# Nonlinear waves and shock formation

In the cases of nonlinear acoustics, why is shock formation unlikely when the dispersion is strong when compared to the nonlinearity of the wave?

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The nonlinear term, $\left( \mathbf{V} \cdot \nabla \right) \mathbf{V}$, determines the steepening of a wave. This can be balanced/offset by loss terms like dispersion, diffusion, viscosity, resistivity, friction, etc. If the loss term dominates over the nonlinear term, then the wave cannot steepen as there is too much damping. If the loss term balances the nonlinear term, a shock wave can form as the steepening wave approaches a gradient catastrophe (i.e., point where wave breaking occurs).