Time Reversing Solitary Waves

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore, we present numerical experiments for the time-reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a new Boussinesq model, both in transmission and in reflection.

Figure 1

Terrain-following (curvilinear) coordinate system.

Figure 2

Time reversal in transmission for a dispersive ($\beta =0.1$) pulse ${\eta }_o(x)=-10x\exp (-x^2/0.08)$. The random topography fluctuations are sampled at mesh points and connected by straight lines. The fluctuation level is $50\%$ of the total depth. The correlation length is $10\%$ of the effective pulse width. The dynamics is described in the vertical direction of the graph. The initial and final configurations have been displayed separately from the evolution set.

Figure 3

Time reversal in transmission for a nonlinear hyperbolic ($\alpha =0.01$) pulse. The critical time (computed from Burgers equation) is $t_c=-1/(\alpha {\eta }_o^{\prime })=1/(10\alpha )$. The initial (unit width) wave profile ${\eta }_o(x)$ considered is the derivative of a Gaussian. The random topography fluctuations (with $\epsilon =0.6$) are of the same type as in the dispersive simulation. The time-reversed data (recorded beyond the shock distance) has been slightly lifted for better display.

Figure 4

Forward propagation of a solitary wave, as indicated by the bottom trace. The dynamics is described in the vertical direction of the graph. The recorded reflected and transmitted signals are presented at the top. The parameters are $\alpha =\beta =0.01$ and the fluctuation level is $50\%$ of the total depth.

Figure 5

Time-reversal experiment for the reflected signal generated by a solitary wave as described in Fig. 4.

Figure 6

Time-reversal experiment for the transmitted signal generated by a solitary wave as described in Fig. 4. The recording time is longer than in Fig. 4, and therefore the random topography was shifted to the left of the computational domain.