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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).

You can think of dispersive effects as acting to "spread out" the spatial scales in the wave while the nonlinear term acts to "focus" the spatial scales. In fully nonlinear systems, the loss terms are often dependent on the gradient in the spatial scale of the wave front. This means that the terms are often small/negligible until the spatial scales of the wave front become small, thus the gradients large.

- from: Sagdeev, [1966] in Rev. Plasma Phys. Vol. 4, pg. 21 (of 69)
Interestingly, dispersion is not an irreversible term because it increases the order of the derivatives in the equations by an even number. The other loss terms I mentioned increase the order of the derivatives in the equations by an odd number, thus introducing irreversibility. A link to Sagdeev's article can be found here. Though the link does not contain the PDF file, unfortunately.

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Hi thanks for the reply. Can you give a link for the reference you have mentioned? I'm not able to find it. –  vijay Oct 9 at 4:55
@vijay - Sorry, for some reason I put down the wrong year in that reference. Sagdeev did have some important work published in 1962 as well, but it was in Russian. I have the PDF file, but not a link. I wasn't sure if I could attach a PDF file of a paper as an image. –  honeste_vivere Oct 9 at 14:53
@vijay - You should also look up any papers/books written by G.B. Whitham. He has a book that goes into gory detail regarding dispersion, steepening, and just about every other topic on waves. –  honeste_vivere Oct 9 at 14:57
Hi, is it possible to email the pdf file to vijaysnitt[at]gmail[dot]com? –  vijay Nov 14 at 5:38

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