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利用傅里叶级数计算任意函数形式的弹性纵波垂直通过非线性结构面的透射波和反射波的速度波形。采用双曲模型描述结构面的变形特性,基于位移不连续模型,结合波振面处动量守恒定律,推导了弹性纵波垂直通过非线性结构面的基本方程。假设应力波在含结构面岩体中传播时,结构面的存在不改变应力波波形函数的最小正周期,运用傅里叶级数理论和周期延拓方法,得到了任意函数形式的弹性纵波垂直入射时透射波和反射波速度波形的傅里叶级数解,并验证了傅里叶级数解是合理的。利用傅里叶级数解,分析了单一频率的正弦谐波入射至结构面时,透射波中各阶谐波的振幅和相位与谐波阶数的关系。研究结果表明,各阶谐波的振幅与阶数呈负指数关系衰减,前7阶谐波振幅的衰减指数为谐波阶数的二次函数,当谐波阶数大于7时,衰减指数为谐波阶数的一次函数;各阶谐波的相位与谐波阶数呈线性关系。
Using Fourier Series to Calculate the Transverse and Reflected Wave Velocity Waveforms of Elastic Longitudinal Waves Traversing Vertical Nonlinear Structured Surfaces. The hyperbolic model is used to describe the deformation characteristics of the structural plane. Based on the discontinuous displacement model and the law of conservation of momentum at the wave vibration plane, the basic equations of the elastic longitudinal wave passing through the nonlinear structural plane are deduced. Assuming that the existence of structural plane does not change the minimum positive periodicity of the stress waveform when the stress wave propagates in the rock with structured surface, the elastic longitudinal wave of arbitrary function is obtained by using the Fourier series theory and the periodic extension method It is reasonable to verify the Fourier series solution of the Fourier series solution of the transmitted and reflected wave velocity waveforms at the time of incident. The Fourier series solution is used to analyze the relationship between the amplitude and the phase of each harmonic in the transmitted wave and the order of the harmonic when the sinusoidal harmonic of a single frequency is incident on the structural plane. The results show that the amplitude and order of each harmonic decrease in a negative exponential relationship, and the attenuation index of the first seven harmonic amplitudes is a quadratic function of the harmonic order. When the harmonic order is greater than 7, the attenuation index is The harmonic order of the first order function; the order of the harmonics of the harmonic order and a linear relationship.