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The universal power law for the spectrum of one-dimensional breaking Riemannwaves is justified for the simple wave equation. The spectrum of spatialamplitudes at the breaking time $t = t_b$ has an asymptotic decay of$k^{-4/3}$, with corresponding energy spectrum decaying as $k^{-8/3}$. Thisspectrum is formed by the singularity of the form $(x-x_b)^{1/3}$ in the waveshape at the breaking time. This result remains valid for arbitrary nonlinearwave speed. In addition, we demonstrate numerically that the universal powerlaw is observed for long time in the range of small wave numbers if smalldissipation or dispersion is accounted in the viscous Burgers or Korteweg-deVries equations.

Universal power law for the energy spectrum of breaking Riemann waves