# On the solution of an acoustic wave equation with variable-order derivative loss operator

- Abdon Atangana
^{1}Email author

**2013**:167

https://doi.org/10.1186/1687-1847-2013-167

© Atangana; licensee Springer 2013

**Received: **17 February 2013

**Accepted: **24 May 2013

**Published: **12 June 2013

## Abstract

When modeling sound propagation, the use of fractional derivatives leads to models that better describe observations of attenuation and dispersion. The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. The acoustic wave equation with loss operator is generalized to the concept of variable-order derivatives in this work. The generalized equation is solved via the Crank-Nicholson scheme. The stability and the convergence of this case are examined in detail.

### Keywords

acoustic wave equation loss operator variable-order derivative Crank-Nicholson scheme stability convergence## 1 Introduction

Fractional derivatives are well suited to describe wave propagation in complex media. When introduced in classical wave equations, they allow the modeling of attenuation and dispersion that better describes sound propagation in biological tissues. Fractional derivatives have been used for modeling heat transfer or diffusion [1, 2], seismic data [3] and sound wave propagation, [4–6] only to name a few. When modeling sound propagation, the use of fractional derivatives leads to models that better describe observations of attenuation and dispersion [7]. The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. For instance, for biological tissues [8] and marine sediments [9], the frequency dependency of attenuation and dispersion is more complicated. Different forms of the wave equation have been proposed to reflect this complexity [4, 7, 10–12].

Several simulators take a modified nonlinear wave equation as a starting point by replacing the traditional loss operator by fractional derivatives [7, 13, 14] or a convolution in time [15, 16]. 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. However, in the case where the medium, through which these sounds are propagating, is variable or heterogeneous, neither the wave equation described by integer order nor that described by constant fractional order are suited for describing the phenomena. To solve the above problems, the variable-order (VO) fractional acoustic wave equation models are suggested for use in this work. This present work is therefore devoted to the discussion underpinning the description of the extension of an acoustic wave equation to the concept of the variational-order derivative and the solution of the generalized equation using the Crank-Nicholson scheme.

## 2 A possible modification of an acoustic wave equation

For the readers that are not acquainted with the concept of the variational-order derivative, we start this section by presenting the basic definition of this derivative.

### 2.1 Variable-order differential operator

The above derivative is called the Caputo variational-order differential operator, in addition, the derivative of the constant is zero.

### 2.2 Statement of the problem

where ${\alpha}_{0}$ and *y* are constants that characterize the medium. Such attenuation can be described by wave equations with particular loss operators.

*L*, which in the general case is a convolution, is

*y*. One such case is classical visco-elasticity, which is used as a first-order model, for instance, air and water. The loss operator is then [17]

*τ*is medium-specific relaxation time. In 1994 Szabo developed a wave equation for $y\in [0;2]$ which for $y=2$ was similar to an approximation to equation (2.3) where the viscoelastic loss term instead is a third-order time derivative [18]

The above proposed loss operator is equivalent to the loss operator of the viscoelastic equation for $y=2$. This operator is interesting as it turns out to be based on an underlying fractional Kelvin-Voigt model despite the impression Wismer [20] gives that it was found by inspection just like the previous operators. Although the wave equations given here may successfully be applied for wave propagation simulation, most of them are nevertheless derived through *ad hoc* procedures which are not directly linked to more basic physical principles.

The above equation is then called the ‘variable-order derivative acoustic wave equation with loss operator’. This new equation does not have obviously an exact analytical solution. In particular, the equation cannot be solved analytically. It is therefore important to examine its solution numerically. The aim of the next section is then devoted to the discussion underpinning the numerical solution of the modified equation using the Crank-Nicholson scheme.

## 3 Numerical solution of a variable-order derivative acoustic wave equation with loss operator

Numerical methods yield approximate solutions to the governing equation through the discretization of space and time. Within the discredited problem domain, the variable internal properties, boundaries and stresses of the system are approximated. Deterministic, distributed-parameter, numerical models can relax the rigid idealized conditions of analytical models or lumped-parameter models, and they can therefore be more realistic and flexible for simulating fields conditions. The finite difference schemes for constant-order time or space fractional diffusion equations have been widely studied [21, 22]. For constant-order time fractional diffusion equations, the implicit difference approximation scheme was proposed in [23]. The weighted average finite difference methods were introduced in [24]. The matrix approach for fractional diffusion equations was proposed in [25], and Hanert proposed a flexible numerical scheme for the discretization of the space-time fractional diffusion equation [26]. Lately, the numerical schemes for a VO space fractional advection-dispersion equation was considered by the author of [27]. An investigation of the explicit scheme for a VO nonlinear space fractional diffusion equation was done in [28]. Before performing the numerical methods, we assume that equation (2.3) has a unique and sufficiently smooth solution. To establish the numerical schemes for the above equation, we let ${x}_{l}=lh$, $0\le l\le M$, $Mh=L$, ${t}_{k}=k\sigma $, $0\le k\le N$, $N\sigma =T$, *h* is the step and *τ* is the time size, *M* and *N* are grid points.

### 3.1 Crank-Nicholson scheme [29]

## 4 Stability analysis of the Crank-Nicholson scheme

In this section, we analyze the stability conditions of the Crank-Nicholson scheme for the generalized acoustic wave equation.

*φ*is a real spatial wave number, now replacing the above equation (4.5) in (3.6) we obtain

To achieve this, we make use of the recurrence technique on the natural number *k*.

which completes the proof.

## 5 Convergence analysis of the Crank-Nicholson scheme

where ${V}_{1}$, ${V}_{2}$, ${V}_{3}$ and *K* are constants. Taking into account the Caputo-type fractional derivative, the detailed error analysis on the above schemes can refer to the work in [31] and further work by [32].

**Lemma 1**${\parallel {\mathrm{\Omega}}^{k+1}\parallel}_{\mathrm{\infty}}\le K({\sigma}^{1+{\rho}_{l}k+1}+2{h}^{2}{\tau}^{{\alpha}_{l}k}){({\mathrm{\Omega}}_{j}^{l,k+1})}^{-1}$

*is true for*($k=0,1,2,\dots ,N-1$),

*where*${\parallel {w}^{k}\parallel}_{\mathrm{\infty}}={max}_{1\le l\le M-1}({\mathrm{\Omega}}^{k})$,

*K*

*is a constant*.

*In addition*,

This can be achieved via the recurrence technique on the natural number *k*.

which completes the proof.

**Theorem 1**

*The Crank*-

*Nicholson scheme is convergent*,

*and there exists a positive constant*

*K*

*such that*

The interested can find the solvability of the Crank-Nicholson scheme in the work done by [29]. Therefore the details of the proof will not be presented in this paper.

## 6 Numerical simulation

*t*for a fixed value of $x=0.8$.

## 7 Conclusion

The wave equation for viscous losses involving integer-order derivatives only leads to an attenuation which is proportional to the square of the frequency. This does not always reflect reality. The acoustic wave equation with loss operator was generalized using some approaches of variational calculus. Since the modified equation is difficult to solve analytically, we make use of the numerical scheme to solve this new equation. The numerical used in solving this new equation is the Crank-Nicholson scheme. The convergence and the stability of this scheme in this case were presented in details.

## Declarations

## Authors’ Affiliations

## References

- Mainardi F: The fundamental solutions for the fractional diffusion-wave equation.
*Appl. Math. Lett.*1996, 9: 23-28. 10.1016/0893-9659(96)00089-4MathSciNetView ArticleGoogle Scholar - Schneider WR, Wyss W: Fractional diffusion and wave equations.
*J. Math. Phys.*1989, 30: 134-144. 10.1063/1.528578MathSciNetView ArticleGoogle Scholar - Koh C, Kelly J: Application of fractional derivatives to seismic analysis of base-isolated models.
*Earthquake Eng. Struct. Dyn.*1990, 19: 229-241. 10.1002/eqe.4290190207View ArticleGoogle Scholar - Szabo T: Time domain wave equations for lossy media obeying a frequency power law.
*J. Acoust. Soc. Am.*1994, 96: 491-500. 10.1121/1.410434View ArticleGoogle Scholar - Buckingham M: Theory of acoustic attenuation, dispersion, and pulse propagation in unconsolidated granular materials including marine sediments.
*J. Acoust. Soc. Am.*1997, 102: 2579-2596. 10.1121/1.420313View ArticleGoogle Scholar - Norton GV, Novarini JC: Including dispersion and attenuation directly in the time domain for wave propagation in isotropic media.
*J. Acoust. Soc. Am.*2003, 113: 3024-3030. 10.1121/1.1572143View ArticleGoogle Scholar - Chen W, Holm S: Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency.
*J. Acoust. Soc. Am.*2004, 115: 1424-1430. 10.1121/1.1646399View ArticleGoogle Scholar - Duck FA: Acoustic properties of tissue at ultrasonic frequencies. In
*Physical Properties of Tissues - A Comprehensive Reference Book*. Academic Press, San Diego; 1990:98-108. chap. 4Google Scholar - Kibblewhite A: Attenuation of sound in marine sediments: a review with emphasis on new low-frequency data.
*J. Acoust. Soc. Am.*1989, 86: 716-738. 10.1121/1.398195View ArticleGoogle Scholar - Wismer M: Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation.
*J. Acoust. Soc. Am.*2006, 120: 3493-3502. 10.1121/1.2354032View ArticleGoogle Scholar - Holm S, Sinkus R: A unifying fractional wave equation for compressional and shear waves.
*J. Acoust. Soc. Am.*2010, 127: 542-548. 10.1121/1.3268508View ArticleGoogle Scholar - Treeby B, Cox B: Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian.
*J. Acoust. Soc. Am.*2010, 127: 2741-2748. 10.1121/1.3377056View ArticleGoogle Scholar - Ochmann M, Makarov S: Representation of the absorption of nonlinear waves by fractional derivatives.
*J. Acoust. Soc. Am.*1993, 94: 3392-3399. 10.1121/1.407192MathSciNetView ArticleGoogle Scholar - Liebler M, Ginter S, Dreyer T, Riedlinger R: Full wave modelling of therapeutic ultrasound: efficient time-domain implementation of the frequency power-law attenuation.
*J. Acoust. Soc. Am.*2004, 116: 2742-2750. 10.1121/1.1798355View ArticleGoogle Scholar - Tavakkoli J, Cathignol D, Souchon R, Sapozhnikov O: Modeling of pulsed finite amplitude focused sound beams in time domain.
*J. Acoust. Soc. Am.*1998, 104: 2061-2072. 10.1121/1.423720View ArticleGoogle Scholar - Remenieras J, Bou Matar O, Labat V, Patat F: Time-domain modelling of nonlinear distortion of pulsed finite amplitude sound beams.
*Ultrasonics*2000, 38: 305-311. 10.1016/S0041-624X(99)00112-2View ArticleGoogle Scholar - Szabo TL: Time domain wave equations for lossy media obeying a frequency power law.
*J. Acoust. Soc. Am.*1994, 96: 491-500. 10.1121/1.410434View ArticleGoogle Scholar - Szabo TL, Wu J: A model for longitudinal and shear wave propagation in viscoelastic media.
*J. Acoust. Soc. Am.*2000, 107: 2437-2446. 10.1121/1.428630View ArticleGoogle Scholar - Chen W, Holm S: Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency.
*J. Acoust. Soc. Am.*2004, 115(4):1424-1430. doi:10.1121/1.1646399 10.1121/1.1646399MathSciNetView ArticleGoogle Scholar - Wismer MG: Finite element analysis of broad-band-acoustic pulses through inhomogenous media with power law attenuation.
*J. Acoust. Soc. Am.*2006, 120: 3493-3502. 10.1121/1.2354032View ArticleGoogle Scholar - Zhang Y: A finite difference method for fractional partial differential equation.
*Appl. Math. Comput.*2009, 215: 524-529. 10.1016/j.amc.2009.05.018MathSciNetView ArticleGoogle Scholar - Tadjeran C, Meerschaert MM, Scheffler HP: A second order accurate numerical approximation for the fractional diffusion equation.
*J. Comput. Phys.*2006, 213: 205-213. 10.1016/j.jcp.2005.08.008MathSciNetView ArticleGoogle Scholar - Meerschaert MM, Tadjeran C: Finite difference approximations for fractional advection dispersion equations.
*J. Comput. Appl. Math.*2004, 172: 65-77. 10.1016/j.cam.2004.01.033MathSciNetView ArticleGoogle Scholar - Yuste SB, Acedo L: An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations.
*SIAM J. Numer. Anal.*2005, 42: 1862-1874. 10.1137/030602666MathSciNetView ArticleGoogle Scholar - Podlubny I, Chechkin A, Skovranek T, Chen YQ, Vinagre Jara BM: Matrix approach to discrete fractional calculus II: partial fractional differential equations.
*J. Comput. Phys.*2009, 228: 3137-3153. 10.1016/j.jcp.2009.01.014MathSciNetView ArticleGoogle Scholar - Hanert E: On the numerical solution of space-time fractional diffusion models.
*Comput. Fluids*2011, 46: 33-39. 10.1016/j.compfluid.2010.08.010MathSciNetView ArticleGoogle Scholar - Zhuang P, Liu F, Anh V, Turner I: Numerical methods for the variable-order fractional advection-dispersion equation with a nonlinear source term.
*SIAM J. Numer. Anal.*2009, 47: 1760-1781. 10.1137/080730597MathSciNetView ArticleGoogle Scholar - Lin R, Liu F, Anh V, Turner I: Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation.
*Appl. Math. Comput.*2009, 212: 435-445. 10.1016/j.amc.2009.02.047MathSciNetView ArticleGoogle Scholar - Crank J, Nicolson P: A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type.
*Proc. Camb. Philos. Soc.*1947, 43(1):50-67. doi:10.1007/BF02127704 10.1017/S0305004100023197MathSciNetView ArticleGoogle Scholar - Langlands TAM, Henry BI: The accuracy and stability of an implicit solution method for the fractional diffusion equation.
*J. Comput. Phys.*2005, 205: 719-736. 10.1016/j.jcp.2004.11.025MathSciNetView ArticleGoogle Scholar - Diethelm K, Ford NJ, Freed AD: Detailed error analysis for a fractional Adams method.
*Numer. Algorithms*2004, 36: 31-52.MathSciNetView ArticleGoogle Scholar - Li CP, Tao CX: On the fractional Adams method.
*Comput. Math. Appl.*2009, 58: 1573-1588. 10.1016/j.camwa.2009.07.050MathSciNetView ArticleGoogle Scholar

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