed method can successfully and effectively detect the location of underground limestone caves compared to the traditional method. The proposed method provides a simple but effective alternative for improving the signal-to-noise ratio (SNR) of SH-wave data generated by vibroseis source.

Propagation characteristics of shear horizontal waves in piezoelectric semiconductor nanoplates incorporating surface effect

Elastic wave-based devices made of piezoelectric semiconductors (PSs) find wide applications in modern scientific and technological fields. Elastic waves propagating in PS nanostructures have a rich variety of propagation characteristics due to the interaction between deformation, polarization, and charge carrier, as well as the influence of surface effect. Based on the Gurtin-Murdoch surface model, we conduct an analysis of horizontally polarized shear (SH) waves propagating in an infinite n-type PS plate with nano-thickness in this paper. For the open-circuit and electrically isolated boundary conditions, the analytical frequency and phase velocity spectrum relations and cut-off frequencies of SH waves with anti-symmetric and symmetric modes are presented. The influence of the surface effect, initial electron concentration, and thickness of the PS plate on the propagation characteristics of SH waves, such as frequency spectrum, phase velocity, and cut-off frequency, are elaborated in detail. Numerical results show that propagation characteristics of SH waves are size-dependent, and the macroscopic effective shear rigidity is softened at the nanoscale due to surface effect, which thus leads to the reduction of the frequency and phase velocity of SH waves. Moreover, the propagation characteristics of SH waves can be manipulated by changing the initial electron concentration.

Griffith crack analysis in nonlocal magneto-elastic strip using Daubechies wavelets

The main focus of this work is to employ the Daubechies wavelet approximation to analyze the Griffith crack in nonlocal magneto-elastic horizontally shear (SH) wave propagation. It is assumed that the crack is located in a homogeneous isotropic nonlocal magneto-elastic strip of finite thickness and infinite length. The governing mixed boundary value problem is converted to a dual integral equation by employing Fourier transformation. This dual integral equation is further converted to a Fredholmn integral equation of the second kind by means of Abel transformation. A wavelet-based approximation scheme is employed to solve the integral equation as it is not solvable analytically. Griffith crack analysis using Daubechies wavelets has not been studied previously in the literature. The stress intensity factor (SIF) at the crack tip is determined numerically after solving the integral equation. In the end, the effect of the magnetic field, strip breath, incident angle of the shear wave and the nonlocal parameter on stress intensity factor (SIF) at the crack tip is explained through various graphical representations.

On the Differences in the Periodic Behavior of Magnetic Helicity Flux in Flaring Active Regions with and without X-class Events

Abstract Observational precursors of large solar flares provide a basis for future operational systems for forecasting. Here, we study the evolution of the normalized emergence (EM), shearing (SH), and total (T) magnetic helicity flux components for 14 flaring (with at least one X-class flare) and 14 nonflaring (<M5-class flares) active regions (ARs) using the Space-weather Helioseismic Magnetic Imager Active Region Patches vector magnetic field data. Each of the selected ARs contain a δ-type spot. The three helicity components of these ARs were analyzed using wavelet analysis. Localized peaks of the wavelet power spectrum (WPS) were identified and statistically investigated. We find that (i) the probability density function of the identified WPS peaks for all the EM/SH/T profiles can be fitted with a set of Gaussian functions centered at distinct periods between ∼3 and 20 hr. (ii) There is a noticeable difference in the distribution of periods found in the EM profiles between the flaring and nonflaring ARs, while no significant difference is found in the SH and T profiles. (iii) In flaring ARs, the distributions of the shorter EM/SH/T periods (<10 hr) split up into two groups after flares, while the longer periods (>10 hr) do not change. (iv) When the EM periodicity does not contain harmonics, the ARs do not host a large energetic flare. (v) Finally, significant power at long periods (∼20 hr) in the T and EM components may serve as a precursor for large energetic flares.

Counterexample-trained neural network model of rate and temperature dependent hardening with dynamic strain aging

Constitutive models dealing with the thermal and visco-plasticity of metals have seen wide applications in the automotive industry. A basic plasticity and fracture characterization of a 1.5 mm thick DP780 dual phase steel sheet based on uniaxial tensile (UT) experiments with seven distinct material orientations is complemented by low (∼0.001/s), intermediate (∼1/s) and high (∼150/s) strain rate experiments on notched tensile (NT) and shear (SH) specimens at temperatures ranging from 20 °C to 500 °C. At low strain rates, we observe a non-monotonic effect of the temperature on the force-displacement curves, with the highest curve obtained at 300 °C. Contrasting low speed tests, a monotonic effect of temperature is observed for intermediate and high strain rate experiments, with the highest curves obtained at 20 °C for both cases. Strain rate jump tests are performed proving the positive strain rate sensitivity of the steel. A machine-learning based plasticity model is developed to capture the observed complex strain rate- and temperature effect. The material is modeled as elasto-plastic, with a Hill’48 yield surface and a non-associated flow rule. The flow resistance is decoupled into a reference strain hardening term and a neural network term, which is a function of the plastic strain, strain rate, temperature, and an additional dynamic strain aging variable. The plasticity model is implemented into a material user subroutine and identified using a counterexample-guided hybrid experimental-numerical approach. The extracted loading paths reveal a complex rate and temperature effect on the ductility of DP780. A neural network based fracture initiation model is therefore adopted to describe the fracture onset across various stress states, strain rates and temperatures considered.

Generalized coupled mode theory in phononic slab waveguides

We develop the generalized coupled mode theory for the phononic slab waveguides, including core and cladding regions. At first, the eigenfrequency analysis is performed to obtain the dispersion curves for shear (SH) and Rayleigh–Lamb (RL) modes that are the orthogonal modes of slab waveguide. Secondly, the coupled mode equations (CMEs) are introduced for SH and RL modes, separately. Then CMEs are utilized to study transferring elastic energy between two coupled waveguides. We also analytically investigate the coupling of SH modes between non-parallel waveguides through calculating the coupling coefficients for small angles between them. Finally, the coupling of non-parallel waveguides in two different situations is demonstrated, when they are getting away from each other, and vice versa.

Green’s function analysis of shear wave propagation in heterogeneous poroelastic sandwiched layer influenced by an impulsive source

The present analysis confers the propagation characteristics of horizontally polarized​ shear (SH) wave through a heterogeneous transversely isotropic fluid-saturated poroelastic sandwiched layer of finite width embedded between two heterogeneous isotropic elastic half-spaces due to the impact of an impulsive line source. The dissipation of energy caused by the relative fluid flow is neglected in transversely isotropic fluid-saturated poroelastic sandwiched layer. The heterogeneities in layer and half-spaces are considered with the respective spatial quadratic and exponential type variations in the elastic constants and densities. Green’s functions are obtained for this sandwiched layered structure by considering a charge-density/line-force (Dirac-delta function) at lower bonding surface of layer and half-space. Dispersion relation for the propagation of SH wave in aforesaid structure has been obtained with well-known properties of Green’s functions, Fourier’s transformation and appropriate boundary conditions. The obtained dispersion relation is reduced for a structure with a heterogeneous transversely isotropic fluid-saturated poroelastic layer of finite width overlying a heterogeneous isotropic elastic half-space in the particular case of problem. Moreover, the obtained dispersion relations are validated and matched with pre-established standard and classical results. The effect of various affecting parameters viz. anisotropy of layer, porosity of layer and heterogeneities of layer and half-spaces on phase velocity of propagating wave in said structure are shown graphically through numerical computations. Further, a comparative study of distinct cases of heterogeneous/homogeneous structures is also performed which serves as the major highlight of the present study.

Quadramode materials: Their design method and wave property

Extremal materials with certain vanishing eigenvalues of their elastic matrix are able to manipulate elastic wave with their extremal static property, the devices designed with such materials can work in a broad frequency band, which is highly demanded in low frequency applications. A type of extremal materials with four zero eigenvalues called quadramode (QM) materials is the subject of this investigation. Wave properties of different QM materials, particularly their capacity on shaping iso-frequency curves, are firstly examined with homogeneous models. A three-dimensional QM material is then designed using a truss lattice model. The designed lattice is validated through comparison between the homogeneous and discrete models on their prediction on iso-frequency curves and polarizations. An out-of-plane shear (SH) wave polarizer is proposed with the designed QM material, it can effectively prevent the mode conversion at interface with fluids. This unique property is further explored for waterborne sound isolation, a prototype of this polarizer is also proposed and demonstrated through numerical simulation. This study paves the way for exploring the exotic wave properties of QM extremal materials, and opens a new route to control low frequency elastic wave.

Shear-Wave Velocity Site Characterization in Oklahoma from Joint Inversion of Multimethod Surface Seismic Measurements: Implications for Central U.S. Ground-Motion Prediction

ABSTRACT We analyze multimethod shear (SH)-wave velocity (VS) site characterization data acquired at three permanent and 25 temporary seismograph stations in Oklahoma that recorded M 4+ earthquakes within a 50 km hypocentral distance of at least one of the 2016 M 5.1 Fairview, M 5.8 Pawnee, or M 5.0 Cushing earthquakes to better constrain earthquake ground-motion modeling in the region. We acquired active-source seismic data for time-averaged VS to 30 m depth (VS30) at 28 seismograph stations near the Fairview, Pawnee, and Cushing epicentral areas. The SH-wave refraction travel times coupled with Rayleigh- and Love-wave phase velocity dispersion were extracted and modeled in a nonlinear least-squares (L2) joint inversion to obtain a best-fit 1D VS versus depth profile for each site. At a subset of sites where the preferred L2 inverse model did not optimally fit each of the Love, Rayleigh, and SH travel-time datasets, we explore application of simulated annealing in a joint inversion to find a more global solution. VS30 values range from 262 to 807 m/s for the preferred measured (in situ) VS profiles, or National Earthquake Hazards Reduction Program (NEHRP) site class D to B, and are broadly comparable with estimates from previous data reports in the region. Site amplification estimates were calculated next from 1D SH transfer functions of the preferred VS profiles and then compared against observed horizontal-to-vertical spectral ratios (HVSRs) from nearby seismograph stations. We generally see good agreement between the predicted in situ model and the observed HVSR resonant frequencies, with nominal amplifications between 2 and 10 within the 2–15 Hz frequency band. Next, using 40 known in situ VS30 measurements in the region, we demonstrate that the in situ VS30 values improve the fit for selected suites of ground-motion models (GMMs) for M 4+ earthquakes within a 50 km hypocentral distance when compared with proxy methods, arguing for future development of GMMs implementing in situ VS profiles.

Consistent Boundary Condition for Horizontally-Polarized Shear (SH) Waves Propagated in Layered Waveguides

무한 매질에서의 파전파 현상은 공학과 자연과학의 여러 분야에서 다양한 물리적 현상을 서술하는데 활용되고 있고, 이 문제에 대한 해를 얻기 위하여 해석적 방법 또는 수치적 방법이 개발되어 사용되고 있다. 이 문제에 대한 정확한 해를 얻기 위해서는 무한 영역으로의 에너지 방사를 정확히 고려해야 하고, 이를 위해 다양한 수치적 또는 역학적 모형 또는 경계조건이 개발되었다. 이 연구에서는 층상 waveguide에서의 scalar wave 또는 SH파 전파 문제에 적용할 수 있는 새로운 경계조건을 제안하고자 한다. 이를 위해 waveguide의 수직방향으로 유한요소 이산화를 적용하여 얻은 SH파의 지배방정식을 변형하여 waveguide의 무한 영역의 영향을 나타내는 경계조건을 유도한다. 층상 waveguide에서의 SH파에 대한 고유모드의 직교성을 이용하여, 새로운 경계조건은 기존의 root-finding absorbing boundary condition와 동등함을 보이고, 이로부터 새로운 경계조건의 차수가 증가할수록 정확성이 증가하고, 또한 이산화된 수준에서도 안정함을 유도할 수 있다. 제안된 경계조건을 층상 waveguide에서의 파전파 문제에 적용하여 그 정확성과 안정성을 검증한다.