We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

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When a charged particle moves parallel and close to the surface of a metasurface, intense Smith-Purcell radiation can be observed at resonant frequencies. Here, we present a systematic investigation on the Smith-Purcell radiation and evanescent-to-propagating wave conversion in metal-groove metasurfaces. Based on a coupled mode theory, analytic formulas are derived for the resonant frequency, Q-factor, and wave conversion efficiency at resonant frequency. The accuracy of the formulas is verified by numerical simulations. It is found that the resonant frequency and Q-factor depend on the depth and filling ratio of the grooves, respectively. A high Q-factor can be obtained by decreasing the filling ratio of the grooves. As the Q-factor increases, the wave conversion efficiency at resonant frequency increase but exhibits an upper limit. Such an upper bound of efficiency (Cr,max = 4) can be approached at a moderate Q-factor (Q = 16) or an optimal filling ratio of the grooves (fs = 0.05). Our results may benefit the construction of compact high-power free-electron light sources.