Elastic waves in fluid-saturated granular media depend on the rheology which includes elements representing the fluid and, if necessary, gas bubbles. We investigated the effect of different rheological schemes, including and excluding the bubbles, on the linear Frenkel–Biot waves of P1 type. For the wave with the bubbles the scheme consists of three segments representing the solid continuum, fluid continuum, and a bubble surrounded by the fluid. We derived the Nikolaevskiy-type equations describing the velocity of the solid matrix in the moving reference system. The equations are linearized to yield the decay rate lambda as a function of the wave number k. We compared the lambda(k)-dependence for the cases with and without the bubbles, using typical values of the input mechanical parameters. For the both cases, the lambda(k)-curve lies entirely below zero, which is in line with the notion of the elastic wave being an essentially passive system. We found that the increase of the radius of the bubbles leads to faster decay, while the increase in the number of the bubbles leads to slower decay of the elastic wave.

中文翻译：

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流变学在模拟颗粒状流体饱和介质中带有气泡的弹性波中的作用

流体饱和颗粒介质中的弹性波取决于流变学，其中包括代表流体的元素，如有必要，还包括气泡。我们研究了不同流变方案（包括和排除气泡）对 P1 型线性 Frenkel-Biot 波的影响。对于带有气泡的波，该方案由三个部分组成，分别代表固体连续体、流体连续体和被流体包围的气泡。我们导出了描述固体矩阵在移动参考系统中的速度的 Nikolaevskiy 型方程。方程被线性化以产生衰减率 lambda 作为波数 k 的函数。我们使用输入机械参数的典型值比较了有气泡和无气泡情况下的 lambda(k) 依赖性。对于这两种情况，lambda(k) 曲线完全低于零，这符合弹性波本质上是被动系统的概念。我们发现气泡半径的增加导致衰减更快，而气泡数量的增加导致弹性波的衰减更慢。