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Given Burgers' equation, $m_{\tau} + mm_x = 0,$ one expects to have discontinuities and thus shock waves in the case the initial conditions are smooth. For example, one may take $m_0(x) = \sin(x), x\in[0,\pi]$ as initial conditions. Then, the characteristic lines would be $x = x_0 + \sin(x_0)\tau$ which will cross and thus we expect shock waves. Assuming we have a shallow water environment, or let's say a home made experiment, why we do not observe the shock wave if we generate a sine wave on the water surface?

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I guess you're misinterpreting something here:

  • Burgers' equation is not the PDE governing shallow water equations

  • PDEs governing shallow water system are a set of $n+1$ equations, begin $n$ the number of space dimensions, see wikipedia

  • Burger equation $\partial_t u + u \partial_x u = 0$ governs a 1-dimensional problem, the analogous of the flow in a channel/pipe, with one dimension much larger than the others (that can be considered as homogeneous). Burgers' equation is usually studied as the first model of a non-linear hyperbolic system, before introducing more complex models consisting in systems of PDEs to describe "more physical" systems, like compressible flows or shallow water

  • how are you prescribing $sin(x)$ initial condition?

  • if you want to see shocks in shallow water you just need to turn on the tap and have a look at the water in the sink. If the jet is strong enough you could see stationary water jumps (the analogous of shock waves) in a system governed by the hyperbolic PDE system of shallow water equations.

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  • $\begingroup$ Thanks. If you use characteristics theory and perturbation method to solve the shallow water equations, you get as a reduced system of equations two Burger equations in case of flat orography. I assumed, it is reasonable to assume periodic initial conditions for a gravity wave. Why is it wrong? $\endgroup$
    – user996159
    Commented May 26 at 15:03
  • $\begingroup$ @user996159 just out of curiosity, could you provide a link where it is done? $\endgroup$
    – Syrocco
    Commented May 26 at 15:09
  • $\begingroup$ @user996159 you can usually diagonalize a hyperbolic system and find a diagonal form in a set of transformed variables that are a combination of the original ones, that are likely to be not so easily interpretable, by just looking at the physical system $\endgroup$
    – basics
    Commented May 27 at 7:20
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Your shock waves can be interpreted as breaking waves which would require new physics (foam etc.). In the context of shallow water waves you can also modify your equations of motion for accuracy. The simplest approach is to simply add a viscous term which regularizes the shocks. Another common approach is to add a dispersion term which helps counteract the shock by spreading the wave packet. Up to rescaling, you recover the Korteweg-De Vries equation: $$ \partial_t\phi+\partial_x^3\phi+6\phi\partial_x\phi =0 $$ It has soliton solutions where both effects balance out. Your sinusoidal initial condition for the KdV equation was actually investigated numerically by Zabisky and Kruskal in 1965. The result is even animated in the corresponding wikipedia article: Korteweg-De Vries equation.

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