Shock wave

A shock wave is a moving discontinuity of a weak solution to a conservation law. Its speed is fixed by the Rankine-Hugoniot condition, while an entropy condition selects the physically admissible direction of compression.

A viscous shock is a smooth travelling transition produced when positive diffusion resolves a shock of a conservation law. For $u_t+q(u{)}_x=ϵ[D(u)u_x{]}_x$ with $D>0$, an inner coordinate $X=(x-ct)/ϵ$ gives $D(U)U^{\prime }=q(U)-q(u_{-})-c(U-u_{-})$. The endpoint conditions impose the Rankine-Hugoniot speed. If $q$ is strictly concave, the right side is positive between its roots, so the admissible profile increases from its rear state to its front state. This supplies the entropy selection directly from a regularizing mechanism.

Two identical cold clouds approach at speeds $\pm V$ in their center-of-momentum frame. Reflection symmetry makes the uniformly compressed gas stationary. A shock moving outward at $S$ sees upstream speed $V+S$ and downstream speed $S$; their ratio equals the strong adiabatic compression $(\gamma +1)/(\gamma -1)$. Solving gives the displayed speed. For a monatomic gas $S=V/3$. Finite-cloud breakout ends the steady upstream supply and requires a subsequent boundary-release calculation.

A shock wave with significant radiation losses in its downstream evolution. Often a narrow dissipative jump initially heats the gas according to adiabatic Rankine-Hugoniot conditions, followed by a much longer radiative cooling layer. The gas's specific entropy increases at the jump and may then fall as radiation carries entropy away. If both distant states have equal temperature, the combined mass and momentum balances give an isothermal shock; the gas's energy flux is not conserved across the whole radiating region.

The distance a downstream fluid element travels while losing a substantial fraction of its thermal energy. Multiplying the immediate downstream velocity by a radiative gas cooling time gives an order-of-magnitude length. A resolved layer instead uses $\ell =\int u(t)dt$, because mass density, temperature and speed change during cooling. It is distinct from the microscopic dissipative thickness of the shock wave.

A blast wave is the expanding disturbance produced by localized rapid release of energy, initially led by a strong shock wave. An energy-conserving self-similar blast wave ties front speed to the mass swept up. Once the front is only weakly supersonic, a strong-shock approximation ceases to apply; a subsequent acoustic wave is not excluded by that transition.

For an explosion confined approximately to a thin atmospheric layer with column mass density $\Sigma =\int \rho dz$, an optimistic conserved-energy scaling is $E\sim \pi D^2\Sigma U^2$. The strong blast wave approximation becomes invalid near front speed $U\sim c_a$, where $c_a$ is the ambient speed of sound, giving the displayed distance scale. Its numerical coefficient depends on the flow profiles, geometry and heat capacity ratio. Vertical energy escape makes full confinement an optimistic approximation. The criterion estimates survival of a strong shock wave, rather than whether any later acoustic wave is detectable.

The Rankine-Hugoniot condition expresses conservation laws across a moving discontinuity of a weak solution. Across a discontinuity $x=\sigma t$ in $u_t+f(u{)}_x=0$, with left and right states $u_{-}$ and $u_{+}$, conservation requiresWriting $[u]=u_{+}-u_{-}$ and $[f(u)]=f(u_{+})-f(u_{-})$ gives $\sigma [u]=[f(u)]$. The same relation holds componentwise for a system of conservation laws; in compressible flow, conservation of mass, momentum and energy supplies the jump conditions for a shock wave. Integrating the weak solution identity over a thin space-time region crossing the discontinuity yields this balance between the swept-up conserved quantity and the difference of fluxes.

For a scalar equation with strictly convex flux and states $u_l>u_r$, the Rankine-Hugoniot condition determines a discontinuity speed strictly between the right and left characteristic speeds. Characteristic curves enter this compressive discontinuity from both sides. Its entropy flux for a scalar conservation law inequalities make it an entropy solution. It is the limit of a phase-normalized decreasing travelling wave in the vanishing viscosity approximation.