A Fast Scalar-Transfer Method for Overtone Dispersion of Rayleigh-Type Channel Waves in Layered Waveguides

Theoretical dispersion curves of Rayleigh-type channel waves are essential for identifying modes and interpreting guided waves in coal-seam waveguides. The present formulation is derived for horizontally layered, isotropic and elastic media, and does not include attenuation or lateral heterogeneity. Classical fast scalar-transfer formulations, however, were developed mainly for free-surface Rayleigh waves and cannot be applied directly to waveguides bounded by both upper and lower half-spaces. In this study, we extend the fast scalar-transfer method to Rayleigh-type channel waves by incorporating the upper-half-space radiation condition into the five-scalar recursive framework. We derive a normalized upper-half-space vector and construct a bilinear dispersion determinant for efficient evaluation of the channel-wave dispersion function on frequency–phase-velocity grids. The formulation is validated using staggered-grid finite-difference simulations for three-layer and five-layer coal–rock waveguide models. The predicted multimode phase-velocity curves agree well with the dispersion-energy ridges extracted from synthetic seismograms. Runtime tests on a common frequency–velocity grid show that, for a 14-layer model, the proposed method is approximately 8, 35 and 68 times faster than the fast delta matrix, generalized transfer matrix and fast generalized reflection–transmission methods, respectively. The proposed formulation provides an efficient tool for overtone dispersion computation of Rayleigh-type channel waves in layered waveguides.