Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation

Heterogeneity can be accounted for by a random potential in the wave equation. For acoustic waves in a fluid with fluctuations of both density and compressibility (as well as for electromagnetic waves in a medium with fluctuation of both permittivity and permeability) the random potential entails a scalar and an operator contribution. For simplicity, the latter is usually overlooked in multiple scattering theory: whatever the type of waves, this simplification amounts to considering the Helmholtz equation with a sound speed $c$ depending on position $\mathbf{r}$. In this work, a radiative transfer equation is derived from the wave equation, in order to study energy transport through a multiple scattering medium. In particular, the influence of the operator term on various transport parameters is studied, based on the diagrammatic approach of multiple scattering. Analytical results are obtained for fundamental quantities of transport theory such as the transport mean-free path ${\ell }^{*}$, scattering phase function $f$, and anisotropy factor $g$. Discarding the operator term in the wave equation is shown to have a significant impact on $f$ and $g$, yet limited to the low-frequency regime, i.e., when the correlation length of the disorder ${\ell }_c$ is smaller than or comparable to the wavelength $\lambda$. More surprisingly, discarding the operator part has a significant impact on the transport mean-free path ${\ell }^{*}$ whatever the frequency regime. When the scalar and operator terms have identical amplitudes, the discrepancy on the transport mean-free path is around $300\%$ in the low-frequency regime, and still above $30\%$ for ${\ell }_c/\lambda ={10}^3$ no matter how weak fluctuations of the disorder are. Analytical results are supported by numerical simulations of the wave equation and Monte Carlo simulations.

Figure 1

Polar plots of the phase functions $f\left(\mathrm{\Theta }\right)$ (solid line) and $f^{\left(\alpha \alpha \right)}\left(\mathrm{\Theta }\right)$ (dashed line) for various values of $k_0{\ell }_c$.

Figure 2

Anisotropy factor $g$ as a function of $k_0{\ell }_c$ when the operator contribution is (solid line) or is not (dashed line) taken into account.

Figure 3

Dimensionless transport mean-free path as a function of dimensionless frequency $k_0{\ell }_c$ when the operator contribution is (solid line) or is not (dashed line) taken into account.

Figure 4

Normalized acoustic flux as a function of time. (a) Sketch of the numerical experiment. The color map represents the spatial fluctuations of the potentials $\alpha$ and $\beta$. A point source emits a Gaussian pulse with a central frequency and bandwidth of $1\mathrm{MHz}$ on one side of a cube of length $L=118{\ell }_c$, with $\sigma =0.15$, ${\ell }_c\sim 4.78\mathrm{mm}$, hence $k_0{\ell }_c=2$. The exiting flux is measured on the opposite face of the cube perpendicular to ${\mathbf{e}}_z$ at $z=L/2$. (b) Measured flux when the operator contribution is (solid line) or is not (dashed line) taken into account. The data are normalized by the energy $W_0$ conveyed by incident pulse. The blue (slightly above) and red (slightly below) curves are the flux calculated from FDTD and Monte Carlo simulations, respectively.

Figure 5

Transmitted flux as a function of time in the case of an infinite slab with thickness $L=8395{\ell }_c$ ($k_0{\ell }_c=0.3$ and $\sigma =0.1$). The incoming wave is plane, with a normal incidence. The flux is computed using the Monte Carlo method, when the operator part is taken into account (solid line) or not (dashed line). Each curve is normalized by its maximum.

Figure 6

Transmitted flux as a function of time in the case of an infinite slab with thickness $L=8\times {10}^4{\ell }_c$ ($k_0{\ell }_c=10$ and $\sigma =0.01$). The flux is computed using the Monte Carlo method, when the operator part is taken into account (solid line) or not (dashed line). Each curve is normalized by its maximum. The straight lines are the asymptotes predicted by diffusion theory.

Figure 7

Solution of the inverse problem. The pairs $(\sigma ,k_0{\ell }_c)$ that are compatible with the value of ${\ell }^{*}$ obtained from Fig. 6 are plotted, taking into account the operator term (continuous line) or not (dashed line). The exact result ($\sigma =0.01,k_0{\ell }_c=10)$ is represented by a circle.

Figure 8

Ratio between $k_r$ and $k_0$, for $\sigma =0.1$ in the scalar (dashed line) and operator (solid line) cases. At this level of fluctuation, $k_0$ and $k_r$ are indistinguishable in the scalar case. On the contrary, in the operator case the low frequency-regime becomes clearly incompatible with the approximation $k_r\approx k_0$ as frequency diminishes, until Eq. (C2) has no longer a real solution for $k_0{\ell }_c<0.0995$.