Modified Szabo's wave equation models for lossy media obeying frequency power law

TL;DR

This paper enhances Szabo's acoustic attenuation models for lossy media by incorporating Caputo fractional derivatives, addressing mathematical and implementation issues while maintaining causality and positive attenuation characteristics.

Contribution

It introduces Caputo fractional derivatives into Szabo's models, resolving hyper-singular integrals and clarifying initial condition implementation.

Findings

Models now better handle non-integer frequency exponents.

Hyper-singular integrals are effectively addressed.

Initial condition implementation is clarified.

Abstract

Szabo's models of acoustic attenuation (Szabo 1994a) comply well with the empirical frequency power law involving non-integer and odd integer exponent coefficients while guaranteering causality, but nevertheless encounter the troublesome issues of hyper-singular improper integral and obscurity in implementing initial conditions. The purpose of this paper is to ease or remove these drawbacks of the Szabo's models via the Caputo fractional derivative concept. The positive time fractional derivative is also first introduced to include the positivity of the attenuation possesses.