= Solution
The <Rankine-Hugoniot conditions> follow by integrating <conservation laws> across a thin moving layer, then letting its macroscopic thickness tend to zero. For example, normal <momentum conservation> in one dimension has the conservative form
$$
\partial_t(\rho u)+\partial_x(\rho u^2+p)=0.
$$
For a <shock wave> moving at speed $V$, integrate over a control interval whose endpoints move with the discontinuity. The <Reynolds transport theorem> gives
$$
\boxed{V[\rho u]=[\rho u^2+p],}
$$
where $[f]=f_R-f_L$ denotes the difference of limiting states. <Mass conservation> similarly gives $V[\rho]=[\rho u]$. If $v=u-V$ is the normal <velocity> in the <shock frame>, combining the two relations yields
$$
\boxed{[\rho v]=0,\qquad[p+\rho v^2]=0.}
$$
The same construction applied to <conservation of energy>, for a non-radiating <normal shock wave> with nonzero mass flux, gives $[h+v^2/2]=0$, where $h$ is the <specific enthalpy>. An <equation of state> closes these relations, and the physical compression branch has increasing <entropy>.
Finite <dynamic viscosity> replaces the ideal discontinuity by a thin smooth transition. A viscous <stress> contributes a derivative to the momentum flux inside the layer and converts directed <kinetic energy> into internal <energy>. If its gradients vanish in the uniform far states, the same limiting <Rankine-Hugoniot conditions> still hold there. Balancing nonlinear transport with viscous diffusion gives a thickness of order $\nu/|u_L-u_R|$ in the <viscous Burgers equation>, where $\nu$ is the <kinematic viscosity>.
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