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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.

Setwhere $s>0$. Applying first-order perturbation of a simple eigenvalue to the flux Jacobian, or expanding its quadratic formula directly, separates the characteristic that changes total concentration from the characteristic that changes composition:At $ϵ=0$, both species move with the common hindered velocity $\hat{W}(1-s)$. Summing their conservation laws givesso the total concentration is a nonlinear kinematic wave:Taking the ratio $c={\varphi }_1/s$ instead gives the composition wave in a bidisperse suspensionand henceThese are the requested leading-order ordinary differential equations along the two characteristic families.

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.

A system of conservation laws evolves a vector of conserved variables $U$ through a flux vector $F(U)$. For a smooth solution it has quasilinear form $U_t+A(U)U_x=0$, where $A(U)=DF(U)$ is the flux Jacobian.