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
WKB方法是一种非常重要的数学方法,在解微分方程中有着广泛的应用,它是数学物理中的一种渐近方法。WKB在物理中也同样有很重要的应用,它通常被用来解决量子力学里面的一些物理问题,比如解薛定谔方程、求Bohr-Sommerfeld量子化条件等。...WKB方法是一种非常重要的数学方法,在解微分方程中有着广泛的应用,它是数学物理中的一种渐近方法。WKB在物理中也同样有很重要的应用,它通常被用来解决量子力学里面的一些物理问题,比如解薛定谔方程、求Bohr-Sommerfeld量子化条件等。近年来人们发现用WKB方法来求解黑洞中准正则模式(QNM)的问题也十分有效。用解出的QNM来近似引力波就可以得到黑洞的一些性质。本文对此展开研究,以一个施瓦西黑洞为例,介绍如何将QNM问题转化为WKB问题,并给出结果。WKB method is a very important mathematical method, which is widely used in solving differential equations. It is a kind of asymptotic method in mathematical physics. WKB also has important applications in physics, where it is often used to solve some physical problems in quantum mechanics, such as solving Schrodinger equation and finding Bohr-Sommerfeld quantization conditions. In recent years, people found that WKB method is also very effective to solve quasi-normal modes (QNM) problems in black holes. Some properties of black holes can be obtained by approximating gravitational waves with the solved QNM. Taking a Schwarzschild black hole as an example, this paper introduces how to transform the QNM problem into the WKB problem and gives the results.展开更多
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.
文摘WKB方法是一种非常重要的数学方法,在解微分方程中有着广泛的应用,它是数学物理中的一种渐近方法。WKB在物理中也同样有很重要的应用,它通常被用来解决量子力学里面的一些物理问题,比如解薛定谔方程、求Bohr-Sommerfeld量子化条件等。近年来人们发现用WKB方法来求解黑洞中准正则模式(QNM)的问题也十分有效。用解出的QNM来近似引力波就可以得到黑洞的一些性质。本文对此展开研究,以一个施瓦西黑洞为例,介绍如何将QNM问题转化为WKB问题,并给出结果。WKB method is a very important mathematical method, which is widely used in solving differential equations. It is a kind of asymptotic method in mathematical physics. WKB also has important applications in physics, where it is often used to solve some physical problems in quantum mechanics, such as solving Schrodinger equation and finding Bohr-Sommerfeld quantization conditions. In recent years, people found that WKB method is also very effective to solve quasi-normal modes (QNM) problems in black holes. Some properties of black holes can be obtained by approximating gravitational waves with the solved QNM. Taking a Schwarzschild black hole as an example, this paper introduces how to transform the QNM problem into the WKB problem and gives the results.
