The mesoscopic phenomenon par excellence is the coherent backscatter. It is due to the constructive interference of partial waves that follow reciprocal paths in a disordered medium and is manifested by a doubling of the intensity scattered in the opposite direction to the direction of propagation of the incident wave. As a function of the scattering angle, the intensity profile develops a peak called "backscattering cone". This cone can be observed in the stationary regime (excitation by a monochromatic wave, Fig. 4) as well as in the dynamic regime (excitation by a short pulse). The width of the stationary cone tells us about the mean transport free path ℓ* - a distance required for a wave to "lose memory" of its original direction of propagation.

Figure 4. preliminary measurements of an ultrasonic backscatter cone for a school of fish in the steady state. The two curves correspond to dense (red circles) and sparse (gray asterisks) schools.

The measured backscatter cone for ultrasound scattering from fish schools is surprisingly broad (Fig. 4), suggesting that ℓ* lambda, the acoustic wavelength. This is consistent with very strong scattering and proximity to the expected Anderson localization transition for ℓ lambda and ℓ* = 0. Furthermore, the cone can be measured in different configurations (emission and detection by all transducers, emission by one line of transducers and detection by the same line or by a perpendicular line, etc.).

These measurements give unexpected results that we would like to understand and interpret theoretically. We will apply diagrammatic methods known in the literature to calculate the stationary and dynamic backscatter cones in the real experimental configuration. In particular, we will take into account
(1) the Mills cross shape of our ultrasonic antenna,
(2) the positioning of our transducers at the surface of the medium and not in the far field,
(3) the strong internal reflection of the sound waves at the water surface,
(4) the avoidance reaction of the fish which moves them away from the antenna.

To verify and complete the analytical calculations we will also perform numerical Monte Carlo simulations which will allow us to get even closer to the real experimental conditions. The expected results of this task are
(1) a complete theoretical description of the experimental results for the coherent backscatter cone
(2) a theory linking the parameters characterizing a school of fish (the numerical density of fish, their average size or weight) to the shape and temporal evolution of the cone.

The results of this task will allow the use of coherent backscatter cone measurements to determine parameters characterizing the scattering of sound by fish schools (mean free path ℓ*, scattering coefficient D, etc.).