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给出了有效表示偶极子声源在井孔中激发的分波的数值方法.文中将支点看作对数,并分析了井孔特征函数即井孔位移势函数的多值性及其黎曼面结构.在理论分析的基础上结合实例,分析了快地层情况下对井孔声场有贡献且位于不同黎曼叶上的极点的分布规律.结果表明,(0,0)黎曼叶上的极点都是实极点,并形成具有频散特性的多个模式分支,其中最低阶的模式也称为弯曲波模式,数学分析表明弯曲波模式无截止频率,这与以往的观点有所不同.其他黎曼叶上的极点都是复极点,并在复波数平面上形成很多分支.就相关的极点和割线对井孔声场的贡献进行分析,数值计算结果表明无论是快地层还是慢地层情况,割线积分的结果都不能准确地表示纵横波等分波成分,只有结合相关极点的贡献才能较好地表示各分波,其中泊松比较大的慢地层情况下在(0,-1)黎曼叶上且靠近纵波垂直割线的复波数平面上存在虚部较小的极点,这些极点对声场中纵波分波的贡献较大.
A numerical method is presented to effectively represent the dipole excitation in the borehole. The fulcrum is taken as the logarithm, and the multi-value of the wellbore displacement function and its Riemann Surface structure.On the basis of theoretical analysis and with examples, the distribution of poles that contribute to the borehole sound field under fast formation conditions and located on different Riemannian leaves is analyzed.The results show that (0,0) The poles are all real poles and form multiple modes branches with dispersion characteristics, of which the lowest order mode is also called the bending wave mode and the mathematical analysis shows that the bending wave mode has no cutoff frequency, which is different from the previous viewpoints. The poles on the Riemannian leaf are all complex poles and form many branches on the complex wave number plane.The contribution of the related poles and secant to the borehole sound field are analyzed.The numerical results show that both the fast and the slow formation, The results of the secant integrals can not accurately represent the components of the sub-wave such as the S-wave and the S-wave. Only the contributions of the relevant poles can represent each sub-wave well, and in the case of the slow formation with large Poisson’s ratio, (0, -1) Man Ye and close to the longitudinal wave The vertical secant complex wave number plane exists on the smaller imaginary pole of the pole, these poles on the sound field of the longitudinal wave sub-wave contribution.