Utility of Burger's equation in the study of shock waves

Question

Consider the viscous Burger's equation,

$$ \frac{\partial u}{\partial t}+u \frac{\partial u}{\partial x}-\nu \frac{\partial^{2} u}{\partial x^{2}}=0, \tag{1} $$

with $\nu>0$ the kinematic viscosity. If we look for a traveling wave solution, i.e., a solution of the form $u(x, t)=u(x-c t)=u(\xi)$, where $c$ is the speed of the traveling wave, then we find that it is given by the following equation (as proved here), $$ u(x, t)=c-\frac{1}{2}\left(u_{1}-u_{2}\right) \tanh \left[\frac{1}{4 \nu}\left(u_{1}-u_{2}\right)(x-c t)\right], \tag{2} $$

where $u_1$ and $u_2$ are two constants representing the height of the solution $u(\xi)$ for $\xi \rightarrow - \infty$ and $\xi \rightarrow + \infty$, next to next. It is said that this solution represents a shock wave, since it joins the asymptotic states $u_1$ and $u_2$ and, for low values of $\nu$, it allows discontinuities to form. It is said that the viscous Burgers' equation is a good model to investigate the behavior of shock waves.

However, besides the fact that it admits the solution $(2)$ in the form of a shock wave, what else could we say in this regard? Does the Burgers equation have any other application to the study of shock waves? Would the above solution have any physical significance (could it perhaps represent a wave along a pipe or something similar)?

Answer 1

The Burgers equation naturally arises in the study of high-amplitude acoustic waves in one dimension (e.g., in a pipe). In that context, it is usually written as \begin{equation} \frac{\partial p}{\partial x} - \frac{\delta}{c_0^3}\frac{\partial^2p}{\partial t^2} = \frac{\beta}{\rho_0c_0^3}p\frac{\partial p}{\partial t}, \end{equation} where $p$ is the acoustic pressure, $x$ is the position, $t$ is the time, $c_0$ is the small-signal wave speed (not the speed of the actual wave or a shock), $\delta$ is the diffusivity (related to the kinematic viscosity), $\rho_0$ is the ambient mass density, and $\beta$ is the coefficient of nonlinearity (a property of the fluid).

The Burgers equation predicts the propagation of any source waveform propagating in a thermoviscous fluid (away from boundaries, so it is not strictly valid for pipes, but there are work-arounds to account for boundary-layer losses). If shocks do not exist in the source waveform but the amplitude is sufficiently high, the Burgers equation models the waveform distortion, steepening, and shock formation. You can see an animation of this behavior on the Wikipedia article.

An analytical solution is available through the Hopf-Cole transformation, but this is difficult to visualize. If you take the limit that either $p\rightarrow\infty$ or $\delta\rightarrow0$, then you approach the inviscid Burgers equation, which has a much nicer implicit solution: \begin{equation} p = f(\phi), \hspace{10mm} \phi = t-\frac{x}{c_0} + \frac{\beta x p}{\rho_0c_0^3}. \end{equation} This solution is called the Earnshaw solution (or possibly the Poisson solution; I am not certain I understand the history at this point), and it may be interpreted as saying the wave propagates at a speed modified by its amplitude. At some point the Earnshaw solution predicts a wave will become multivalued, or have multiple values of the pressure at a single location. If this equation is supplemented by weak shock theory, then the multivalued points may be accounted for and predict the presence and propagation of shock waves.

I hope this discussion is helpful. I believe that the Burgers equation has more applications, but this is the one I am most familiar with. I have used it extensively in acoustical modeling for nearly a decade now. If you would like more information about this application, I would refer you to a textbook. There are many, but my favorite is "Nonlinear Acoustics" by Hamilton and Blackstock.