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A characteristic speed ${\lambda }_j(U)$ is an eigenvalue of the flux Jacobian $A(U)=DF(U)$ for a system of conservation laws. In a classical solution of $U_t+A(U)U_x=0$, let ${\ell }_j(U)$ be a left eigenvector. Along $\dot{x}={\lambda }_j(U(t,x))$, the chain rule givesFor a constant matrix, $w_j={\ell }_j\cdot U$ therefore obeys $(w_j{)}_t+{\lambda }_j(w_j{)}_x=0$. For example, $A=c\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ gives $w_{+}=U_1+U_2$ moving at $c$ and $w_{-}=U_1-U_2$ moving at $-c$.

For a nonlinear system, ${\ell }_j(U)\cdot dU$ need not be the differential of a scalar function, even after multiplication by a nonzero scalar factor. A scalar Riemann invariant requires such an additional integrability property. Consequently the projected identity alone does not supply a globally defined transported scalar for every wave family.

The flux Jacobian is the Jacobian matrix of the flux in a system of conservation laws. Its eigenvalues are the local characteristic speeds, and its left and right eigenvectors select the corresponding wave variables and directions in state space.

For a strictly hyperbolic system of conservation laws $U_t+F(U{)}_x=0$ with ordered characteristic speeds ${\lambda }_1<\cdots <{\lambda }_n$, a Lax family-$j$ shock has ${\lambda }_j(U_R)<s<{\lambda }_j(U_L)$. The remaining families must stay on the same side of $s$: ${\lambda }_{j-1}(U_L)<s$ and $s<{\lambda }_{j+1}(U_R)$ where those neighbors exist. These inequalities imply that the lower families have speed below $s$ on both sides and the higher families speed above $s$ on both sides. There are $n+1$ incoming characteristic directions: two from the compressed family and one from each other family. The Rankine-Hugoniot conditions supply the jump balance, while these inequalities supply admissibility. For a convex scalar flux this reduces to the entropy shock condition.

For a strictly hyperbolic two-component system of conservation laws with ordered speeds ${\lambda }_{-}<{\lambda }_{+}$, a compressive family-$j$ shock has its family characteristics entering from both sides: ${\lambda }_j(U_R)<s<{\lambda }_j(U_L)$. The other family crosses the interface with the same sign of relative speed on both sides. These are the Lax shock admissibility conditions and give three incoming characteristic directions for two conservation laws.

In the present problem,Since $u_1=s-s/h_1$, we always have $u_1-\sqrt{h_1}<s$. Hence a positive-speed shock of this form cannot be a family-minus shock. For a family-plus shock the remaining strict inequalities areThe first is equivalent to $h_1(h_1+1)>2$, or $h_1>1$. The second is equivalent to $s<h_1^{3/2}$, whose square reduces to $h_1+1<2h_1^2$, again equivalent to $h_1>1$ for positive $h_1$. The minus speed on the right is $-1<s$ automatically. Thus all causal inequalities hold exactly whenDepth increases behind the right-moving shock. For $0<h_1<1$, the algebraic jump solution has the wrong characteristic orientation and is an inadmissible expansion shock; equality is the no-jump limit.

Local mass conservation of each particle species gives the system of conservation lawsFor $\mathbf{\varphi }=({\varphi }_1,{\varphi }_2{)}^T$ and flux $F_i={\varphi }_i{W}_i$, the flux Jacobian isWriting its entries as $A_{ij}$, the two characteristic speeds are its eigenvaluesThe discriminant is nonnegative because $A_{12}{A}_{21}={\varphi }_1{\varphi }_2{\hat{W}}_1{\hat{W}}_2\ge 0$, so positive concentrations of two settling species give real characteristics.

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.

A Riemann problem for a one-dimensional system of conservation laws ${\partial }_{t}U+{\partial }_{x}f(U)=0$ has two constant initial states, $U_L$ for $x<0$ and $U_R$ for $x>0$. The data and equation are invariant under $(x,t)\mapsto (\lambda x,\lambda t)$. When the physically admissible entropy solution is unique, this invariance makes it a self-similar solution depending on $x/t$. For ideal-gas compressible flow, the characteristic speeds are $u-c,u,u+c$. The two acoustic families carry shock waves or rarefaction waves, while the middle family carries a contact discontinuity. Sufficiently strong separating velocities can instead create a vacuum region.