2011/10/01 by Sverre Holm, Sven Peter Näsholm · 1 citation
Engineering · Mathematics · #Acoustic Wave Phenomena Research #Constitutive equation #Finite element method #Fractional Differential Equations Solutions #Fractional calculus #Lossy compression #Mathematical analysis #Mathematics #Momentum (technical analysis) #Numerical methods in engineering #Physics #Quantum electrodynamics #Standard linear solid model #Thermodynamics #Time derivative #Wave equation
paper · doi:10.1121/1.3631626
openalex publication_date 2011/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25
This work presents a lossy partial differential acoustic wave equation including fractional derivative terms. It is derived from first principles of physics (mass and momentum conservation) and an equation of state given by the fractional Zener stress-strain constitutive relation. For a derivative order α in the fractional Zener relation, the resulting absorption α(k) obeys frequency power-laws as α(k) ∝ ω(1+α) in a low-frequency regime, α(k) ∝ ω(1-α/2) in an intermediate-frequency regime, and α(k) ∝ ω(1-α) in a high-frequency regime. The value α=1 corresponds to the case of a single relaxation process. The wave equation is causal for all frequencies. In addition the sound speed does not diverge as the frequency approaches infinity. This is an improvement over a previously published wave equation building on the fractional Kelvin-Voigt constitutive relation.