Total Wave Field Research Articles

A semi-analytical approach for site-city interaction under oblique incident SH waves

In this paper, a semi-analytical approach was proposed for investigating the site-city interactions (SCI). The site was modelled as a horizontal layered half-space, while the building was represented as a collection of shear wall structures fixed on the surface of half-space. The total wave field was considered as a combination of free field and scattering wave field generated by buildings, simulated by direct stiffness method and Green's function, respectively. The coupling between the site and the city is achieved through the displacement continuity conditions at the bottom of the building. Finally, the impact and some influencing factors of SCI effect on site and buildings were studied. It is found that the SCI effect has a significant impact on surface displacement and building response. The SCI effect typically reduces surface displacement near cities and the amplitude of relative displacement of buildings, while increases the seismic duration of buildings. However, this influence will be affected by the incident angle, building layout, building height, and building spacing, so these factors should be comprehensively considered when analyzing the SCI effect. Through the case studies, it is demonstrated that this method can quickly calculate SCI models with complex factors without being limited by the number and thickness of soil layers, incidence angles, and different sizes and spacing of buildings.

Boat effects on lake water clarity and the efficacy of a no-wake zone policy

Boating activity in shallow waters of lakes can lead to sediment resuspension and potential nutrient release, affecting water quality parameters. No-wake zones, to minimize boating-induced effects, are often defined based on distance from shore. This study assessed the effects of boating on shallow waters and the efficacy of an existing no-wake zone policy in a midsize lake. Two experiments were conducted: (1) Turbidity and pressure data were collected along a broad shallow shelf on the southern shore of Lake Tahoe in the summer of 2019 and (2) turbidity and pressure observations were conducted while boats traveled at speeds between 8 kmh−1 (5 mph) and 40 kmh−1 over a water depth of ≈ 2.5 m. Results showed that boat-induced sediment resuspension occurred at speeds exceeding 8 kmh−1, with turbidity increases of up to ∼7 NTU. Boat-induced filtered pressure signal was a maximum of 40% of the total daily wave activity, with peaks observed between 10:00 and 12:00. Nevertheless, during the summer and fall of 2019, winds were the dominant driver of sediment resuspension, contributing around 80% of the total wavefield. An evaluation of the current no-wake zone policy at Lake Tahoe suggested that an 8 kmh−1 (5 mph) speed limit is suitable for preventing wake effects. However, the policy falls short in preventing sediment resuspension in the northwestern and southern shores of Lake Tahoe. This calls for the consideration of multiple variables, such as lakebed sediment type and water depth, when defining no-wake zone policies to effectively mitigate boating-induced effects.

Fundamental breathers and their physical spectra in vector fields with self-steepening

We study the dynamics of fundamental breathers in vector fields with self-steepening. Exact multi-parametric vector solutions of Akhmediev breathers, Kuznetsov-Ma solitons, and beating solitons are obtained. Their existence conditions are analyzed by using the square spectrum parameters. The analytic physical spectra of breathers are presented by using a residue theorem associated with second-order singularity. In contrast to the Manakov limit without the self-steepening term, these spectra exhibit asymmetry in both the vector wave component and the total wave field. We characterize explicitly such asymmetry. The correctness of our exact solutions is confirmed by numerical simulations.

Dynamic inverse plane characteristics of circular holes in a strip-shaped medium under line source excitation

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of circular holes in the strip domain under line source excitation were studied. The total wave field in the strip domain is constructed using the concept of infinite mirroring, which eliminates the mutual impact of the wave field under the action of the upper and lower limits of the strip domain. To determine the overall wave field, solve the wave field's unknown coefficients in accordance with the boundary conditions. The right number of mirror images is found using the dynamic stress distribution of a circular hole in the band-shaped domain as a specific calculation example. The effect of altering the incident frequency and band-shaped domain width on the dynamic stress concentration coefficient surrounding the circular hole is also explored. As a consequence of the findings, the dynamic stress concentration coefficient around the circular hole in the zone domain at low frequencies is substantially bigger than it is at medium and high frequencies. Under identical conditions, the dynamic stress concentration coefficient will exhibit periodic extreme values and converge as the distance increases between the lower boundary and the circular hole’s center.

QS‐wave decoupling equation for wavefield separation in transversely isotropic media

AbstractConsidering the anisotropy of the earth media is helpful in reducing the depth error between seismic and drilling and providing reliable imaging data for seismic interpretation and inversion. Transversely isotropic media with a vertical axis of symmetry are the most common type of anisotropic media and have been under constant study. The separation of the P‐ and S‐wavefields, which affects the accuracy of elastic wave imaging and the inversion in transversely isotropic media with a vertical axis of symmetry, is a hot research topic. Among the commonly used wavefield decoupling methods, the S‐wave is typically obtained by subtracting the P‐wave from the total wavefield. However, this kind of wavefield decoupling method often leads to severe P‐wave crosstalk in the separated S‐wavefield; thus, it needs further development. In this paper, the principle that the divergence of the S‐wave is zero is employed to solve the unresolved S‐wave elastic parameters of the S‐wave stiffness matrix by utilizing the modified zero‐order pseudo‐Helmholtz decomposition operator. The obtained S‐wave elastic parameters are employed to construct the qS‐wave stress and facilitate the derivation of the relationship between the qS‐wave particle velocity and the qS‐wave stress. Furthermore, a qS‐wave decoupling first‐order velocity–stress equation, which matches the qP‐wave decoupling first‐order velocity–stress equation, is derived. By jointly using the decoupling equations of qS‐ and qP‐wave, the separation of the P‐ and S‐wavefields in transversely isotropic media with a vertical axis of symmetry is realized. The efficiency of the proposed method is demonstrated via numerical tests conducted to assess the wavefield separation. Moreover, a complex model is employed to perform elastic reverse‐time migration, resulting in the acquisition of accurate imaging results for PP, PS, SP, and SS waves, without the presence of significant artefacts. The correctness of the qS‐wave decoupling equation in transversely isotropic media with a vertical axis of symmetry is confirmed by the comparative tests using decoupling methods in isotropic media.

DiffraPy: An open-source Python software for seismic diffraction imaging

Seismic diffraction events, often overlooked as noise in conventional seismic imaging methods, contain valuable high-resolution information about subsurface structures and heterogeneities. Furthermore, the diffractive component of the wavefield is masked by the stronger reflection events. Although the literature presents innovative methods for separating diffracted events from the total wavefield, there are currently limited options of open-source alternatives in seismic diffraction processing and imaging. In this study, we introduce DiffraPy, an open-source Python package specifically designed for performing both conventional and diffraction-oriented migrations. The main workflow involves implementing an anti-stationary phase filter in the Kirchhoff migration for enhancing energy outside the Fresnel zone. Additionally, we propose using the semblance matrix obtained from the dip field calculation as an additional weight in conventional migration. The code provides functionality for forward modeling seismic data, allowing for new testing and experimentation. We demonstrate the general workflow of the code using a toy model and evaluate its capabilities on a synthetic salt model. In both cases, the program successfully suppressed the majority of reflected energy in the diffraction-oriented image. The final image highlights the responses of small lenses, discontinuities, and faults, providing valuable insights into the subsurface for interpretation. Despite the method sensitivity to variations in the correct velocity model, we demonstrate that our code is capable of imaging diffractors even when utilizing incorrect migration velocities of up to 5%. Our wavenumber analysis reveals the diffraction image preservation of smaller wavelengths, indicating higher resolution compared to conventional products. Future implementations may include extension to 3D datasets and improve the computational efficiency of our code. We expect our code to offer the geosciences community the first freely accessible and well-documented Python alternative for testing, reproducing, and exploring diffraction imaging examples.

Dynamic anti-planar characteristics of circular holes in laminated structures under the action of line source

Based on the complex variable function method and the wave function expansion method, the dynamic anti-plane characteristics of the circular hole in the laminated structure under line source excitation were studied. To deal with the interface problem within the laminated structure, the idea of “fit” is adopted to construct continuity conditions that satisfy displacement and stress. The circular hole boundary stress-free condition is used, the wave field is constructed using the mirror method, and the unknown coefficients of the scattered wave field are solved through Fourier series expansion, and finally the total wave field in the laminated structure is obtained. Taking the dynamic stress distribution of a circular hole in a laminated structure as a specific calculation example, the effects of changes in line source position, layer thickness, and medium hard-soft ratio on the dynamic stress concentration coefficient around the circular hole are discussed. The results show that when the line source load is applied directly above the circular hole and the lower layer is narrow, the dynamic stress concentration coefficient will have a sudden change compared with other situations. When other conditions are the same, under low, medium, and high-frequency wave fields, the dynamic stress concentration caused by different soft-to-hard ratios shows completely different rules. These rules can be used to better detect defects in laminated structures.

Semi-analytical solution for the 3D response of a tunnel embedded in an elastic half-space subject to seismic waves

This paper presents a semi-analytical solution for the 3D problem of a cylindrical tunnel embedded in an elastic half-space subject to plane harmonic compressional and shear waves. Both the tunnel and soil are modelled as an elastic continuum. Conformal mapping is employed to transform the original physical domain with boundary surfaces of two different types onto an image domain with surfaces of the same type, which makes the problem easier to solve. The total wave field in the half-space consists of incident and reflected (from the half-space surface) plane waves, as well as directly and secondary scattered cylindrical waves, while the total wave field in the tunnel consists of refracted cylindrical waves. The secondary scattered waves, generated when the cylindrical waves directly scattered from the tunnel meet the half-space surface, are represented by cylindrical waves that originate from an image source, which is in line with the spirit of the method of images. The unknown amplitude coefficients of the cylindrical waves are determined from the boundary and continuity conditions of the tunnel–soil system by projecting those onto the set of circumferential modes, which results in a set of algebraic equations. Results show that the present method converges for a small number of circumferential modes. We observe very good agreement between the obtained results and those in literature. In a systematic evaluation, we demonstrate that the method works well for the frequency band of seismic waves, as well as for the complete considered ranges of the tunnel/soil stiffness ratio, the embedded depth of the tunnel, the vertical incident angle and the tunnel thickness. However, the results obtained for a moderate tunnel–soil stiffness contrast under the incident compressional wave are inaccurate when Hankel functions are used to represent the cylindrical waves in the tunnel, which is due to the refracted shear waves in the tunnel transitioning from propagating to evanescent (in the 3D case). These inaccuracies can be perfectly overcome by representing the waves in the tunnel by Bessel functions. We also find that the present method generally works better for the incident compressional wave than for the incident shear wave, as the condition number of the matrix (related to the mentioned algebraic equations) is often larger in the latter case. In view of engineering practice, we conclude that the tunnel is safer when the surrounding soil is stiffer, the tunnel is thicker and the vertical incident angle is larger. Finally, the present method, which is in general fast, elegant and accurate, can be used in preliminary design so as to avoid pronounced resonances, and to assess stress distributions and ground vibrations.

Multiphase wavefield inversion methodology for seismic analysis of soil-structure interaction systems

Currently available inversion methods for providing seismic excitation for soil–structure interaction (SSI) systems regard all ground motions as the body wave component and invert only the body wavefields. However, the ground motions are dominated by body waves and surface waves, and ignorance of the latter has been demonstrated to be conservative for structural responses. To provide more accurate wavefields for the seismic input of SSI systems located on the half-space of rock mediums, a multiphase wavefield inversion (MPWI) methodology considering both body waves and Rayleigh surface waves is proposed in this study. The methodology applies forward modeling and inversion theory of body waves and Snell's law for complex angles to determine the exact Rayleigh wave components and construct the Rayleigh wavefields. The residual body waves are used to invert the body wavefields, focusing on optimizing the determination of incident angles by a least-squares iteration approach. Rayleigh and body wavefields are finally combined to construct the total wavefields. The efficiency of the proposed methodology is verified and validated with analytical solutions and real downhole recordings, and the applicability is examined by response comparisons with the traditional method in the free field and an urban complex building. Simulation results demonstrate that due to the consideration of the Rayleigh waves, the MPWI methodology results in higher free-field stresses and accordingly induces more intense seismic excitation. Correspondingly, the structural responses present an amplification effect.

A simple method for fast simulation of total reflection waves of single-well imaging with eccentric tools

Abstract In the exploration of unconventional oil and gas reservoirs, single-well acoustic imaging has become an important technique to detect geophysical structures around the borehole. In this study, we propose a simple method to calculate the total reflection wavefields of pressure and displacement for single-well imaging (SWI) with eccentric monopole or dipoles. To calculate the pressure at the eccentric site for incident waves with different angles of incidence and azimuth, reciprocity is applied to the radiation from the eccentric monopole source and the far-field forces generating the waves corresponding to the incident waves. Displacement solutions are derived by calculating the gradient of the pressure fields. When considering eccentric monopole or dipole sources for an inclined reflector outside the borehole, reciprocity is again used to derive the total reflection waves, including the same type of waves (P-P/SV-SV/SH- SH) and the converted waves (P- SV/SV-P). The analytical results agree well with the reference solutions of the 3D finite-difference time-domain method, illustrating the validation of the method. The analysis shows that the pressure amplitude in the eccentric position is greater than that in the borehole axis, proving that the eccentric monopole can increase the weak amplitude of reflection waves. We also simulate the wavefields for the acoustic imaging model for eccentric monopole and dipoles. The results show that the pressure and displacement wavefields in the eccentric position are sensitive to the azimuth angle of the reflector, proving that SWI with eccentric logging tools is capable of identifying the azimuth angle.

Semi-analytical method for solving the anti-plane seismic dynamic response of steep slope site in a layered half-space

In order to calculate the anti-plane seismic dynamic response of steep slope in a layered site, a semi-analytical method is proposed. The total wave fields are decomposed into known free-field waves with regular boundary conditions and scattered wave fields caused by the known wave fields acting at the irregular boundary of the steep slope. The problem of solving scattering fields is then transformed from the external input problem of the seismic waves to the internal radiation problem by applying the equivalent seismic loads on the irregular boundary of a steep slope site. Then, a modified scaled boundary finite element method (SBFEM) using splicing lines rather than a point as scaling center to computer the internal radiation problem of the steep slope site analytically along the radial direction and numerically along the curvilinear direction. Numerical examples are used to verify the accuracy, effectiveness, and convergence of this method. Finally, a parametric study of the seismic response of a steep slope in a layered half-space is performed in frequency domain. The results show that the amplification peak occurs at the incident wave angle is 30° for vertical slope. For non-vertical slope, the maximum displacement all occur at the position away from one times slope height form the crest of the slope. Moreover, compared with vertical slope, the topography effect of inclined steep slope is more obvious, and this effect tends to decrease with the increase of the slope angle. Therefore, it can be concluded that the angle of the incident wave and slope angle have a significant effect on the dynamic response of the steep slope.

Tunable metasurfaces for seismic Love wave manipulation: A theoretical study

Seismic Love waves are horizontally polarized surface waves propagating within an overlying layer, which have been shown to pose potential hazards to infrastructures. In this study, we propose an original analytical formulation to investigate the interaction between seismic Love waves and a metasurface that is composed of a cluster of tunable anti-plane resonators. The formulation utilizes the Green's function of a layered half-space subjected to a surface SH source, to describe both the source-induced incident field and the resonator-excited scattered field. Based on the multiple scattering theory, we obtain the total wavefield of the layered half-space equipped with surface resonators by coupling the incident field and the scattered field. Besides, a closed-form dispersion relation governing the hybrid Love waves is derived by utilizing the effective medium approximation. We demonstrate the capabilities of our analytical formulation by modelling Love waves propagating across a pair and a cluster of resonators, evidenced by some complex phenomena (e.g., surface-to-bulk wave conversion and higher-order hybridized surface modes). The hybridization features of different Love modes and metasurface resonances are clearly revealed. We anticipate the proposed tunable metasurface and the formulation can shed light on versatile control of Love wave propagation.

Spatiotemporal distributions of air-sea CO2 flux modulated by windseas in the Southern Indian Ocean

The Southern Indian Ocean is a major reservoir for rapid carbon exchange with the atmosphere, plays a key role in the world’s carbon cycle. To understand the importance of anthropogenic CO2 uptake in the Southern Indian Ocean, a variety of methods have been used to quantify the magnitude of the CO2 flux between air and sea. The basic approach is based on the bulk formula—the air-sea CO2 flux is commonly calculated by the difference in the CO2 partial pressure between the ocean and the atmosphere, the gas transfer velocity, the surface wind speed, and the CO2 solubility in seawater. However, relying solely on wind speed to measure the gas transfer velocity at the sea surface increases the uncertainty of CO2 flux estimation. Recent studies have shown that the generation and breaking of ocean waves also significantly affect the gas transfer process at the air-sea interface. In this study, we highlight the impact of windseas on the process of air-sea CO2 exchange and address its important role in CO2 uptake in the Southern Indian Ocean. We run the WAVEWATCH III model to simulate surface waves in this region over the period from January 1st 2002 to December 31st 2021. Then, we use the spectral partitioning method to isolate windseas and swells from total wave fields. Finally, we calculate the CO2 flux based on the new semiempirical equation for gas transfer velocity considering only windseas. We found that after considering windseas’ impact, the seasonal mean zonal flux (mmol/m2·d) increased approximately 10%-20% compared with that calculated solely on wind speed in all seasons. Evolution of air-sea net carbon flux (PgC) increased around 5.87%-32.12% in the latest 5 years with the most significant seasonal improvement appeared in summer. Long-term trend analysis also indicated that the CO2 absorption capacity of the whole Southern Indian Ocean gradually increased during the past 20 years. These findings extend the understanding of the roles of the Southern Indian Ocean in the global carbon cycle and are useful for making management policies associated with marine environmental protection and global climatic change mitigation.

Homogenization of sound-soft and high-contrast acoustic metamaterials in subcritical regimes

We propose a quantitative effective medium theory for two types of acoustic metamaterials constituted of a large number N of small heterogeneities of characteristic size s, randomly and independently distributed in a bounded domain. We first consider a “sound-soft” material, in which the total wave field satisfies a Dirichlet boundary condition on the acoustic obstacles. In the “sub-critical” regime sN = O(1), we obtain that the effective medium is governed by a dissipative Lippmann–Schwinger equation which approximates the total field with a relative mean-square error of order O(max((sN)2N-1/3, N-1/2)). We retrieve the critical size s ~ 1/N of the literature at which the effects of the obstacles can be modelled by a “strange term” added to the Helmholtz equation. Second, we consider high-contrast acoustic metamaterials, in which each of the N heterogeneities are packets of K inclusions filled with a material of density much lower than the one of the background medium. As the contrast parameter vanishes, δ → 0, the effective medium admits K resonant characteristic sizes (si(δ))1≤i≤K and is governed by a Lippmann–Schwinger equation, which is diffusive or dispersive (with negative refractive index) for frequencies ω respectively slightly larger or slightly smaller than the corresponding K resonant frequencies (ωi (δ))1≤i≤K. These conclusions are obtained under the condition that (i) the resonance is of monopole type, and (ii) lies in the “subcritical regime” where the contrast parameter is small enough, i.e. δ = o(N−2)), while the considered frequency is “not too close” to the resonance, i.e. Nδ1/2 = O(|1 - s/si(δ)|). Our mathematical analysis and the current literature allow us to conjecture that “solidification” phenomena are expected to occur in the “super-critical” regime Nδ1/2|1 - s/si(δ)|-1 → + ∞.

Tunable elastic metasurface based on adjustable impedances for Gaussian beam manipulation

Different from the common tunable elastic metasurface (TEM), like piezoelectric active metasurface, in this paper, we propose a TEM composed of impedance-adjustable subunits (IASs) to realize tunable manipulations of flexural waves. Based on the impedance theory, the adjustable mechanism of the phase shift of the IAS is clarified according to the derivations for the relationship between the force/moment interface impedances and complex reflection coefficients. The boundary effect is considered, and the difference between the impedances of the flexural and acoustic waves is demonstrated for the subunits with free and clamped boundary conditions. By using the diffraction theory and coupled-mode theory, an analytical model is established for the metasurface subjected to a Gaussian flexural beam incidence, based on which, the total wavefield consisting of all propagating diffraction orders is solved. Its accuracy and the modulation function adjustability of the TEM are demonstrated by the finite element (FE) simulations and experiments. The proposed IAS possesses a flexible and reliable layout for TEM design and has potential engineering applications in vibration control, health monitoring, etc.

Ground motion around a semi-cylindrical canyon in saturated soil: P wave incidence

Based on Biot wave theory in saturated two-phase medium, the analytical solution of plane P-wave scattering by semi-circular canyon in half space in saturated two-phase medium is given by using complex function method and Hankel function integral transformation method. Firstly, a semi-circular depression model in saturated half space is constructed, the wave field potential function in the medium is solved, and its general solution form is obtained. Secondly, the unknown coefficient value in the potential function is obtained by substituting the corresponding stress boundary conditions. With the help of Hankel function integral transformation method, the wave potential function in complex coordinates is transformed to meet the horizontal boundary conditions of half space in rectangular coordinates. Finally, the surface displacement amplitude at the semi-circular depression and the surrounding horizontal boundary is obtained through the total wave field potential function, and the variation law of surface displacement amplitude under the influence of different parameters is analyzed.

Global field performance of monopile-supported offshore wind turbines with sinusoidal layouts using innovative combined grating conditions

For the global performance of monopile wind farms, the desired wave field distribution using traditional layout methods is hard to obtain. In this study, the investigation aims to efficiently explore the potential wave response reduction of the multiple layer design of wind farm layouts using novel grating conditions. It is very important and necessary to optimize the layouts of monopile-supported OWTs (offshore wind turbines) by analyzing the wave field performance, especially considering scour protection and avoiding the proximity of the wave frequency to natural frequency of OWTs. This paper presents a layout and a design method of monopile-supported OWTs using combined grating theorems, which can take space modulation into account to deal with various issues in existing layouts. The results show that the present method can modulate the wave field responses more evenly than the conventional cases. More specifically, total wave field distribution sensitivities were discussed under different wavelengths, amplitudes, layouts, pile-radius, and the angles of incident waves. It can be illustrated that the monopile-supported OTWs with sinusoidal configurations can have more modulation effects on wave fields in an appropriate wavelength band. As indicated, this method not only provides wave space modulation control but also sheds light on the wave field reduction mechanisms.

Rigorous solution for kinematic response of end-bearing pile under vertically incident P-waves

Considering the pile–soil interaction, this study investigates a displacement model for the steady-state kinematic response of a single pile to vertically incident P-waves based on a continuum model. A rigorous solution for the vertical kinematic response of an end-bearing pile is presented herein. The scattered waves were uncoupled from the total wavefield using a reasonable displacement assumption. According to the governing equations and boundary conditions, scattered waves can be easily expressed in an infinite series form, representing the superposition of each vibration order. The generalized expression of the pile–soil dynamic resistance is obtained using Hamilton’s principle. The pile–soil resistance is divided into the Winkler-spring part (produced by elastic shear force) and the dynamic-membrane-spring part (produced by normal elastic stress and inertial force) along the pile–soil interface. Similar to scattered waves, the displacement of the pile and free-field can be expressed in an infinite series form. According to the assumption of the displacement field, all undetermined constants are determined, and the final series solution is provided. The presented has a more concise and definite physical meaning and has excellent application prospects in pile seismic design. Based on the given rigorous analysed solution, a parametric study was conducted to investigate the effects of the pile and soil parameters on the dynamic response of the pile–soil system. Comparing the responses of piles with different stiffnesses, the characteristic stiffness represented a rigid pile (ℜ≤2) and a flexible pile (ℜ≥3).

Diffraction Imaging of the Fractured Crystalline Basement, Lancaster Field, UK

In this paper, we demonstrate the application of seismic diffraction imaging technology to map the distribution of faults within Hurricane Energy’s operated, fractured basement fields, offshore West of Shetland. After the reprocessing of legacy 3D seismic data in 2018, it became apparent that the reflection seismic image had limited ability to image the intra-basement structure. Intra-basement imaging is crucial for understanding the distribution, frequency and geometry of intra-basement structures which directly influence production trends in fractured basement reservoirs. Diffraction imaging is a novel seismic processing technique that can produce high-resolution images of small-scale scattering subsurface events by extracting and focusing the diffractive component of the total wavefield to its subsurface origin. Analysis of the diffraction imaging seismic volume alongside well data and other seismic products can complement conventional seismic attributes and enhance the interpretational value of the seismic data. This paper shows how the intensity of the diffraction image varies from location to location across the Rona Ridge and is seen to have a strong, positive correlation with observed well productivity. As such, the application of this technology shows great promise to provide additional information for detailed analysis of Hurricane Energy’s fractured basement reservoirs and can facilitate new well placement and trajectory design.

Kinematic Response of End-Bearing Piles under the Excitation of Vertical P-Waves Considering the Construction Effect

Under most engineering conditions, soil disturbance due to pile installation may cause soil properties to vary within the region adjacent to the pile in the radial direction. This paper derives a rigorous solution to investigate the kinematic response of end-bearing piles under the excitation of vertical P-waves considering the construction effect. The displacement responses of piles and soil, governed by the dynamic equilibrium equation, are theoretically derived with the separation of variables method. The scattered waves induced by the pile–soil system, which is the key factor of the problem, are decoupled from the total wavefields. Moreover, the friction occurring at the interface of the soil and pile shaft are directly obtained. Thus, the present solution can accurately account for the pile–soil interaction. Comparisons between the numerical results of the present method and the available results are performed. A detailed discussion on the kinematic response coefficient, amplification factor, and soil motion is provided.