The Viscous Grain Shearing (VGS) theory predicts the existence of a compressional wave and a shear wave in an unconsolidated marine sediment. Although it is known that, subject to certain constraints, the shear wave satisfies causality, the causal nature of the compressional wave is less well understood. In this paper, the VGS compressional-wave speed and attenuation are examined in three frequency regimes, where it is shown that they follow approximately frequency power laws. It is then proved that the VGS propagation factor, which is a combination of the phase speed and attenuation, is a causal transform: its inverse Fourier transform is zero for all times prior to the onset of the source. The derivation of this result, which is a necessary condition if the VGS compressional wave equation is to satisfy causality, includes the development of a technique for evaluating a class of previously unknown integrals. This integration procedure relies on a limiting argument combined with certain Fourier transforms, the latter taking the form of “improper” integrals, which, it is shown, can be expressed explicitly based on the properties of generalized functions. An expression for the impulse response of the VGS compressional wave is also developed and shown to satisfy causality, although the transition from zero to the peak level is abrupt, quite unlike the perfectly smooth behavior exhibited by the impulse response of the VGS shear wave, which is maximally flat everywhere in the medium at the instant the source is activated.

中文翻译：

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松散海相沉积物中压缩波的因果性质

粘性颗粒剪切 (VGS) 理论预测在松散的海洋沉积物中存在纵波和横波。尽管众所周知，在某些限制条件下，横波满足因果关系，但对纵波的因果性质却知之甚少。在本文中，VGS 压缩波速度和衰减在三个频率范围内进行了检查，结果表明它们大致遵循频率功率定律。然后证明，作为相位速度和衰减的组合的 VGS 传播因子是一个因果变换：它的傅里叶逆变换在源开始之前的所有时间都为零。这个结果的推导，如果 VGS 压缩波动方程要满足因果关系，这是一个必要条件，包括开发一种用于评估一类以前未知的积分的技术。该积分过程依赖于结合某些傅里叶变换的限制参数，后者采用“不适当”积分的形式，可以根据广义函数的性质明确表示。还开发了 VGS 压缩波的脉冲响应的表达式并显示满足因果关系，尽管从零电平到峰值电平的过渡是突然的，这与 VGS 剪切波的脉冲响应表现出的完美平滑行为完全不同，后者在源被激活的那一刻，在介质中的任何地方都最大程度地平坦。该积分过程依赖于结合某些傅里叶变换的限制参数，后者采用“不适当”积分的形式，可以根据广义函数的性质明确表示。还开发了 VGS 压缩波的脉冲响应的表达式并显示满足因果关系，尽管从零电平到峰值电平的过渡是突然的，这与 VGS 剪切波的脉冲响应表现出的完美平滑行为完全不同，后者在源被激活的那一刻，在介质中的任何地方都最大程度地平坦。该积分过程依赖于结合某些傅里叶变换的限制参数，后者采用“不适当”积分的形式，可以根据广义函数的性质明确表示。还开发了 VGS 压缩波的脉冲响应的表达式并显示满足因果关系，尽管从零电平到峰值电平的过渡是突然的，这与 VGS 剪切波的脉冲响应表现出的完美平滑行为完全不同，后者在源被激活的那一刻，在介质中的任何地方都最大程度地平坦。