Nonlinear acoustic wave equations with fractional loss operators Fabrice Prieura) and Sverre Holm Department of Informatics, University of Oslo, P.O. Box 1080, NO-0316 Oslo, Norway (Received 22 September 2010; revised 25 January 2011; accepted 1 July 2011) 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. VC 2011 Acoustical Society of America. [DOI: 10.1121/1.3614550] PACS number(s): 43.25.Dc, 43.20.Bi, 43.35.Cg, 43.35.Mr [ROC] Pages: 1125–1132 I. INTRODUCTION dispersion. A wave equation based on fractional constitutive equations gives an alternative to modeling absorption and Fractional derivatives, whether the formal name is used dispersion in complex media like biological tissues. or not, have been used for modeling heat transfer or diffu- In this article, we aim at finding the source of the frac- sion,1,2 seismic data,3 and sound wave propagation,4–6 only to tional derivative in the nonlinear wave equation. We derive a name a few. They allow the description of the physics of com- nonlinear wave equation using constitutive equations as a plex media in solid and fluid mechanics. When modeling starting point. The purpose of the article is to relate nonin- sound propagation, the use of fractional derivatives leads to teger power absorption laws to more fundamental physical models that better describe observations of attenuation and phenomena, rather than just the measured absorption charac- dispersion.7 The wave equation for viscous losses involving teristics. It also establishes a connection between fractional integer order derivatives only, leads to an attenuation which is constitutive equations coming from different fields of physics proportional to the square of the frequency. This does not describing mechanical stress or heat transfer. The constitutive always reflect reality. In, e.g., biological tissues8 and marine equations come from the fractional Kelvin–Voigt model from sediments,9 the frequency dependency of attenuation and dis- solid mechanics,27,28 and a fractional extension of the Gur- persion is more complicated. Different forms of the wave tin–Pipkin model from heat conduction,29,30 while the other equationhavebeenproposedtoreflectthiscomplexity.4,7,10–12 building equations come from fluid mechanics.31,32 Nonlinear effects in sound wave propagation, may also We start by briefly recalling the definition and properties be taken into account during numerical simulation. This is 13,14 15–17 of the fractional derivative. Then,wederiveamodifiedver- the case for the BERGEN code, the KZKTEXAS code, sion of Euler’s equation, and of the entropy equation, introduc- and the angular spectrum method of Christopher and ing fractional derivatives. We explain what these Parker.18,19 In the case of the angular spectrum approach, the modifications are based on, using solid mechanics and heat attenuation is modeled as proportional to xy, with x the diffusion theory. Combining these two equations, we get a angular frequency and y non-integer, allowing one to simu- wave equation by following the steps and approximations late attenuation in media like biological tissue. Time domain done in fluid mechanics theory. Thereafter, we show that the simulators can use multiple relaxation processes to approxi- obtained wave equation is a generalization of the Westervelt mate such attenuation both in the linear case20 and in the equation, and that the dispersion equation can describe attenu- nonlinear case.21 Typically, this requires two or more relaxa- ation and dispersion for propagation in complex media such as tion processes to model a power law over a restricted fre- biological tissues. Finally, generalized forms of the Khokhlov– quency range. Each process requires two parameters to be Zabolotskaya–Kuznetsov (KZK) and Burgers’ equations using found from a curve fit. These parameters describe the fractional derivatives are obtained. physics in the case of propagation in sea water or air. In more complex media, the link to the physics is not so direct. II. FRACTIONAL DERIVATIVE Several simulators take a modified nonlinear wave equa- tion as a starting point by replacing the traditional loss oper- The fractional derivative is an extension to integer order ator by fractional derivatives,7,22,23 or a convolution in derivatives, and is best understood by looking at its Fourier time.24–26 Their justification for modifying the standard transform in the frequency domain. For any positive integer wave equations is the ability of fractional derivatives to lead n, the temporal Fourier transform of the nth order derivative to a dispersion equation that better describes attenuation and of a function f (t) satisfies the relation n d f n a)Author to whom correspondence should be addressed. Electronic mail: F n ; x ¼ðjxÞ Fff g: (1) fabrice@ifi.uio.no dt J. Acoust. Soc. Am. 130 (3), September 2011 0001-4966/2011/130(3)/1125/8/$30.00 VC 2011 Acoustical Society of America 1125 Author's complimentary copy The fractional derivative of order c, for c real, can be seen as Euler’s equation, and the entropy equation, are the equations the operator whose Fourier transform satisfies Eq. (1), where that we will modify by introducing fractional derivatives. n is replaced by c. In the time domain, this corresponds to a convolution A. Euler’s equation ð dcf 1 t 1 dn In this section, we describe how the expression of the c ¼ r n f ðsÞds; (2) stress tensor can be described by the fractional Kelvin–Voigt dt Cð1 À rÞ 0 ðt À sÞ dt model, and how this leads to a form of Euler’s equation with where 0 n À 1 < c < n; r ¼ c À n þ 1; and C(1 À r) is the fractional derivatives. Following the expression of Euler’s 31 c function. Equation (2) is the definition of the fractional de- equation in Eq. (15.5) of Landau and Lifshitz, we have rivative given by Caputo.33,34 Fractional derivatives intro- @t @t @r @p @r0 duce a memory effect in the physical process they q i þ t i ¼ ik ¼ þ ik ; (7) 28,35 k describe. The nth order derivative is convolved with a @t @xk @xk @xi @xk memory function where q is the density, ti the components of the particle 1 1 speed vector, t and xi the temporal and spatial coordinates, : (3) 0 Cð1 À rÞ tr and p the total pressure. rik and rik represent the stress ten- sor, and viscous stress tensor, respectively. Using Eqs. (15.2) In the case where r ! 1 (no memory), the memory function and (15.3) of Ref. 31 tends toward a Dirac impulse function, and the order of the r ¼pd þ r0 fractional derivative tends toward the integer n. In the case ik ik ik where r ! 0 (infinite memory), Eq. (2) tends toward an inte- @ti @tk 2 @tl @tl gration of the nth order derivative resulting in the (n 1)th ¼pdik þ g þ À dik þ fdik ; (8) À @xk @xi 3 @xl @xl order derivative. Subsequently, the fractional integral of order a can also where g and f are the shear and bulk viscosity coefficients, 34 be defined as respectively, and are independent of velocity. This is the ð same relation established by Markham et al.27 in their Eq. 1 t Ia½f ðtÞ ¼ ðt À sÞðaÀ1Þf ðsÞds for 0 < a: (4) (13.3). In their article, they refer to the physical model as the CðaÞ 0 Stokes’s model. Further on, approximating the static total pressure by the inviscid total pressure, they get the relation Its Fourier transform satisfies the relation qe a Àa p K ; (9) FfI ½f ðtÞ; xg¼ðjxÞ Fff g: (5) q0 Hence fractional integrals and derivatives allow to model where K is the Young’s modulus, qe the excess density, and any power law in the frequency domain. Fractional integrals q0 the equilibrium density. And, finally, they get Eq. (14.2): and derivatives can be combined, giving the property: qe @ti @tk 2 @tl cÀa r K d g d dc d ik ¼ ik þ þ À ik a dtcÀa if 0 < a < c; q0 @xk @xi 3 @xl c ½I ¼ aÀc (6) dt I if 0 < c < a: @tl þ fdik : (10) @xl Fractional derivatives have been introduced in solid mechan- ics to more appropriately describe the stress-strain rela- Using a linear form of the equation of continuity (a full non- 28 1 tions, or heat transfers in viscoelastic media. This will be linear form is presented in Sec. IV) used here as a starting point to modify the constitutive equations. @qe @ti ¼q0 ; (11) @t @xi we get the constitutive equation III. FROM FRACTIONAL CONSTITUTIVE EQUATIONS TO FRACTIONAL EULER’S AND ENTROPY @ui @ti @tk 2 @tl EQUATIONS rik ¼Kdik þ g þ À dik @xk @xk @xi 3 @xl The basic equations that the nonlinear wave equation @tl þ fdik ; (12) derived in this paper is built upon are: the equation of conti- @xl nuity, expressing the conservation of mass; the equation of state, expressing the thermodynamic state of the fluid; where ui are the components of the displacement vector field Euler’s equation, that translates the conservation of momen- and ti ¼ @ui=@t.