黏弹介质中勒夫波频散问题的统一解及其动态特征分析

目前完全弹性介质中面波频散特征的研究已较为完善,多道面波分析技术(MASW)在近地表勘探领域也取得了较好的效果,但黏弹介质中面波的频散特征研究依然较少.本文基于解析函数零点求解技术,给出了完全弹性、常Q黏弹和Kelvin-Voigt黏弹层状介质中勒夫波频散特征方程的统一求解方法.对于每个待计算频率,首先根据传递矩阵理论得到勒夫波复频散函数及其偏导的解析递推式,然后在复相速度平面上利用矩形围道积分和牛顿恒等式将勒夫波频散特征复数方程的求根问题转化为等价的连带多项式求解问题,最后通过求解该连带多项式的零点得到多模式勒夫波频散曲线与衰减系数曲线.总结了地层速度随深度递增和夹低速层条件下勒夫波频散特征根在复相速度平面上的运动规律和差异.证明了频散曲线交叉现象在复相速度平面上表现为:随频率增加,某个模式特征根的移动轨迹跨越了另一个模式特征根所在的圆,并给出了这个圆的解析表达式.研究还表明,常Q黏弹地层中的基阶模式勒夫波衰减程度随频率近似线性增加,而Kelvin-Voigt黏弹地层中的基阶模式勒夫波衰减程度随频率近似指数增加,且所有模式总体衰减程度强于常Q黏弹地层中的情况.

Unified solution of the Love-wave dispersion problem and its dynamic features in a viscoelastic medium

At present, the dispersion characteristics of surface waves in an elastic medium has been well studied, and the multi-channel analysis of surface waves (MASW) has achieved good results in the near surface exploration field. However, the research in a viscoelastic medium is still relatively few. In this paper, we present a unified solution to the Love wave dispersion equation in the elastic medium, viscoelastic medium with constant Q and Kelvin-Voigt viscoelastic medium based on zeros solving technique of analytic function. For each frequency to be calculated, the analytic recurrence formula of the Love wave complex dispersion function and its partial derivation can be obtained in terms of the transfer matrix theory. Then applying rectangle contour integral and Newton identity in the plane of complex phase velocity, the root solving of the Love wave complex dispersion equation is transformed into the equivalent problem of solving associated polynomials. Finally, the Love wave dispersion curves and attenuation coefficient curves of multi-modes can be obtained by solving the zeros of the polynomials. The movement laws and differences of Love-wave dispersion characteristic roots in two cases in the plane of complex phase velocity are summarized:layer velocity increases with depth and with low velocity layer embedded. It proves that when dispersion curves cross, the trajectory of a mode characteristic root will span the circle of another mode root in the plane of complex phase velocity. The analytic expression of this circle is given, too. The study also shows that the attenuation degree of fundamental mode Love waves in the viscoelastic layer with constant Q increases approximately linearly with frequency; while it increases approximately exponentially with frequency in the Kelvin-Voigt viscoelastic layer. And the overall attenuation of all modes is stronger than that in the viscoelastic layer with constant Q.