Studies on Magneto-Hydrodynamic Waves and other Anisotropic wave motions
Studies on Magneto-Hydrodynamic Waves and other Anisotropic wave motions
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DOI：
10.1098/rsta.1960.0010
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发表时间：
1960-03
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影响因子：
--
通讯作者：
M. Lighthill
中科院分区：
文献类型：
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作者：
M. Lighthill
本文有两个独立但紧密交织的论点;一个主要是数学，一个主要是物理。数学链开始于一种方法的渐近评估傅立叶积分在许多方面，为大价值的论点。这是用来研究偏微分方程的四个变量，x，y，z和t，这是线性的常数系数，但它可以是任何顺序，并表示波动是各向异性或色散或两者兼而有之。它给出了这些方程的解的渐近行为（在大的距离），表示由有限或无穷小空间范围的源产生的波。本文特别集中于固定频率的源和满足辐射条件的解;但附录专门讨论在初始静止介质中由有限持续时间的源产生的波，以及不稳定的系统。数学结果给出了部分物理解释参数确定的能量传播速度的平面波穿越各向异性介质。这些表明，在其他事实中没有普遍认识到，即使对于非色散（例如弹性）波，能量传播速度一般不垂直于波前，尽管其垂直于波前的分量是相速度。第二个主要是物理方面的论证，从磁流体动力学波在不可压缩、无粘性和理想导电介质中的重要而惊人的特性开始，即只在一个方向上传播--给定的扰动只沿着穿过它的磁力线传播，因此不会随着距离而衰减。在天体物理学中，有一些重要的例子，密度是如此之低，以致于碰撞效应（例如，电介电性）引起的衰减在相关的长度尺度上应该可以忽略不计。因此，我们要问，在简单理论中被忽略的非碰撞性质的影响，特别是可压缩性和霍尔电流，会在多大程度上改变波的单向、无衰减传播。这些效应以前已经包括在磁流体动力学波动理论中，但是没有得到来自局部源的波的方向分布。这个问题解释了需要的数学理论刚刚描述，并给出了一个全面的说明其应用。
There are two separate but closely interwoven strands of argument in this paper; one mainly mathematical, and one mainly physical. The mathematical strand begins with a method of asymptotically evaluating Fourier integrals in many dimensions, for large values of their arguments. This is used to investigate partial differential equations in four variables, x, y, z and t, which are linear with constant coefficients, but which may be of any order and represent wave motions that are anisotropic or dispersive or both. It gives the asymptotic behaviour (at large distances) of solutions of these equations, representing waves generated by a source of finite or infinitesimal spatial extent. The paper concentrates particularly on sources of fixed frequency, and solutions satisfying the radiation condition; but an Appendix is devoted to waves generated by a source of finite duration in an initially quiescent medium, and to unstable systems. The mathematical results are given a partial physical interpretation by arguments determining the velocity of energy propagation in a plane wave traversing an anisotropic medium. These show, among other facts not generally realized, that even for non-dispersive (e.g. elastic) waves, the energy propagation velocity is not in general normal to the wave fronts, although its component normal to them is the phase velocity. The second, mainly physical, strand of argument starts from the important and striking property of magneto-hydrodynamic waves in an incompressible, inviscid and perfectly conducting medium, of propagation in one direction only—a given disturbance propagates only along the magnetic lines of force which pass through it, and therefore suffers no attenuation with distance. There are cases of astrophysical importance where densities are so low that attenuation due to collisional effects—for example, electrical resistivity—should be negligible over relevant length scales. We therefore ask how far the effects of a non-collisional nature which are neglected in the simple theory, particularly compressibility and Hall current, would alter the unidirectional, attenuation-less propagation of the waves. These effects have been included previously in magneto-hydrodynamic wave theory, but the directional distribution of waves from a local source was not obtained. This problem explains the need for the mathematical theory just described, and gives a comprehensive illustration of its application.