On an attenuation obeying a frequency power law

Abstract

An attenuation obeying a frequency power law scales as |ω|^β, where ω is angular frequency and β is a real constant. According to the strain-hardening shear-wave equation in the grain-shearing theory, the attenuation of the shear wave supported by a marine sediment follows a frequency power law. The associated dispersion formula, which is a causal transform, predicts that the phase speed is also a power law and that the product of the phase speed and the attenuation divided by ω is a constant, independent of frequency. From the dispersion formula, along with the conditions that the phase speed and attenuation must both be positive and the propagation factor must exhibit conjugate symmetry, it follows that the exponent β can take only certain values that fall in well-defined intervals extending indefinitely with no upper or lower limit. However, to satisfy the requirement that Green’s function be causal, β is further constrained to lie in the interval [0.5 1), under which condition the Green’s function is maximally flat at the time the source is activated. With other values of β the Green’s function is non-causal, exhibiting non-zero values at times preceding the onset of the source. [Research supported by ONR.]An attenuation obeying a frequency power law scales as |ω|^β, where ω is angular frequency and β is a real constant. According to the strain-hardening shear-wave equation in the grain-shearing theory, the attenuation of the shear wave supported by a marine sediment follows a frequency power law. The associated dispersion formula, which is a causal transform, predicts that the phase speed is also a power law and that the product of the phase speed and the attenuation divided by ω is a constant, independent of frequency. From the dispersion formula, along with the conditions that the phase speed and attenuation must both be positive and the propagation factor must exhibit conjugate symmetry, it follows that the exponent β can take only certain values that fall in well-defined intervals extending indefinitely with no upper or lower limit. However, to satisfy the requirement that Green’s function be causal, β is further constrained to lie in the interval [0.5 1), under which condition the Green’s function is ...