The Westervelt equation with viscous attenuation versus a causal propagation operator: A numerical comparison

Norton, Guy V.; Purrington, Robert D.

doi:10.1016/j.jsv.2009.05.031

Cited by 18 publications

(12 citation statements)

References 14 publications

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“…Traditionally, the Westervelt equation has been used to model acoustic propagation in inhomogeneous media such as tissue, treating the medium as a thermoviscous fluid. As mentioned previously, Norton and Purrington [8,11] have replaced the traditional (thermoviscous) loss mechanism in the Westervelt equation with a causal TDPF that takes into account the full dispersive nature of the medium, (i.e. both frequency dependent velocity and attenuation).…”

Section: Guy V Norton and Robert D Purringtonmentioning

confidence: 99%

“…In previous work [8,11] we have studied the propagation of sound in a dispersive medium by solving the Westervelt equation in the time domain, treating the dispersion by modeling the medium as a thermoviscous fluid, or, alternatively, by employing the time domain propagation factor (TDPF) [14,15,16,17]. It was found that employing the causal TDPF yields results that differ significantly from employing the Westervelt equation with the traditional loss mechanism, which is due to the fact that the latter fails to take into account the full dispersive characteristics of the medium.…”

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confidence: 97%

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The Westervelt equation with a causal propagation operator coupled to the bioheat equation.

Norton¹,

Purrington²

2016

EECT

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The Westervelt wave equation is frequently used to describe nonlinear propagation of finite amplitude sound. If one assumes that the medium can be treated as a thermoviscous fluid, a loss mechanism can be incorporated. In this as in previous work the authors replaced the typical loss mechanism incorporated in the Westervelt equation with a causal Time Domain Propagation Factor (TDPF) which incorporates the full dispersive effects (both frequency dependent phase velocity and attenuation) in the numerical solution while remaining in the time-domain. In the present work we investigate heat deposition due to finite amplitude propagation through a dispersive medium (e.g., human tissue). To this end, the Westervelt equation with and without the TDPF is coupled to the Pennes bioheat equation and the coupled equations are solved using the method of finite differences to determine the resulting heat deposition. We show that non-linear effects are large and that proper treatment of dispersion results in significant changes as compared to modeling the medium as a thermoviscous fluid.

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Section: Guy V Norton and Robert D Purringtonmentioning

confidence: 99%

mentioning

confidence: 97%

The Westervelt equation with a causal propagation operator coupled to the bioheat equation.

Norton¹,

Purrington²

2016

EECT

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“…The same assumption is implicitly done in the derivation of the fractional forms of the Westervelt and Burgers' equations. 22,26 This leads to the fractional Westervelt equation of the second form:…”

Section: Fractional Wave Equationmentioning

confidence: 99%

“…[24][25][26] Their justification for modifying the standard wave equations is the ability of fractional derivatives to lead to a dispersion equation that better describes attenuation and dispersion. A wave equation based on fractional constitutive equations gives an alternative to modeling absorption and dispersion in complex media like biological tissues.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear acoustic wave equations with fractional loss operators

Prieur

Holm

2011

The Journal of the Acoustical Society of America

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Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow a modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Traditional constitutive equations from solid mechanics and heat conduction are modified using fractional derivatives. They are used to derive a nonlinear wave equation which describes attenuation and dispersion laws that match observations. This wave equation is a generalization of the Westervelt equation, and also leads to a fractional version of the Khokhlov-Zabolotskaya-Kuznetsov and Burgers' equations.

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“…Hilbert transform or Fractional Calculus are some of the main mathematical tools which have been used to develop theoretical models respecting causality (Hertz & Al, 1991) (D.T, 1969) (He, 1999). In the case of viscous media for which the attenuation of the waves is proportional to the frequency squared, the methods based on Laplace transform allow the derivation of impulse responses respecting the causality (Thomas.L, 1995) (D.T, 1969) (Norton & Purrington, 2009). …”

Section: Introductionmentioning

confidence: 99%