Attenuation in 3D fractal media

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Attenuation in 3D fractal media

The energy spectrum, $E^{(1)}_{H,b}(k)$ , of von Kármán's autocorrelation function $N_{H,b}(r)$ in equation 7 is real and even:

$\displaystyle {E^{(1)}_{H,b}(k)}~{\sigma^{-2}}$	$\textstyle =$	$\displaystyle \int_{-\infty}^{+\infty}\hspace{-3mm}N_{H,b}(r)\,e^{ikr}dr\,,$	(25)
	$\textstyle =$	$\displaystyle C^{(1)}_{H}~\frac{2b}{\left(1+b^2k^2\right)^{H+\frac{1}{2}}}\,.$	(26)

Values of

defined by equation 22 are

$\displaystyle S(0)$	$\textstyle =$	$\displaystyle C^{(1)}_{H}\,b\,,$	(27)
$\displaystyle Re[S(2k_0)]$	$\textstyle =$	$\displaystyle \frac{C^{(1)}_{H}\,b}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}}\,.$	(28)

Coefficient $C^{(1)}_{H}$ , defined by equation 8, is an increasing function of exponent

and has to be calculated numerically, except for some specific values:

$\begin{displaymath} \begin{array}{lll} C^{(1)}_{H}\sim{\pi}/{\Gamma(H)}\rightarr... ...5}=1.3317\ldots\,, & C^{(1)}_{1}=\pi/2\,. \nonumber \end{array}\end{displaymath}$

The dispersion relation of equation 21 solves for an explicit solution of attenuation and dispersion:

$\displaystyle Re[k/k_0]$	$\textstyle =$	$\displaystyle 1+\frac{\sigma^2}{2}\left(1+2k_0~Im[S(2k_0)]\right)\,,$	(29)
$\displaystyle Im[k/k_0]$	$\textstyle =$	$\displaystyle \sigma^2k_0b~C^{(1)}_{H} \left[1-\frac{1}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}} \right].$	(30)

When

, the derivation produces simple expressions as detailed in Appendix B. The use of $S^{\ast}(k)=S(-k)$ with the Kramers-Krönig relation can be used to determine the real part of

. In the context of the second-order approximation, scattering attenuation in a von Kármán isotropic medium is

$\displaystyle \frac{1}{Q} = \frac{2\,Im[k]}{Re[k]} = 2~\sigma^2\,k_0b~C^{(1)}_{H} \left[1-\frac{1}{(1+4\,b^2k_0^2)^{H+\frac{1}{2}}}\right].$

(31)

For $k_0b\,\ll\,1$ , the scattering attenuation reduces to the Rayleigh diffusion regime:

$\displaystyle \frac{1}{Q}$

$\textstyle \simeq$

$\displaystyle 8\,\sigma^2\,C^{(1)}_{H}\left(H+\frac{1}{2}\right)\left(k_0b\right)^3.$

(32)

Subsections

Next: Penetration depth Up: Scattering attenuation in 3D Previous: Low-frequency waves in 3D

2013-07-26