This study investigates the effects of ocean boundaries on modal shapes in very-low-frequency(VLF,1–10 Hz)sound propagation through the deep ocean.Utilizing a normal mode solution formulated in terms of parabolic cyl...This study investigates the effects of ocean boundaries on modal shapes in very-low-frequency(VLF,1–10 Hz)sound propagation through the deep ocean.Utilizing a normal mode solution formulated in terms of parabolic cylinder functions(PCF),we demonstrate that boundary interactions induce a phase change reduction below-πat frequencies of several hertz.This reduction,in turn,forces a key transition in the solution,shifting the order of the PCF from integer to non-integer values.Analysis of the characteristic shape of the PCF versus its order reveals that these boundary-influenced modes exhibit an energy shift toward deeper regions and a weakened axial convergence of the underwater sound field.展开更多
A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the re...A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the reference functions method to describe the solution to the depth-separated wave equation approximately using parabolic cylinder functions.The conditions for the validity of this approximation are also discussed.Furthermore,a formula that incorporates waveguide effects for the modal group velocity is derived,revealing that boundary effects at very low frequencies can have a significant impact on the propagation characteristics of even low-order normal modes.The present method not only offers improved accuracy compared to the classical WKB approximation and the uniform asymptotic approximation based on Airy functions,but also provides a wider range of depth applicability.Additionally,this method exhibits strong agreement with numerical methods and offers valuable physical insights.Finally,the method is applied to the study of very-low-frequency sound propagation in the South China Sea,leading to sound transmission loss predictions that closely align with experimental observations.展开更多
A general analytical method to calculate the passive rigid retaining wall pressure was deduced considering all displacement modes. First, the general displacement mode function was setup, then the hypotheses were made...A general analytical method to calculate the passive rigid retaining wall pressure was deduced considering all displacement modes. First, the general displacement mode function was setup, then the hypotheses were made that the lateral passive pressure is linear to the corresponding horizontal displacement and the soil behind retaining wall is composed of a set of springs and ideal rigid plasticity body, the general analytical method was proposed to calculate the passive rigid retaining wall pressure based on Coulomb theory. The analytical results show that the resultant forces of the passive earth pressure are equal to those of Coulomb＇s theory, but the distribution of the passive pressure and the position of the resultant force depend on the passive displacement mode parameter, and the former is a parabolic function of the soil depth. The analytical results are also in good agreement with the experimental ones.展开更多
Abstract In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain Rn ×RM （n, m ≥2） are introduced. Lp bounds of such operators are obtained under weak conditions on the ke...Abstract In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain Rn ×RM （n, m ≥2） are introduced. Lp bounds of such operators are obtained under weak conditions on the kernels.展开更多
We study a class of nonlocal-diffusion equations with drifts,and derive a priori-Hölder estimate for the solutions by using a purely probabilistic argument,whereis an intrinsic scaling function for the equation.
基金Project supported by the National Natural Science Foundation of China(Grant No.12204128)。
文摘This study investigates the effects of ocean boundaries on modal shapes in very-low-frequency(VLF,1–10 Hz)sound propagation through the deep ocean.Utilizing a normal mode solution formulated in terms of parabolic cylinder functions(PCF),we demonstrate that boundary interactions induce a phase change reduction below-πat frequencies of several hertz.This reduction,in turn,forces a key transition in the solution,shifting the order of the PCF from integer to non-integer values.Analysis of the characteristic shape of the PCF versus its order reveals that these boundary-influenced modes exhibit an energy shift toward deeper regions and a weakened axial convergence of the underwater sound field.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12174048 and 12204128)。
文摘A Wentzel-Kramers-Brillouin(WKB)method is introduced for obtaining a uniform asymptotic solution for underwater sound propagation at very low frequencies in deep ocean.The method utilizes a mode sum and employs the reference functions method to describe the solution to the depth-separated wave equation approximately using parabolic cylinder functions.The conditions for the validity of this approximation are also discussed.Furthermore,a formula that incorporates waveguide effects for the modal group velocity is derived,revealing that boundary effects at very low frequencies can have a significant impact on the propagation characteristics of even low-order normal modes.The present method not only offers improved accuracy compared to the classical WKB approximation and the uniform asymptotic approximation based on Airy functions,but also provides a wider range of depth applicability.Additionally,this method exhibits strong agreement with numerical methods and offers valuable physical insights.Finally,the method is applied to the study of very-low-frequency sound propagation in the South China Sea,leading to sound transmission loss predictions that closely align with experimental observations.
基金Project (201012200094) supported by the Freedom Exploration Program of Central South University of ChinaProject (20090461022) supported by the China Postdoctoral Science FoundationProject (2010ZJ05) supported by the Science and Technology supporting Program of Xinjiang Production and Construction Corps in China
文摘A general analytical method to calculate the passive rigid retaining wall pressure was deduced considering all displacement modes. First, the general displacement mode function was setup, then the hypotheses were made that the lateral passive pressure is linear to the corresponding horizontal displacement and the soil behind retaining wall is composed of a set of springs and ideal rigid plasticity body, the general analytical method was proposed to calculate the passive rigid retaining wall pressure based on Coulomb theory. The analytical results show that the resultant forces of the passive earth pressure are equal to those of Coulomb＇s theory, but the distribution of the passive pressure and the position of the resultant force depend on the passive displacement mode parameter, and the former is a parabolic function of the soil depth. The analytical results are also in good agreement with the experimental ones.
文摘Abstract In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain Rn ×RM （n, m ≥2） are introduced. Lp bounds of such operators are obtained under weak conditions on the kernels.
基金The research of ZC was partially supported by NSF Grant DMS-1206276.The research of XZ was partially supported by NSFC Grant of China(Nos.11271294,11325105).
文摘We study a class of nonlocal-diffusion equations with drifts,and derive a priori-Hölder estimate for the solutions by using a purely probabilistic argument,whereis an intrinsic scaling function for the equation.
