With the transition to deep coal mining, the transparent detection of hidden geological hazards in the floor strata is fundamental for production safety. In mine seismic exploration, gliding waves—inhomogeneous plane waves propagating along the coal–rock interface—offer a unique advantage for penetrating high-velocity floors via the skin effect, overcoming the total reflection limitations of conventional in-seam waves. This study investigates the propagation laws and anomaly response characteristics of floor gliding waves using super-critical incidence theory and high-order staggered-grid finite difference simulations. The results demonstrate that the apparent velocities of gliding P and S-waves are bounded by those of the coal and host rock, exhibiting minimal dispersion. Quantitative analysis using a penetration depth model reveals that while penetration depth is frequency-dependent—with lower frequencies providing deeper reach—high-frequency components remain essential for high-resolution imaging. Crucially, the proposed method was validated through a field Case Study at the 11123 working face. By utilizing a specialized deep-hole excitation strategy to ensure super-critical incidence, the inversion successfully identified a hidden fault extending up to 60 m below the floor, which was subsequently confirmed by rock roadway excavation. These findings establish a robust physical basis for designing underground floor-detection systems and provide a significant theoretical reference for addressing detection blind spots in deep mining environments. 1. Introduction With the increasing depth of coal mining in China, geological conditions have become increasingly complex. Hidden hazards, such as high-pressure water in the coal seam floor, severely restrict the safe and efficient production of mines [ 1, 2]. In the context of deep mining, the fine detection of floor aquifuge integrity and hidden water-conducting structures is key to ensuring production safety [ 3, 4]. Furthermore, the structural stability of the floor rock mass serves as a fundamental cornerstone for guaranteeing operational safety in underground mines, especially when mining activities transition into deep formations prone to catastrophic high-pressure water hazards. Utilizing high-precision wavefield numerical simulations to investigate the dynamic characteristics of elastic waves within complex, multi-layered, or heterogeneous strata provides critical reference and support for evaluating rock mass stability, predicting potential joint/discontinuity failure mechanisms, and formulating proactive hazard mitigation strategies. Consequently, executing rigorous computational wavefield modeling is essential to overcome parameter subjectivities, thereby enabling the reliable identification of hidden geodynamic risks [ 5, 6, 7]. Currently, mine detection methods for floor disaster sources mainly include direct current (DC) electrical methods, transient electromagnetic methods (TEMs), and ground-penetrating radar (GPR), which have certain advantages in identifying water-rich areas [ 8, 9, 10]. However, limited by interference from mine metal components and resolution bottlenecks, electrical methods often struggle to finely delineate deep thin layers [ 11]. In contrast, seismic exploration, based on elastic mechanics, possesses higher physical resolution. Given that surface seismic methods find it difficult to meet underground exploration needs, developing high-resolution underground seismic technology based on near-source excitation—particularly fine detection methods using diffraction wavefield imaging—has become an important path for the accurate identification of hidden floor hazards [ 12, 13]. However, due to the total reflection confinement mechanism of the coal seam waveguide, energy in conventional channel wave exploration is mainly concentrated within the coal seam, making it difficult to radiate deep into the floor. This leads to insufficient penetration and exploration capability for floor structures [ 26, 27]. Although studies have used full-wavefield numerical simulation to analyze the dynamic characteristics of phases such as roadway Rayleigh waves, it remains difficult to break through the waveguide constraint to identify deep floor structures [ 28]. Gliding waves are inhomogeneous plane waves propagating along the interface, and their amplitude distribution follows the “skin effect,” providing a physical basis for energy penetration into high-velocity floors. Numerical simulations confirm that these interface phases exhibit controllably enhanced dynamic characteristics in restricted spaces such as while drilling boreholes. Moreover, in complex coal seams where fundamental channel waves are not well-developed, using gliding wave refraction components to replace traditional waveguide modes for structural positioning has shown theoretical feasibility [ 29, 30, 31, 32, 33]. Nevertheless, research on the response mechanism, penetration depth, and evolution laws of gliding waves specifically for the particular wave impedance of coal measures remains insufficient. Therefore, this paper systematically explores the propagation mechanism and effective detection range of coal seam floor gliding waves based on the supercritical incidence mechanism of seismic waves at the coal–rock interface. The wave equation for gliding waves is derived using elasto dynamics theory, and high-order staggered-grid finite difference simulations are employed to quantitatively analyze the non-linear response between detection depth and factors such as excitation frequency and incidence angle. This study aims to reveal the dynamic propagation characteristics of gliding waves in floor strata, providing a theoretical basis and numerical method support for utilizing non-seam-wave phases to address floor detection blind spots. 2. Materials and Methods 2.1. Theoretical Model and Reflection Coefficient Assume that the coal seam and the floor constitute a semi-infinite double-layer medium model ( Figure 1). The upper layer represents the coal seam medium (low-velocity layer) with density ρ 1 and S-wave velocity v 1 . The lower layer represents the floor rock (high-velocity layer) with density ρ 2 and S-wave velocity v 2 , satisfying the condition v 2 > v 1 . When the wave is incident from the coal seam to the coal–rock interface at an angle β i , according to the boundary continuity conditions, the reflection coefficient R and transmission coefficient T can be expressed as R = ρ 1 v 1 c o s β i − ρ 2 v 2 c o s β t ρ 1 v 1 c o s β i + ρ 2 v 2 c o s β t (1) where β t is the transmission angle in the floor medium. According to Snell’s law, the relationship between the incident angle and the transmission angle satisfies s i n β i v 1 = s i n β t v 2 (2) Since v 2 > v 1 , when the incident angle β i increases to the critical angle β c = a r c s i n v 1 / v 2 , the transmission angle β t = 90 ∘ . At this point, the transmitted wave glides along the interface. 2.2. Supercritical Incidence and Gliding Wave Generation As the incident angle continues to increase into the supercritical state ( β c 1 , and the transmission angle becomes a complex number. Based on the theory of complex variables and the analytical solution of the wave equation, the displacement field expression of the transmitted wave in the floor is expressed as u 2 x , z , t = T A 0 e x p − k 0 z v 2 2 v 1 2 s i n 2 β i − 1 e x p i ω t − x c x A 0 = 2 cos β i cos β 2 i + m 2 δ 2 m = ρ 2 v s 2 ρ 1 v s 1 (3) where A 0 is the incident wave amplitude, k 0 is the wave number, ω is the angular frequency, x is the propagation direction along the interface, and z is the depth direction perpendicular to the interface and pointing downwards. 2.3. Propagation Characteristics Analysis Analyzing the displacement field expression (3) above, the following propagation characteristics of the floor gliding wave can be derived: Inhomogeneous plane wave characteristics: The real exponential term in the first part of the expression characterizes the distribution law of energy in the direction perpendicular to the interface ( z -axis), indicating that the amplitude drops exponentially with depth. Meanwhile the wavefield maintains a harmonic form along the direction of the interface ( x -axis). This type of wave, where the direction of amplitude attenuation is perpendicular to the direction of phase propagation, is defined in physics as an inhomogeneous plane wave e . Phase velocity characteristics: c x in the expression represents the apparent velocity (phase velocity) of the gliding wave propagating along the interface, which is expressed as c x = v 1 s i n β i (4) Since β c 100 Hz), the detection depth of floor gliding waves is typically less than 5 m. To achieve the detection of deep hidden hazards at depths of 10–20 m below the floor, the principal frequency of the source must be reduced to below 20 Hz. To analyze the influence of the incident angle, the frequency is kept constant, and the variation in detection depth is examined as the incident angle increases to 90 ∘ ( Figure 6). The curves show that as the incident angle approaches the critical angle, the denominator tends to zero, and the theoretical detection depth approaches infinity. As the incident angle increases toward 90°, the detection depth decreases rapidly. This implies that in the design of actual observation systems, the detection depth can be improved by optimizing the offset to control the incident angle so that it remains close to the critical angle. 4. Numerical Simulation Verification 4.1. Model Construction and Parameter Settings To verify the universality of the theoretical laws, this chapter extends the research object from a single SH-wave field to a P-SV elastic wavefield, which is more consistent with actual working conditions. A three-layer strata model consisting of “surrounding rock–coal seam–surrounding rock” was constructed ( Figure 7). The coal seam thickness is set to 3 m, and the upper and lower surrounding rocks are simulated as semi-infinite space media. The specific physical parameters of the strata are as follows: the P-wave and S-wave velocities of the surrounding rock are 2808 m/s and 1800 m/s, respectively; the P-wave and S-wave velocities of the coal seam are 1710 m/s and 900 m/s, respectively; and the densities of coal seam and surrounding rock are 1300 kg/m 3 and 2600 kg/m 3, respectively. Regarding the observation system settings, the shot point is located at the center of the coal seam, and the source employs a zero-phase Ricker wavelet with a principal frequency of 100 Hz. A horizontal receiving line is arranged at the coal seam–floor interface, with both the minimum offset and the trace spacing set to 5 m. In this study, the elastic wave equation is used for numerical simulation. The model dimensions are nx = 728 and nz = 768. The spatial sampling intervals are Δx = Δz = 0.75 m, and the temporal sampling interval is Δt = 0.1 ms. A PML absorbing boundary condition was used, with an absorbing layer thickness of 30 grid cells. 4.2. Kinematic and Wavefield Characteristic Analysis 4.2.1. Wave Group and Travel-Time Characteristics Based on the aforementioned model parameters, the seismic records obtained from the floor interface survey line are shown in Figure 8. The wavefield excited by the 100 Hz source undergoes phase superposition at the floor boundary, primarily consisting of the direct P-wave, S-wave, and dispersed channel waves. The records indicate that the energy distribution of different phases varies across the two components. In the horizontal component, the direct P-wave energy is relatively strong, with an amplitude higher than the subsequent channel waves. In the vertical component, the amplitudes of the P-wave and channel waves are substantially comparable. Restricted by multiple reflections and the total reflection mechanism within the coal seam, the energy response of the S-wave is weak in both components. The comparison suggests that the horizontal component has a higher detection sensitivity for the direct waves at the floor interface. Travel–time picking was performed for the P-waves and S-waves in the seismic records, and the results of the event extraction are shown in Figure 9. Due to the small thickness of the coal seam and the significant velocity difference between the coal and the floor, the travel–time curves exhibit an approximately linear form. Based on the travel–time relationship, the calculated apparent velocities for the P- and S-waves are 2632 m/s and 1494 m/s, respectively. Combined with the physical parameters of the strata defined in Section 4.1, it can be observed that the velocity of the first-arrival P-wave lies between the P-wave velocity of the coal seam and that of the surrounding rock, and is slightly lower than the latter. This is consistent with the propagation velocity domain characteristics of gliding waves derived in the previous section. Similarly, the propagation velocity of the S-wave also satisfies the velocity constraint for gliding waves. Accordingly, it is inferred that these P- and S-waves are not direct waves, but rather gliding P-waves and gliding S-waves propagating along the coal seam floor. 4.2.2. F-K Spectra and Energy Characteristics To further separate different phases from the superimposed wavefield and verify their kinematic properties, a frequency-wavenumber (F-K) transform was performed on the aforementioned two-component seismic records ( Figure 10). In the F-K domain, the records exhibit two sets of energy clusters with distinct differences in apparent velocity. Extraction via apparent velocity scanning reveals that the apparent velocity of the P-wave is distributed between 2566 and 2700 m/s, while that of the S-wave is distributed between 1398 and 1468 m/s. These spectral calculation results are consistent with the previous travel–time analysis conclusions. The results confirm that the apparent velocities of both the gliding P-waves and S-waves are slightly lower than the body wave velocities of the surrounding rock, which conforms to the physical characteristics of gliding waves propagating along an interface. Based on the spectral distribution characteristics determined in Section 4.2.2, this study designed corresponding F-K filters to perform wavefield separation. The extracted gliding P-waves and S-waves are shown in Figure 11. The records show that the events of the separated gliding P-waves and S-waves are basically parallel, and no obvious dispersion phenomenon was observed. Due to the influence of energy overlap between the S-wave and part of the channel waves in the F-K domain, a small amount of residual channel wave signals remain in the extracted S-wave wavefield. To quantitatively analyze the energy attenuation law of gliding waves, the event amplitudes of the gliding P-waves and S-waves were extracted from the two-component records, and the energy variation curves with offset were plotted ( Figure 12). The curve characteristics indicate that in both the horizontal and vertical components, the energy of the gliding P-waves and S-waves tends to decrease gradually as the offset increases. Furthermore, the energy attenuation rate of the S-wave is significantly greater than that of the P-wave. 4.3. Verification of the Relationship Between Detection Depth and Frequency To verify the detection depth characteristics of gliding waves, a vertical survey line was added to the previously described three-layer strata model. This survey line originates at the coal seam floor and extends into the floor strata, with a trace spacing of 1 m and a total of 200 receiving nodes. The two-component seismic records obtained from the vertical survey line through numerical simulation are shown in Figure 13. The aforementioned seismic records were filtered to extract the $P $-wave and $S $-wave records with principal frequencies of 20 Hz and 100 Hz, respectively. On this basis, the wavefield energy was extracted and normalized to obtain the distribution curves of gliding wave amplitude as a function of depth ( Figure 14). Comparing the curve characteristics reveals that at the same depth in the floor, the energy retention of the 20 Hz signal is consistently higher than that of the 100 Hz signal. According to the definition of penetration depth introduced earlier, the theoretical penetration depth values corresponding to the two frequencies in this model were calculated (indicated by the dashed lines in Figure 14). The comparison results demonstrate that the actual penetration depth of the 20 Hz wavefield is greater than that of the 100 Hz wavefield. This numerical simulation result validates the theoretical derivation that low-frequency gliding waves possess a wider vertical propagation range. The records from the vertical survey line were extracted to analyze the variation in gliding wave energy with respect to the distance from the interface (depth). The results demonstrate that as the receiver depth increases (moving further from the interface), the gliding wave amplitude exhibits a significant attenuation trend. This observation is highly consistent with the exponential decay law derived from the theoretical formulas in the previous section, thereby validating the propagation characteristics of gliding waves as inhomogeneous plane waves. 5. Field Validation and Application Example 5.1. Field Experiment and Data Acquisition The seismic survey was conducted at the 11123 working face to verify the effectiveness of the proposed super-critical incident wave detection theory. This section details the spatial geometry and the specialized excitation strategy employed during the experiment. 5.1.1. Geometry and Survey Design The survey utilized a transmission (through-seam) acquisition geometry between the intake and return tailgates. As shown in Figure 15, the shot points were deployed along the intake tailgate, while the geophones were installed along the return tailgate to ensure a dense ray-path coverage across the targeted detection area. The specific technical parameters of the acquisition system are summarized in Table 1. 5.1.2. Deep-Hole Excitation Strategy To specifically capture the response of deep-seated floor structures, a targeted deep-hole excitation strategy was implemented. The shot holes were precisely drilled to a depth of 3 m from the floor of the No. 3 coal seam, ensuring that the seismic source was excited within the underlying No. 1 coal seam. This configuration was designed to ensure that the incident waves interact with the high-velocity floor interface at a super-critical incident angle. This specialized setup facilitates the generation of gliding waves that propagate along the floor strata, allowing for the characterization of anomalies well below the immediate coal seam floor, which are often inaccessible via conventional in-seam seismic methods. 5.2. Data Processing and Inversion 5.2.1. Signal Conditioning and Wavefield Identification The seismic data processing for the 11123 working face was performed to enhance the signal-to-noise ratio and isolate the floor-sensitive gliding wave components. After initial pre-processing steps, including bad-trace removal and energy compensation, the quality of the wavefield was examined through typical shot records. As shown in Figure 16, the pre-processed records exhibit clear wave arrivals with distinct energy packets, where the high-frequency channel waves and floor gliding waves are well-preserved after suppressing low-frequency noise. This high-quality data provided a solid foundation for subsequent wave identification and inversion. 5.2.2. SIRT-Based Energy CT Imaging First, the observation system was established, and the imaging area was discretized into grids with a grid size of Δx = Δy = 5 m, resulting in 139 grids in the x-direction and 39 grids in the y-direction. To accurately characterize energy attenuation anomalies caused by geological structures, spherical divergence compensation was applied to the original records for true-amplitude recovery. Subsequently, filtering was performed to suppress noise and obtain seismic records with a high signal-to-noise ratio. Energy attenuation CT imaging was then carried out using the SIRT algorithm. The maximum number of iterations was set to 20, and the convergence criterion was defined as an inversion energy difference of ≤1%. Figure 17 presents the energy attenuation CT image obtained using the SIRT algorithm and its comparison with the geological interpretation results. The imaging results show good agreement with the three-dimensional seismic interpretation and the geological information revealed by roadway. 5.3. Results and Interpretation To obtain energy attenuation CT images at different depths within the coal seam floor, the preprocessed seismic records were decomposed into different frequency bands, and energy attenuation imaging was performed for each frequency band. Based on the gliding-wave penetration-depth formula derived in this study, energy attenuation sections corresponding to depths ranging from 15 m to 60 m were finally obtained, as shown in Figure 18. As shown in Figure 18, a pronounced zone of strong energy attenuation is observed in the coal seam floor, particularly in the lower-right part of the study area. The three-dimensional overlay view indicates that the geological anomaly can be clearly identified throughout the depth range of 15–60 m. Moreover, the attenuation intensity exhibits noticeable variations at different depths. The multi-depth perspective clearly demonstrates the vertical persistence of geological features, providing a “transparent” view of the floor strata that conventional channel wave methods cannot achieve. 6. Discussion 6.1. Comparison with Conventional Reflected Channel Wave Technology Conventional reflected channel-wave exploration utilizes the energy of multiple total reflections within the coal seam and is effective for detecting structures such as faults and collapse columns within the seam. However, its sensitivity to geological anomalies beneath the coal seam floor is limited. The floor gliding wave investigated in this study is generated by the supercritical incidence of SH waves, with energy concentrated near the coal seam–floor interface and penetrating into the floor strata. Owing to its sensitivity to variations in the physical properties near the interface, gliding-wave exploration can provide useful information on geological anomalies in the floor strata and serves as a complement to conventional channel-wave exploration. 6.2. Observation System Layout and Acquisition Recommendations The configuration of the observation geometry can affect the reception of gliding-wave energy. Considering the kinematic characteristics of gliding waves, larger source–receiver offsets may facilitate the generation and observation of wavefields associated with supercritical incidence. Furthermore, the simulation results indicate that gliding wave penetration depth declines as frequency rises. For deep structural detection of the coal seam floor, broadband seismometers are recommended to record effective low-frequency signals. 6.3. Limitation This work derives the propagation equation of gliding waves from supercritical SH wave incidence. The extension to P—SV gliding waves requires P—and SV—wave displacement potentials and relevant supercritical incidence analysis, which leads to greater complexity in theoretical derivation. A systematic study on P-SV gliding waves will be studied in the future. 7. Conclusions Based on elastic wave theory and finite difference numerical simulations, this study investigated the propagation characteristics and penetration behavior of coal seam floor gliding waves. The main conclusions are as follows: In a three-layer coal-bearing strata model, the apparent velocity of gliding waves is bounded by the body wave velocities of the coal seam and the surrounding rock. Compared with channel waves, gliding waves exhibit minimal dispersion. Theoretical analysis and numerical simulations demonstrate that the wave amplitude decreases exponentially with depth. The penetration depth of gliding waves is controlled by frequency and the physical properties of the media on both sides of the interface. The effective penetration depth increases with decreasing frequency. In practical applications, by conducting sliding wave energy imaging across different frequency bands, geological anomaly bodies within a 60 m range beneath coal seams were successfully detected. The results indicate that the proposed method shows promising prospects for the detection of geological structures in the coal seam floor. Author Contributions Conceptualization, T.D. and J.Y.; methodology, T.D.; software, T.D.; validation, T.D. and H.W.; formal analysis, T.D.; investigation, T.D. and H.W.; resources, J.Y.; data curation, T.D.; writing—original draft preparation, T.D.; writing—review and editing, J.Y.; visualization, T.D.; supervision, J.Y.; project administration, J.Y.; funding acquisition, J.Y. All authors have read and agreed to the published version of the manuscript. Funding This research was funded by the Coal-Major Project, grant number 2025ZD1700801; and the Guizhou Provincial Science and Technology Major Project, grant number [2024] 029; National Science and Technology Major Project of China, grant number 2025ZD1700701; China Coal Technology and Engineering Group, CCTEG, grant number 2021-2-GH001. Institutional Review Board Statement Not applicable. Informed Consent Statement Not applicable. Data Availability Statement The data presented in this study are available within the article. Acknowledgments The authors would like to thank Associate Researcher Xiaohui Yang from Xuzhou University of Technology for her support and assistance with the forward modeling algorithm in this paper. Conflicts of Interest Author Huricha Wang was employed by the company China Coal Technology and Engineering Group. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. References Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. © 2026 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license. MDPI and ACS Style Duan, T.; Yu, J.; Wang, H. Investigation of the Formation Mechanism and Propagation Characteristics of Gliding Waves in the Coal Seam Floor. Appl. Sci. 2026, 16, 5798. https://doi.org/10.3390/app16125798 AMA Style Duan T, Yu J, Wang H. Investigation of the Formation Mechanism and Propagation Characteristics of Gliding Waves in the Coal Seam Floor. Applied Sciences. 2026; 16(12):5798. https://doi.org/10.3390/app16125798 Chicago/Turabian Style Duan, Tianzhu, Jingcun Yu, and Huricha Wang. 2026. "Investigation of the Formation Mechanism and Propagation Characteristics of Gliding Waves in the Coal Seam Floor" Applied Sciences 16, no. 12: 5798. https://doi.org/10.3390/app16125798 APA Style Duan, T., Yu, J., & Wang, H. (2026). Investigation of the Formation Mechanism and Propagation Characteristics of Gliding Waves in the Coal Seam Floor. Applied Sciences, 16(12), 5798. https://doi.org/10.3390/app16125798
