Gravity waves in a stratified fluid
Gravity waves in a stratified fluid
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分层流体中的重力波
DOI：
10.1017/s002211206000075x
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发表时间：
1960
影响因子：
3.7
通讯作者：
C. Yih
中科院分区：
文献类型：
--
作者：
C. Yih
通过将控制微分系统简化为Sturm-Liouville系统，统一处理了分层流体中的波动问题，无论有无密度间断。借助于Sturm比较定理，发现（无需详细计算），对于任何分层，相速度随波数减小而增大，对于相同的波数，相速度随各处密度梯度增大而增大，随各处密度增大而减小。斯特姆振荡定理提供了一个给定的分层，一个给定的波数，和一个给定数量的本征函数的零点（或流体中的固定表面的给定数量）的相速度的上限和下限。给出这些界限的不等式被用来解释众所周知的倾向，表面的密度不连续性表现为刚性边界时，在每一层的分层是轻微的。在这种情况下，界面的刚性边界行为使人们能够通过将各个层（界面被视为刚性）的谱叠加在界面（或自由表面）波的谱上来获得近似的本征值谱，通过忽略每层中轻微的连续分层来获得。指出即使在密度不连续的情况下，Ritz方法也可用于计算特征值，并举例说明了Ritz方法的精度。当深度为无限大时，谱的性质也得到了澄清。在理论的发展过程中，明确地确定并给出了压缩性和三维性的影响，不稳定分层的增长率与稳定分层中波的相速度有关，并证明了能量均分。本文讨论了造波机引起的运动，以揭示控制偏微分方程的类型与扰动的性质（局部性或非局部性）之间的联系。本文还讨论了表面张力的影响和垂直振荡下分层流体的稳定性。
A unified treatment of wave motion in a stratified fluid, with or without density discontinuities, is achieved by reducing the governing differential system to a Sturm-Liouville system. With the aid of Sturm's comparison theorem, it is found (without detailed calculations) that, for any stratification, the phase velocity increases as the wave-number decreases and that, for the same wave-number, the phase velocity increases as the density gradient is increased everywhere and decreases as the density is increased everywhere by a constant amount. Sturm's oscillation theorem provides upper and lower bounds for the phase velocity for a given stratification, a given wave-number, and a given number of zeros of the eigenfunction (or a given number of stationary surfaces in the fluid). The inequalities giving these bounds are used to explain the well-known tendency for surfaces of density discontinuities to behave as rigid boundaries when the stratification in each layer is slight. The rigid-boundary behaviour of interfaces in such cases enables one to obtain the approximate eigenvalue spectrum by superimposing the spectra of the individual layers (with the interfaces treated as rigid) on the spectrum of the interfacial (or free surface) waves, obtained by ignoring the slight continuous stratification in each layer. It is pointed out that the Ritz method can be used for calculating the eigenvalues even when the density is discontinuous, and examples are given to show the accuracy of the Ritz method. The nature of the spectrum when the depth is infinite is also clarified. In the course of the development of the theory, the effects of compressibility and of three-dimensionality are determined and given explicitly, the rate of growth of unstable stratifications is related to the phase velocity of waves in stable ones, and equipartition of energy is proved. Motion due to a wave-maker is discussed in order to bring out the connexion between the type of the governing partial differential equation and the nature (local or not local) of the disturbances. The effect of surface tension and the stability of a stratified fluid under vertical oscillation are also discussed.