基于能量守恒框架下的波动力学理论研究  

Research on elastodynamic theory based on the framework of energy conservation

作　　者：王秀明[1,2,3] 周吟秋 Wang Xiu-Ming;Zhou Yin-Qiu(National Lab.of Acoustics,Institute of Acoustics in Chinese Academy of Sciences,Beijing 100190,China;School of Physics Sciences,the University of the Chinese Academy of Sciences,Beijing 100149,China;Beijing Engineering Research Center for Offshore Drilling Exploration and Measurement,Beijing 100190,China)

机构地区：[1]中国科学院声学研究所声场与声信息国家重点实验室,北京100190 [2]中国科学院大学物理学院,北京100149 [3]北京市海洋深部钻探测量工程技术研究中心,北京100190

出　　处：《物理学报》2023年第7期256-266,共11页Acta Physica Sinica

摘　　要：基于波动力学的基本概念,提出了在能量守恒框架下建立波动力学方程的新思路与方法.首先,回顾了用牛顿第二定律推导波动力学方程,同时回顾并分析了利用Hamilton变分原理,推导了在连续介质中的Lagrange方程、Hamilton正则方程,以及相应的波动力学方程;其次,在能量守恒的框架下,建立了连续介质的Lagrange方程、Hamilton正则方程和波动力学方程,并证明其结果与利用经典力学推导的结果的一致性,特别地,澄清了用Hamilton变分原理建立保守系统下连续介质的Lagrange方程和Hamilton正则方程时在边界条件应用时的一些模糊认识.在能量守恒框架下建立一系列动力学方程,为我们在不涉及泛函求极值的变分原理等基础上刻画和表述复杂介质中波动现象的演化规律提供了另一种途径,也深入探讨了最小作用原理的物理本质.最后,在能量守恒的框架下给出了建立黏弹性介质中的波动力学微分方程的应用.Based on the analysis of establishing dynamic equations by using Newton's mechanics,Lagrange's,and Hamilton's mechanics,a new idea of establishing elastodynamic equations under the framework of energy conservation is proposed.Firstly,Newton’s second law is used to derive wave equations of motion.Secondly,Lagrange's equation,Hamilton's canonical equations,and the corresponding dynamical equations in a continuum medium are derived by using Hamilton’s variational principle.Thirdly,under the framework of energy conservation,Lagrange's equation,Hamilton's canonical equations,and the acoustic dynamic equations of the continuum are established,and the results are proved to be consistent with those derived from classical mechanics.Some fuzzy understandings when using Hamilton's variational principle to establish Lagrange’s equation and Hamilton’s canonical equation,are clarified.A series of dynamical equations established under the framework of energy conservation provides an alternative way to characterize and represent the propagation characteristics of wave motions in various complex media without involving the variational principle of functional extremum.Finally,as an application example,the differential equation of elastodynamics in a viscoelastic medium is given under the framework of energy conservation.

关 键 词：波动力学方程 HAMILTON 变分原理 LAGRANGE 方程 能量守恒 

分 类 号：O302[理学—力学]