Connecting the Viscous Grain-shearing Mechanism of Wave Propagation in Marine Sediments to Fractional Calculus

Pandey, Vikash; Holm, Sverre

doi:10.3997/2214-4609.201600862

Cited by 3 publications

(3 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From that it follows that the shear wave model builds on a fractional Newton constitutive equation, and the compressional wave solution corresponds to a fractional Kelvin-Voigt model [31]. Furthermore its variants such as the viscous grain shearing model [32] can also be derived from a combination of ordinary and fractional spring damper models [33], so all of these models satisfy complete monotonicity.…”

Section: Discussionmentioning

confidence: 99%

Restrictions on wave equations for passive media

Holm

2017

The Journal of the Acoustical Society of America

View full text Add to dashboard Cite

Most derivations of acoustic wave equations involve ensuring that causality is satisfied. Here, the consequences of also requiring that the medium should be passive are explored. This is a stricter criterion than causality for a linear system and implies that there are restrictions on the relaxation modulus and its first few derivatives. The viscous and relaxation models of acoustics satisfy passivity and have restrictions on not only a few, but all derivatives of the relaxation modulus. These models are described as a system of springs and dampers with positive parameters and belong to the important class of completely monotone systems. It is shown here that the attenuation as a function of frequency for such media has to increase slower than a linear function. Likewise, the phase velocity has to increase monotonically. This gives criteria on which one may judge whether a proposed wave equation is passive or not, as illustrated by comparing two different versions of the viscous wave equation.

show abstract

Section: Discussionmentioning

confidence: 99%

Restrictions on wave equations for passive media

Holm

2017

The Journal of the Acoustical Society of America

View full text Add to dashboard Cite

show abstract

“…Furthermore, in appropriate limiting conditions, fractional derivatives asymptotically converge to the integer-order derivatives. In recent years, a connection between the fractional derivatives and the physics of complex media has also been established [14,[16][17][18]. It is worth noting that the expressions for, Q C and I C , in Eqs.…”

mentioning

confidence: 99%

Charge-voltage relation for a universal capacitor

Pandey

2020

Preprint

View full text Add to dashboard Cite

Most capacitors of practical use deviate from the assumption of a constant capacitance. They exhibit memory and are often described by a time-varying capacitance. It is shown that a direct implementation of the classical relation, Q (t) = CV (t), that relates the charge, Q (t), with the constant capacitance, C, and the voltage, V (t), is not applicable when the capacitance is timevarying. The resulting equivalent circuit that emerges from the substitution of, C, by, C (t), is found to be inconsistent. Since, C (t), leads to a time-variant system, the current, Q, that is obtained from the product rule of the differentiation is not valid either. The search for a solution to this problem led to the expression for the charge, that is given by the convolution of the time-varying capacitance with the first-order derivative of the voltage, as, Q (t) = C (t) * V (t). Coincidentally, this equation also corresponds to the charge-voltage relation for a fractional-capacitor which is probably first reported in this Letter.

show abstract

“…This is also evident from the fact that despite having more than three hundred years of history, the order of the fractional derivative is still mostly obtained from curve-fitting the experimental data with the theoretically predicted curves. This comment is presented with the motivation that an utmost care should be given in the application of fractional derivatives in describing anomalous, complex, and memory-driven physical phenomena [2][3][4].…”

mentioning

confidence: 99%