A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field (Couder et al. 2005a;Protière et al. 2006). This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory (de Broglie 1930(de Broglie , 1956(de Broglie , 1987. We here develop a numerical model of pilot-wave hydrodynamics by coupling recent models of the droplet's bouncing dynamics (Moláček & Bush 2013a,b) with a more realistic model of weakly viscous wave generation and evolution. (Lamb 1932;Dias et al. 2008). The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect, and walker-walker interactions.

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The purpose of this work is to explore in detail the structure of the interior flow generated by periodic surface waves on a fluid with constant vorticity. The problem is mapped conformally to a strip and solved numerically using spectral methods. Once the solution is known, the streamlines, pressure and particle paths can be found and mapped back to the physical domain. We find that the flow beneath the waves contains zero, one, two or three stagnation points in a frame moving with the wave speed, and describe the bifurcations between these flows. When the vorticity is sufficiently strong, the pressure in the flow and on the bottom boundary also has very different features from the usual irrotational wave case.

Boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids. Several studies have revealed that standard numerical approximations tend to lead to unstable methods, and various remedies have been introduced and tested. In this paper, we conduct a stability analysis of the linearized equations with a particular objective in mind | the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes. We discover two reasons for the typical failure in numerical discretizations: one is the inability of the mesh to represent the vorticity created by surface tension e ects on the nest scale; and the other is the inaccuracies in the evaluation of the boundary integral for the velocity. With the insight gained from our linear analysis, we propose a new method that is spectrally accurate and linearly stable. Further, the exact dispersion relation is obtained for all the available discrete modes. Numerical tests suggest that the method is also stable in the nonlinear regime. However, our method runs into di culties generic to methods based on Lagrangian motion. The markers accumulate near a stagnation point on the interface, forcing us to use an ever decreasing timestep in our explicit method. We introduce a redistribution of markers to overcome this di culty. When we redistribute according to equal arclength, we nd excellent agreement with a method based on preserving equal spacing in arclength.

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