Overturning of nonlinear acoustic waves in media with power-law attenuation

Transcription

1 Ultrasound: Paper ICA Overturning of nonlinear acoustic waves in media with power-law attenuation John M. Cormack (a) and Mark F. Hamilton (a) (a) Applied Research Laboratories, University of Texas at Austin, Austin, Texas, USA, Abstract Without incorporation of weak-shock theory, the lossless Burgers equation predicts a multivalued waveform for nonlinear propagation beyond a certain distance. Inclusion of thermoviscous attenuation, which increases quadratically with frequency, prevents the occurrence of multivalued waveforms. The same is true for any attenuation law that is proportional to frequency raised to an exponent greater than unity. For exponents less than unity the situation is less clear. For example, when attenuation is constant with frequency (exponent equal zero) there is a critical value of the source amplitude below which a multivalued waveform is predicted and above which it is not. Prediction of multivalued waveforms indicates that the mathematical model is inadequate and must be either supplemented by weak-shock theory, or augmented to include an additional loss factor. To investigate the prediction of multivalued waveforms subject to power-law attenuation with exponents between zero and unity a Burgers equation with the loss term expressed as a fractional derivative is used [Prieur and Holm, J. Acoust. Soc. Am. 13, (211)]. Transformation of the equation into intrinsic coordinates following Hammerton and Crighton [J. Fluid Mech. 252, (1993)] permits simulation of waveforms beyond the point at which they become multivalued. These solutions are used to determine the parameter space in which initially sinusoidal plane waves are predicted to evolve into multivalued waveforms for power-law attenuation with exponents less than unity. Keywords: Power-law attenuation, multivalued solutions

2 Overturning of nonlinear acoustic waves in media with power-law attenuation 1 Introduction Heterogeneous media such as biological tissues and marine sediments can exhibit acoustic attenuation in certain frequency ranges that varies approximately as frequency raised to some power [1, 2]. Time-domain models of wave propagation in media with power-law attenuation often employ a loss operator that is a fractional derivative with respect to either time or space [3]. The fractional derivative can be used as an ad hoc way to match measurements of power-law attenuation [4], or it can manifest as the cumulative effect of various random [5], viscoelastic [6, 7], or relaxation [8] processes. Attenuation and dispersion associated with the fractional derivative satisfy the Kramers-Kronig relations [9, 1]. Propagation of finite-amplitude sound in media with power-law attenuation has been modeled with nonlinear evolution equations containing a fractional derivative [11, 12, 13]. Thermoviscous attenuation, which increases quadratically with frequency, balances waveform steepening due to nonlinearity in such a way that solutions are guaranteed to be single valued regardless of wave amplitude. Conversely, attenuation that is assumed to be independent of frequency (i.e., frequency raised to the power zero) is insufficient by itself to prevent the occurrence of multivalued solutions when the wave amplitude exceeds a certain threshold [14]. Relaxing fluids exhibit both power laws, attenuation with quadratic frequency dependence at low frequencies and constant attenuation at high frequencies. Relaxation alone is also incapable of preventing the existence of multivalued solutions for sufficiently high wave amplitudes [15, 16]. Multivalued solutions thus point to the fact that essential physics is missing from the mathematical model for nonlinear propagation at the wave amplitudes under consideration. The purpose of the present paper is to determine the critical source amplitude (or alternatively, critical attenuation coefficient for a given source amplitude) required to ensure single-valued solutions for a source radiating at a single frequency into a nonlinear medium with power-law attenuation. The nonlinear evolution equation used for this study is a Burgers equation with a fractional derivative [11]. Intrinsic coordinates are introduced to enable numerical simulation of multivalued waveforms [17]. The form of the fractional loss operator in intrinsic coordinates reveals that power-law attenuation for which the exponent is greater than unity ensures singlevalued solutions. Numerical simulations are used to determine the source amplitude above which solutions become multivalued for power-law attenuation with exponents between zero and unity. Overturning of nonlinear acoustic waves, the terminology used in the title, was used by Hammerton and Crighton [16, 17] and is helpful for visualizing the simulated evolution of an initially sinusoidal waveform into a multivalued waveform, which resembles the actual overturning of a water wave. The terminology may also be helpful for distinguishing the present analysis from a focus on whether simulated propagation of a preexisting shock can produce a multivalued waveform. 2

3 2 Power-law attenuation The model equation used in the present study is a Burgers equation with a loss operator in the form of a fractional derivative of order η as given by Prieur and Holm [11]: p x = β p p ρ c 3 τ δ η p τ η, (1) where p(x, τ) is the acoustic pressure, x the propagation coordinate, β the coefficient of nonlinearity, ρ the equilibrium density, c a reference small-signal sound speed, τ = t x/c a retarded time, and δ a constant coefficient, possibly negative as explained below, with units [T η L 1 ]. The defining relation for the fractional derivative is taken to be { η } p F τ η = ( jω) η p ω, (2) where p ω = F {p} is the temporal Fourier transform of p. Ignoring the nonlinear term in Eq. (1) and taking the Fourier transform yields where d p ω dx = α(ω)p ω, (3) α(ω) = δ( jω) η = δω η cos(ηπ/2) + jδω η sin(ηπ/2) (4) is a complex attenuation coefficient that accounts for both attenuation and dispersion associated with the fractional derivative [18]. Since p ω (x) e αx is a solution of Eq. (3), linear solutions of Eq. (1) at frequency ω are of the form p(x,τ) e αx e jωτ, such that the real attenuation coefficient α(ω) and phase speed c(ω) are defined by the relation ( 1 α(ω) = α(ω) + jω c(ω) 1 ). (5) c Equating the right-hand sides of Eqs. (4) and (5) yields ( ηπ ) α(ω) = δω η 1 cos, 2 c(ω) = 1 ( ηπ + δω η 1 sin c 2 ). (6) The attenuation coefficient possesses the desired power-law dependence on ω, and the expressions for α(ω) and c(ω) more precisely, the real and imaginary parts of Eq. (4) satisfy the Kramers-Kronig relations [9, 1]. It follows from the expression for the attenuation coefficient that δ > is required for < η < 1, and δ < for 1 < η < 3, etc. The case η = 1 warrants special consideration. For this case α(ω) = and 1/c(ω) = 1/c + δ, and there is neither attenuation nor dispersion. Indeed, if c 1 is the phase speed when η = 1 (1/c 1 = 1/c + δ), then transformation of Eq. (1) from (x,τ) coordinates to (x 1,τ 1 ) coordinates where x 1 = x and τ 1 = t x/c 1 = τ δx yields p x = β p p ρ c 3 for η = 1, (7) τ 1 3

4 which is the lossless form of the Burgers equation. Another case of interest is η = 1/2 for thermoviscous boundary-layer attenuation and dispersion in a duct. In this case, with δ = 2B in Blackstock s notation [18], α(ω) = B ω and 1/c(ω) = 1/c + B/ ω. A model for nonlinear propagation in viscoelastic media developed by Varley and Rogers [14] corresponds to η =, for which α(ω) = δ and c(ω) = c. For η = 2, Eq. (1) is the classical Burgers equation for thermoviscous fluids, in which case α(ω) = δ ω 2 and c(ω) = c. Attenuation in real media typically follows a specific power law only over a limited range of frequencies. It is therefore common practice to identify a reference angular frequency ω in this range together with the corresponding attenuation coefficient α = α(ω ) = δω η cos(ηπ/2), in terms of which Eqs. (6) may be expressed as α(ω) = α ( ω ω ) η, 1 c(ω) = 1 + α ( ) ω η 1 ( ηπ tan c ω ω 2 ) for η 1. (8) Attenuation in the megahertz range for biological tissues and in the kilohertz range for marine sediments is often associated with values of η in the neighborhood of unity. The restriction η 1 results from the fact that to obtain Eqs. (8) from Eqs. (6) for η = 1 requires division by zero. If in Eqs. (8) one stipulates that α(ω) = α (ω/ω ) for η = 1, then the phase speed obtained from the Kramers-Kronig relations depends logarithmically on frequency [1]. 3 Multivalued solutions for η 1 For both analytical and numerical purposes it is convenient to express Eq. (1) in dimensionless notation based on a characteristic source pressure p and frequency ω: P σ = P P θ A η P θ η, (9) where P = p/p, σ = x/x, θ = ωτ, and A = δω η x are dimensionless sound pressure, distance, retarded time, and absorption parameter, respectively, with x = ρ c 3 /β p ω the shock formation distance in an ideal fluid (δ = ) for the source condition p = p sinωt at x =. Note that A is inversely proportional to source amplitude, and it is a measure of nonlinear effects in relation to dissipative effects. The focus of the present work is determination of the parameter range in which Eq. (9) admits a multivalued solution. For example, if P = sin θ at σ =, multivalued waveforms are predicted for A = and σ > 1 due to the absence of any energy dissipation mechanism that balances waveform steepening. Equation (9) is an inadequate model in such parameter ranges and must be either supplemented by weak shock theory, or augmented to include an additional loss factor. Specifically of interest is the existence of multivalued solutions for η 1 and A > given P = sinθ at σ =. Implicit analytical solutions of Eq. (9) are available for η = and η = 1; η = : P = e Aσ sin[θ + A 1 (e Aσ 1)P] σ vt = A 1 ln(1 A), (1) η = 1: P = sin[θ + σ(p A)] σ vt = 1, (11) 4

5 where σ vt is the distance at which a vertical tangent first appears in the waveform, and beyond which the waveform is multivalued. Equations (6) show that there is no dispersion in either case, with the constant phase speeds given by c(ω) = c for η = and c(ω) = c 1 for η = 1. For η = there is attenuation, and as A increases from to 1, σ vt increases from 1 to. A multivalued waveform is thus described by Eq. (1) for A < 1, but not for A 1. For η = 1 there is no attenuation, and Eq. (11) describes a multivalued waveform beyond σ = 1 for any value of A. Numerical solutions of Eq. (9) are used to investigate all other values of η. Following Hammerton and Crighton [17], intrinsic coordinates are introduced that permit numerical simulation of nonlinear propagation beyond the distance at which a solution becomes multivalued. The intrinsic coordinates describing a waveform P(θ) at location σ are the angle ψ(θ) of the tangent to the waveform and the corresponding arc length s(θ) along the waveform from P() to P(θ): θ ψ = tan 1 ( P/ θ), s = 1 + ( P/ θ) 2 dθ. (12) Waveform evolution as a function of distance is then described by ψ(s,σ) rather than P(θ,σ). Transformation of the solution for ψ(s,σ) back to a pressure waveform is accomplished via the integrals s s P = sinψ ds, θ = cosψ ds. (13) Hammerton and Crighton [16] used this method to determine conditions for the occurrence of multivalued solutions for periodic waves in a relaxing fluid without thermal or viscous losses. Equation (9) becomes, in intrinsic coordinates, where ψ σ = sin2 ψ + ψ s sinψ cosψ ds + f s s + ψ s f (ψ) ψ ds, (14) s s f (ψ) = A(cosψ) 1 η η = A(cosψ) s η s 2 η η 1 sinψ ds η 1, (15) tanψ 1 η 2. (16) sη 1 Inception of a multivalued solution occurs when ψ achieves the value π/2 at some point on the waveform, indicating a vertical tangent, P/ θ =. In the limit ψ π/2, Eq. (15) reveals that f (ψ) remains finite for η 1, whereas Eq. (16) shows that f (ψ) becomes unbounded for 1 < η 2. Thus multivalued solutions may exist for certain values of η 1 and A >, whereas for any 1 < η 2 and A < a solution cannot become multivalued because the value of ψ can never exceed π/2 (recall δ <, and therefore A <, is required for positive attenuation with 1 < η 2). These general observations are consistent with the conclusion reached by Kashcheeva et al. [19], using a frequency-domain analysis, that preexisting discontinuities in a waveform are stable for η < 1 and unstable for η > 1. 5

6 1 (a) (b).5 π/2 P ψ.5 π/2 1 π π θ Figure 1: Waveforms calculated numerically in intrinsic coordinates for η = and A =.5 at σ/σ vt =,.5, 1, and 2, presented in (a) physical coordinates and (b) intrinsic coordinates. The pressure waveforms in (a) are indistinguishable from the analytic solution given by Eq. (1). s Figure 1 illustrates results of the numerical procedure for η = and A =.5. The blue curve in Fig. 1(a) is the sinusoidal pressure waveform at σ =, which after transformation into intrinsic coordinates via Eqs. (12) becomes the blue curve in Fig. 1(b). Equation (14) is then solved numerically by marching forward in σ, with the fractional derivative in Eq. (15) evaluated using the left-sided Caputo definition [2]. The desired solutions for the pressure waveforms are recovered from Eqs. (13). At distance σ = σ vt (red curves) the solution in intrinsic coordinates achieves the value ψ = π/2, and the corresponding pressure waveform exhibits a vertical tangent at θ =. For σ > σ vt the solution in intrinsic coordinates exceeds ψ = π/2 and the pressure waveform is multivalued. The pressure waveforms obtained numerically in this manner are indistinguishable from the analytical solution given by Eq. (1). For < η < 1 a critical source amplitude exists above which the solution describing propagation of an initially sinusoidal waveform can become multivalued and below which it remains single valued. To characterize this threshold the following dimensionless parameter is introduced: a = α(ω)x = Acos(ηπ/2), (17) where α(ω) is given in Eqs. (6). The reciprocal of α(ω)x may be recognized as the Gol dberg number associated with Eq. (9) for η = 2. Like A, the parameter a is proportional to the attenuation coefficient and inversely proportional to source amplitude. The critical value of a below which multivalued solutions can arise and above which they cannot is designated a cr (η). The value a cr () = 1 follows directly from Eq. (1), and the value a cr (1) = is concluded from the discussion following Eq. (16). Values between these endpoints are obtained from numerical solutions of Eq. (14). 6

7 acr Multivalued solutions (a) Single valued solutions acr (b) η η Figure 2: Parameter space in which multivalued solutions are obtained at distances σ > σ vt for radiation from a monofrequency source. The curve a cr (η) for η 1 is presented in Fig. 2(a). Below this curve is the parameter space in which the amplitude of a monofrequency source is sufficiently high relative to attenuation that a multivalued solution is obtained at finite distance from the source. In this region, power-law attenuation produces insufficient energy dissipation to balance waveform steepening due to nonlinearity and maintain a stable shock front; Eq. (9) is an inadequate model under these circumstances. Above and to the right of the curve, including the region η > 1, solutions are single-valued at all distances. Figure 2(b) provides an expanded view of the region near η = 1 where a cr (η) decreases rapidly to zero as attenuation vanishes in this limit. Although not as rapidly, the critical source amplitude also vanishes in this limit, with multivalued solutions being inevitable at η = 1 for any nonzero source amplitude [recall Eq. (11)]. For η <.95 the curve is relatively smooth and can be approximated to within 3% by a cr (η) 1.93η +.19η 2. In the parameter space below the curve in Fig. (2) the solutions are multivalued at distances σ > σ vt. Curves showing the dependence σ vt (η) for several values of a are presented in Fig. 3. From Eqs. (1) and (17) one obtains σ vt () = a 1 ln(1 a) for the intercepts along the vertical axis. The markers along the horizontal axis indicate the value of η at which σ vt becomes infinite for each value of a. Holding a fixed while increasing η corresponds to holding both source amplitude and attenuation at the source frequency fixed while increasing attenuation of the harmonics. Greater energy dissipation at high frequencies postpones the distance at which a vertical tangent is formed until ultimately σ vt =. For higher values of η, i.e., to the right of a marker in Fig. 3 for the corresponding value of a, attenuation of the harmonics is sufficient to prevent waveform steepening from ever creating a vertical tangent. 7

8 σvt a = Figure 3: Distances away from a monofrequency source at which a vertical tangent is first encountered in the waveform, and beyond which the solution is multivalued. Markers along the horizontal axis indicate the values of η at which σ vt becomes infinite for the corresponding value of a. η Acknowledgements This work was supported by the McKinney Fellowship in Acoustics at Applied Research Laboratories, and the Thrust 2 Graduate Fellowship in the Cockrell School of Engineering, at The University of Texas at Austin. References [1] F. A. Duck, Physical Properties of Tissues A Comprehensive Reference Book, Chap. 4: Acoustic Properties of Tissue at Ultrasonic Frequencies (Academic Press, San Diego, 199), pp [2] A. Kibblewhite, Attenuation of sound in marine sediments: A review with emphasis on new low-frequency data, J. Acoust. Soc. Am. 86, (1989). [3] S. Holm and S. P. Näsholm, Comparison of fractional wave equations for power law attenuation in ultrasound and elastography, Ultrasound in Med. and Biol. 4, (214). [4] T. L. Szabo, Time domain wave equations for lossy media obeying a frequency power law, J. Acoust. Soc. Am. 96, (1994). [5] P. Grigolini, A. Rocco, and B. J. West, Fractional calculus as a macroscopic manifestation of randomness, Phys. Rev. E 59, (1999). [6] M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. R. Astr. Soc. 13, (1967). 8

9 [7] J. F. Kelly and R. J. McGough, Fractal ladder models and power law wave equations, J. Acoust. Soc. Am. 126, (29). [8] S. P. Näsholm and S. Holm, Linking multiple relaxation, power-law attenuation, and fractional wave equations, J. Acoust. Soc. Am. 13, (211). [9] C. W. Horton, Sr., Dispersion relationships in sediments and sea water, J. Acoust. Soc. Am. 55, (1974). [1] K. R. Waters, M. S. Hughes, J. Mobley, G. H. Brandenburger, and J. G. Miller, On the applicability of Kramers-Krönig relations for ultrasonic attenuation obeying a frequency power law, J. Acoust. Soc. Am. 18, (2). [11] F. Prieur and S. Holm, Nonlinear acoustic wave equations with fractional loss operators, J. Acoust. Soc. Am. 13, (211). [12] M. Leibler, S. Ginter, T. Dreyer, R. E. Riedlinger, Full wave modeling of therapeutic ultrasound: Efficient time-domain implementation of the frequency power-law attenuation, J. Acoust. Soc. Am. 116, (24). [13] M. Ochmann and S. Makarov, Representation of the absorption of nonlinear waves by fractional derivatives, J. Acoust. Soc. Am. 94, (1993). [14] E. Varley and T. G. Rogers, The propagation of high frequency, finite, acceleration pulses and shocks in viscoelastic materials, Roy. Soc. London. Series A, Math. and Phys. Sci. 296, (1967). [15] A. L. Polyakova, S. I. Soluyan, and R. V. Khokhlov, Propagation of finite disturbances in a relaxing medium, Sov. Phys. Acoust. 8, (1962). [16] P. W. Hammerton and D. G. Crighton, Overturning of nonlinear acoustic waves. Part 2: Relaxing gas dynamics, J. Fluid Mech. 252, (1993). [17] P. W. Hammerton and D. G. Crighton, Overturning of nonlinear acoustic waves. Part 1: A general method, J. Fluid Mech. 252, (1993). [18] D. T. Blackstock, Generalized Burgers equation for plane waves, J. Acoust. Soc. Am. 77, (1985). [19] S. S. Kashcheeva, O. A. Sapozhnikov, V. A. Khokhlova, M. A. Averkiou, and L. A. Crum, Nonlinear distortion and attenuation of intense acoustic waves in lossy media obeying a frequency power law, Acoust. Phys. 46, (2). [2] E. C. de Oliveira and J. A. T. Machado, A review of definitions for fractional derivatives and integrals, Math. Problems in Eng. Vol. 214, Article ID , 6 pages (214). 9