We investigate energy exchanges through scales occurring when a surface wave reflects on a harmonically oscillating wall. We first experimentally evidence the creation of Doppler-shifted waves and measure their height as a function of the oscillation amplitude. Then, we theoretically compute the amplitudes of these new waves in the gravity regime. Both results show that even without bulk non-linearities, oscillating paddles in a fluid container lead to a complex wave energy spectrum competing with the one predicted by wave turbulence. To exemplify this point, we characterize a simple one-dimensional model consisting of a linear wave equation in an oscillating cavity with distinct injection and dissipation mechanisms. It displays features usually associated with non-linearities, as self-similarity in a spectral domain (the so-called inertial range), appearance of energy at larger and/or lower scales than the forcing one and creation of shock waves.