Nonlinear acoustic wave equations with fractional loss operators

Area of Science:

• Physics
• Acoustics
• Applied Mathematics

Background:

• Classical wave equations struggle to model complex media, particularly sound propagation in biological tissues.
• Existing models lack sufficient accuracy in representing attenuation and dispersion phenomena.
• Traditional constitutive equations in mechanics and heat conduction require advanced mathematical tools for complex media.

Purpose of the Study:

• To introduce fractional derivatives into classical wave equations for improved modeling of wave propagation.
• To develop a nonlinear wave equation that accurately captures attenuation and dispersion in complex media.
• To generalize existing wave equations, such as the Westervelt equation, using fractional calculus.

Main Methods:

• Modification of traditional constitutive equations using fractional derivatives.
• Derivation of a nonlinear wave equation incorporating fractional calculus.
• Application of fractional calculus to model attenuation and dispersion laws.

Main Results:

• Fractional derivatives effectively model wave propagation in complex media.
• The derived nonlinear wave equation accurately describes observed attenuation and dispersion.
• The study presents a fractional generalization of the Westervelt, Khokhlov-Zabolotskaya-Kuznetsov, and Burgers' equations.

Conclusions:

• Fractional derivatives offer a powerful framework for modeling complex wave phenomena.
• The developed fractional wave equation provides a more accurate description of sound propagation in biological tissues.
• This approach advances the understanding and modeling of nonlinear acoustics and wave physics.