Characteristic speed (${\lambda }_j$)

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.

A kinematic wave is a disturbance whose propagation speed is determined by the local relation between a conserved density and its flux. In a scalar conservation law $u_t+f(u{)}_x=0$, smooth values travel at speed $f^{\prime }(u)$.

Let $H(t)={\int }_0^{t}E(r)dr$, increasing until $t^{\ast }$ and decreasing afterwards. For dry initial data and zero upstream depth, a characteristic emitted at time $\tau$ has $h=H(t)-H(\tau )$ and position $s=\frac{5}{3}{\int }_{\tau }^t[H(r)-H(\tau ){]}^{2/3}dr$, for as long as its depth stays nonnegative. The height follows by integrating the source along a characteristic, and the position follows from the derivative of the Manning flux.

When $E<0$ and $H(t)>0$, define the earlier emission time ${\tau }_d(t)\le t^{\ast }$ by $H({\tau }_d)=H(t)$. The front where this water is just exhausted hasIt moves downstream because $s_d^{\prime }=-(10/9)E(t){\int }_{{\tau }_d}^t[H(r)-H(t){]}^{-1/3}dr>0$. Behind it depth is zero; ahead, unexhausted characteristic labels determine the wet profile. This construction presumes removal limited by water availability: continuing a negative source unchanged at zero depth would violate nonnegativity. If $H$ reaches zero, the remaining initially generated water has also vanished.

With constant lateral supply $M$, a discharge law of the displayed form gives equilibrium area $[(m+1)(Q_{\mathrm{in}}+Ms)/c{]}^{1/(m+1)}$ and characteristic speed $c{A}^m$. Under a Chezy law, a wetted-perimeter relation $P_b\propto A^q$ gives $m=(1-q)/2$. Small area perturbations are transported downstream with amplitude inversely proportional to this characteristic speed; finite compressive disturbances can still form shocks because the flux is convex.