Propagation of transient leaking modes in a stratified elastic waveguide
Propagation of transient leaking modes in a stratified elastic waveguide
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分层弹性波导中瞬态泄漏模式的传播
DOI：
10.1029/rg002i001p00123
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发表时间：
1964
期刊：
影响因子：
--
通讯作者：
F. Gilbert
中科院分区：
文献类型：
--
作者：
F. Gilbert
利用Haskell首次应用于地震学问题的技术，当波导由具有平面平行边界的均匀层组成时，获得任何给定弹性波导的久期函数是一件简单的事情。长期函数的根通常表现为色散曲线。本文以无量纲波数K为自变量，以无量纲频率F为因变量。复F对K的曲线图产生诊断（F，K）α色散曲线。所采用的程序是将久期函数在K = 0附近展开，然后跟踪每个根F（K）随K值增加的行为。在所处理的每一个具体问题中，每个根的初始位置F（0）都允许有一个简单的物理描述。例如，在一个简单的大陆波导（固体层/固体半空间）中，有三组根：兰姆根，由和模式组成;剪切“风琴管”根;和压缩“风琴管”根。这两种管风琴的管根有无限多。低频PL波起源于其中一种根，而法向剪切模则起源于所有三种根的过渡。在一个简单的海洋波导（流体层/固体半空间）的剪切器官管根是不存在的。低频PL波再次从其中一个根部产生，并且简正模从Lamb根部和压缩器官管根部的过渡产生。在一个简单的声波导（流体层/流体半空间），只有压缩器官管根存在，和正常模式从这些根的过渡。PL波不存在。当海洋波导的半空间接近泊松比为0.5时，它的消失被清楚地跟踪。
处理由一个以上的层组成的波导只提供了额外的困难，找到器官管根的初始位置。当找到这些位置时，分析以类似于简单波导的方式进行。器官管根的初始位置是复杂的，这种情况可以物理地解释为辐射到波导的半空间中。正是这种辐射的存在导致人们谈论泄漏模式。如果半空间足够软，根也有复杂的初始位置。除了复杂之外，泄漏模式的色散曲线有时具有负群速度区域。如果将群速度等同于沿着波导的能量传输速度，则必须得出结论：当群速度为负时，存在向内的能量通量。但当群速度明显不是能量传输的速度时，有时就无法获得负群速度的简单物理图像。群速度与F的关系图在某些情况下显示带状结构，但通常相当复杂。这种情况可以澄清一些拒绝那些模式与弱激发功能，大的衰减参数，或两者兼而有之。（F，K）图更清晰。现在，地震阵列处理程序正在开发中，人们希望实验（F，K）图将成为地震分析中的标准工具，从而更清楚地了解地震频散和传播特性。
By using a technique first applied to seismological problems by Haskell, it is a simple matter to obtain the secular function for any given elastic waveguide, when the waveguide is composed of homogeneous layers with plane parallel boundaries. Roots of the secular function are usually exhibited as dispersion curves. In this paper the independent variable is dimensionless wave number, K, and the dependent variable is dimensionless frequency, F. A plot of complex F versus K yields the diagnostic (F, K) diagram—a dispersion curve. The procedure employed is to expand the secular function about K = 0 and then to trace the behavior of each root, F(K), for increasing values of K. In each of the specific problems treated the initial position of each root, F(0), admits of a simple physical description. For example, in a simple continental waveguide (solid layer/solid half-space), there are three sets of roots: the Lamb roots, consisting of and modes; the shear ‘Organ pipe’ roots; and the compressional ‘organ pipe’ roots. There is an infinite number of the two kinds of organ pipe roots. The low-frequency PL wave arises from one of the roots, and the normal shear modes arise from transitions of all three kinds of roots. In a simple oceanic waveguide (fluid layer/solid half-space) the shear organ pipe roots are absent. The low-frequency PL wave again arises from one of the roots, and the normal modes arise from transitions of the Lamb roots and the compressional organ pipe roots. In a simple acoustic waveguide (fluid layer/fluid half-space) only the compressional organ pipe roots are present, and the normal modes arise from transitions of these roots. The PL wave is absent. Its disappearance is clearly traced as the half-space of the oceanic waveguide approaches a Poisson ratio of 0.5.
The treatment of waveguides composed of more than one layer offers only the additional difficulty of finding the initial positions of the organ pipe roots. When these positions have been found, the analysis proceeds in a manner similar to that for simple waveguides. The initial positions of the organ pipe roots are complex, a situation that may be interpreted physically as radiation into the half-space of the waveguide. It is the presence of such radiation that leads one to speak of leaking modes. The roots also have complex initial positions, if the half-space is soft enough. In addition to being complex, the dispersion curves for the leaking modes sometimes have regions of negative group velocity. If one equates group velocity to velocity of energy transport along the waveguide, then one must conclude that there is an inward energy flux when the group velocity is negative. But when the group velocity is demonstrably not the velocity of energy transport, a simple physical picture of negative group velocity is sometimes unavailable. Plots of group velocity versus F show a banded structure in some cases but are generally quite complex. The situation can be clarified somewhat by rejecting those modes with weak excitation functions, large decay parameters, or both. Still, the (F, K) diagram is clearer. Now that seismic array processing procedures are being developed, one hopes that experimental (F, K) diagrams will become a standard tool in seismic analysis, leading to a clearer picture of seismic dispersion and propagation characteristics.