Nonlinear waves emitted from a moving source are studied. A meandering spiralin a reaction-diffusion medium provides an example, where waves originate froma source exhibiting a back-and-forth movement in radial direction. The periodicmotion of the source induces a Doppler effect that causes a modulation inwavelength and amplitude of the waves (``superspiral''). Using the complexGinzburg-Landau equation, we show that waves subject to a convective Eckhausinstability can exhibit monotonous growth or decay as well as saturation ofthese modulations away from the source depending on the perturbation frequency.Our findings allow a consistent interpretation of recent experimentalobservations concerning superspirals and their decay to spatio-temporal chaos.
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