TIME-DOMAIN WAVE-EQUATIONS FOR LOSSY MEDIA OBEYING A FREQUENCY POWER-LAW
TIME-DOMAIN WAVE-EQUATIONS FOR LOSSY MEDIA OBEYING A FREQUENCY POWER-LAW
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DOI：
10.1121/1.410434
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发表时间：
1994-07-01
影响因子：
2.4
通讯作者：
SZABO, TL
中科院分区：
文献类型：
--
作者：
SZABO, TL
对于由频率为alpha=alpha(0)\omega\(y)的缓慢变化幂律函数描述的衰减，经典的有耗时域波动方程只存在于y=0或y=2的限制情况下。对于经常出现的衰减远小于波数的实际情况，导出了具有所需衰减一般幂律相关性的有耗色散特性。为了得到相应的时域有耗波动方程，利用广义函数建立了函数与现有导数算子相似的时域损失算子。根据y是偶数、奇数还是非整数，找到了三种形式的有耗波动方程。因果关系的时域表达式在函数上类似于频域的Kramers-Kronig关系，用于推导因果波动方程。即使在y大于或等于1的情况下，也可以得到时域波动方程的最终因果版本，根据Paley-Wiener定理，这种情况是无法从Kramers-Kronig关系中获得的。推导了不同形式的波动方程，包括正常时间，延迟时间和抛物线（一维和三维）。这些方程与对应于y=0、0.5、1和2的文献中的方程相比是有利的。
For attenuation described by a slowly varying power law function of frequency, alpha=alpha(0)\omega\(y), classical lossy time domain wave equations exist only for the restricted cases where y=0 or y=2. For the frequently occurring practical situation in which attenuation is much smaller than the wave number, a lossy dispersion characteristic is derived that has the desired attenuation general power law dependence. In order to obtain the corresponding time domain lossy wave equation, time domain loss operators similar in function to existing derivative operators are developed through the use of generalized functions. Three forms of lossy wave equations are found, depending on whether y is an even or odd integer or a noninteger. A time domain expression of causality analogous in function to the Kramers-Kronig relations in the frequency domain is used to derive the causal wave equations. Final causal versions of the time domain wave equations are obtained even for the cases where y greater than or equal to 1, which, according to the Paley-Wiener theorem, are unobtainable from the Kramers-Kronig relations. Different forms of the wave equation are derived including normal time, retarded time, and parabolic (one and three dimensional). These equations compare favorably with those from the literature corresponding to y=0, 0.5, 1, and 2.