地震波的场方程矩阵和能量的正定二次型及其意义

场方程是能够表述半空间地震波场的整体特征.波动方程、速度方程和能量方程.通过分析可知每个场方程都具有各自的“场方程矩阵”.能量方程能够对所有场方程矩阵进行综合和贯通，给出了能量方程以“弹性矩阵”为核心的普适性表达形式.最后，运用矩阵的正定二次型理论阐述了“能量矩阵与弹性矩阵”之间一致的对称性和正定性.能量矩阵蕴含的动态力的平衡关系、速度的时间_空间分布和能量的传播及变化的物理意义，能够从能量矩阵的正定二次型特性表述出来.本文研究分析问题的方法完全适用于复杂介质模型，相关的认识和结论可以拓展到均匀黏弹性各向同性介质、均匀弹性各向异性介质、均匀黏弹性各向异性介质以及比奥饱和流体介质.

The significance of seismic wave field equations matrices and the energy positive definite quadratic form

"Field equations"-wave equations,velocity equations and energy equations-can represent the overall characteristic of the half space seismic wave field.This paper firstly pointed out that there is a field equation matrix for each of the equations.Secondly we analyzed and revealed that energy equations can be integrated into all field equation matrixes,and presented the generally adaptive expression of the energy equations based on the elastic matrix.At last the consistently symmetrical and positive definite properties between the energy matrices and the elastic matrices are clarified by using the theory of positive definite quadratic form of matrices.The physical meaning of the dynamical equilibrium relationships,velocity space-time distribution and energy propagation involved in the energy matrices can be described by the positive definite quadratic form of the energy matrices.The analytical methods and conclusions used and presented in this paper can be applied to such complex models as porous,viscoelastic and anisotropic medium.