Department of Physics and Astronomy and National Center for Physical Acoustics, University of Mississippi, University, Mississippi 38677, USA.
J Acoust Soc Am. 2010 Jan;127(1):166-73. doi: 10.1121/1.3268512.
The Kramers-Kronig (KK) relations are a large class of integral transformations that exploit the broad principle of simple causality in order to link the physical properties of matter and materials. In applications to the complex-valued wavenumber for acoustic propagation, the method of subtractions is used to form convergent integral relations between the phase velocity and the attenuation coefficient. When the method of subtractions is applied in the usual manner, the integrands in the relations become unnecessarily complicated. In this work, an expanded form of the subtracted relations is presented, which is essentially a truncated Taylor series expansion of the Hilbert transforms. The implementation of the relations only requires the explicit evaluation of two simply expressed integrals involving the Hilbert transform kernel. These two integrals determine the values of the other terms in the subtracted relations, demonstrating the computational efficiency of the technique. The method is illustrated analytically through its application to power-law attenuation coefficients and its associated dispersion, which are observed in a wide variety of materials. This approach explicitly shows the central role of the Hilbert transform kernel in the KK relations, which can become obscured in other formulations.
克喇末-克龙尼克（KK）关系是一大类积分变换，它们利用简单因果关系的广泛原理来连接物质和材料的物理性质。在应用于声波传播的复波数时，采用减法方法在相速度和衰减系数之间形成收敛的积分关系。当以通常的方式应用减法方法时，关系中的积分变得不必要地复杂。在这项工作中，提出了一个扩展的相减关系形式，它本质上是希尔伯特变换的截断泰勒级数展开。关系的实现只需要明确评估两个涉及希尔伯特变换核的简单表示积分。这两个积分确定了相减关系中其他项的值，展示了该技术的计算效率。该方法通过应用于幂律衰减系数及其相关色散来进行分析，这些衰减系数在各种材料中都有观察到。这种方法明确地显示了希尔伯特变换核在 KK 关系中的核心作用，而在其他形式中可能会变得模糊。
