In many areas of physics such as hydrodynamics, modulational instabilities of traveling waves often occur as a first step in the route to spatio-temporal complexity and turbulence. In this paper, we investigate the structure of waves whose amplitude is spatially and/or temporally modulated. For a nonmodulated traveling wave u(x,t) that travels uniformly at velocity c, a very simple space-time symmetry occurs, namely u(x-x_0,t) = u(x,t-t_0) with x_0 = ct_0. A consequence of this symmetry is that the wave can be decomposed into spatial and temporal orthogonal modes that are merely Fourier modes. In this case, the Fourier decomposition is the natural modal decomposition to adopt. When such a wave undergoes a spatio-temporal modulation of its amplitude, sidebands form around the various Fourier wave numbers and/or frequencies of the carrier wave in the Fourier spectrum. If the modulation is of long wavelength, sidebands do not intersect and it is possible to give a precise meaning to the new space-time symmetry in this case. As a result, such a wave is the superposition of orthogonal spatial and temporal modes that are slight (and continuous) deformations of the original Fourier modes of the carrier wave. In this paper, we explore the case of modulation of short wavelengths causing neighboring sidebands to intersect in Fourier space. We show how the latter may have the effect of breaking the space-time symmetry of the carrier wave, thus destroying the coherent displacement of the orthogonal modes composing the wave. We also fully determine analytically the spatial and temporal orthogonal modes (eigenfunctions) of the traveling wave, which are now far from being Fourier modes, and write explicitly the corresponding eigenspectrum law. In turn, since the eigenfunctions can easily be computed from space-time signals, our results can be used to characterize the dynamical characteristics of the latter.
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