A modified warping operator based on BDRM theory in homogeneous shallow water

In this paper, a modified warping operator for homogeneous shallow water based on the Beam-Displacement Ray-Mode （BDRM） theory is presented. According to the BDRM theory, the contribution of the beam displacement and the time delay to the group velocity can be easily considered in a shallow water waveguide. A more accurate dispersion formula is derived by using the cycle distance formula to calculate the group velocity of normal modes. The derived dispersion formula can be ap- plied to the homogeneous shallow water waveguide. Theoretically, the formula is related to the phase of the reflection coeffi- cient and suitable for various bottom models. Furthermore, based on the derived dispersion relation, the modified warping op- erator is developed to obtain linear modal structures. For the Pekeris model, the formulae for the phase of the reflection coeffi- cient are derived in this work. By taking account of the effect of the bottom attenuation on the reflection coefficient, the for- mula for the phase of the reflection coefficient including the bottom attenuation is obtained for the Pekeris model with a lossy bottom. Performance and accuracy of different formulae are evaluated and compared. The numerical simulations indicate that the derived dispersion formulae and the modified warping operator are more accurate.