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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.

A quasilinear system is hyperbolic where its principal coefficient matrix has real eigenvalues and enough independent eigenvectors. The eigenvalues are its characteristic speeds.

Define the concentration-dependent reduced gravityUnder the Boussinesq approximation, the three shallow water equations can be written in advective form asThe coefficient matrix of this quasilinear system has characteristic speedsThus this is a hyperbolic system when $g^{\prime }h>0$. Along the intermediate characteristic curve $dx/dt=u$, the concentration obeysIt decreases because ambient entrainment dilutes the chemical; detrainment does not change the concentration of a well-mixed parcel.

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.