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Linearizing the shallow water equations about depth $h$ at rest gives wave speed $c=\sqrt{gh}$. Replacing $h$ by $c$ in the nonlinear equations yields $c_t+u{c}_x+(c/2)u_x=0$ and $u_t+u{u}_x+2c{c}_x=0$. Direct addition and subtraction giveThus the Riemann invariants $J_{\pm }=u\pm 2(c-c_0)$ are constant on the characteristics $C_{\pm }$ with speeds $u\pm c$. In this problem $c_0=\sqrt{g{h}_0}$.

For isentropic gas with $P=K{\rho }^{\gamma }$, $\gamma >1$, the sound speed obeys $c^2=P^{\prime }(\rho )$. The one-dimensional mass and momentum equations areSince $c\propto {\rho }^{(\gamma -1)/2}$, these implyAdding and subtracting yields the Riemann invariantsA simple wave has one invariant constant throughout. For a right-moving wave into a resting state with sound speed $c_0$, $R_{-}=0$, so $c=c_0+(\gamma -1)u/2$ and the right characteristic speed is $c_0+bu$, $\boxed{b=(\gamma +1)/2}$.

For the damped scalar equation, characteristics from $\xi$ satisfyAssume smooth bounded initial data with bounded derivative, and the stated maximal compressive slope exists. ThenFor the damping threshold for a simple wave, if $m={\max }_{\xi }[-u_0^{\prime }(\xi )]>0$, put $\boxed{{\alpha }_c=bm}$. For $0<\alpha <{\alpha }_c$, a gradient blow-up first occurs atFor $\alpha >{\alpha }_c$, $J$ stays uniformly positive. At $\alpha ={\alpha }_c$ it still satisfies $J\ge e^{-\alpha t}>0$ for every finite time. At a point attaining the minimum slope, $u_x=u_0^{\prime }$ for all time, rather than blowing up. Hence the precise conclusion is no finite-time shock occurs when $\alpha \ge {\alpha }_c$, not the printed strict “if and only if $\alpha >{\alpha }_c$”. Equality only makes the limiting characteristic map degenerate as $t\to \infty$. The profile $u_0(\xi )=-\sin \xi$ directly verifies this endpoint distinction. If there is no compressive slope, take ${\alpha }_c=0$; no finite-time shock occurs for positive damping. A nonattained maximal slope is interpreted with a supremum and the corresponding infimum of breakdown times. The gas interpretation also requires $c_0+(\gamma -1)u/2>0$.

The one-dimensional Isentropic Euler equations areFor the polytropic equation of state,The continuity equation therefore implieswhile the momentum equation becomesAdding and subtracting these equations givesThus the Riemann invariantsare constant along the characteristic curves

For one-dimensional homentropic flow,and $dp=c^{2}d\rho$. For a perfect gas, $dc/c=(\gamma -1)d\rho /(2\rho )$. Combining the two equations therefore shows that the Riemann invariantssatisfyTheir characteristic curves are parametrized byand $R_{\pm }$ is constant along the corresponding curve until characteristic intersection creates a shock.

Mass and momentum conservation areFor homentropic ideal-gas flow, $c^2=dp/d\rho$ andCombining the two conservation equations then givesSubtracting the harmless constant $\pm 2c_0/(\gamma -1)$ proves that each Riemann invariantis constant on $C_{\pm }$, where $dx/dt=u\pm c$.

Put $c=\sqrt{g^{\prime }h}$. The three real eigenvalues areso the system is strictly hyperbolic for $g^{\prime }h>0$. Along $dx/dt=u$,Left eigenvectors for ${\lambda }_{\pm }$ are $(\pm h/(2c),\pm c/h,1)$, hencewhere $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$.

When $w_e,w_d\to 0$, reduced gravity is materially conserved. If it is initially uniform, it remains constant and the other two relations integrate to the standard Riemann invariants

For the prismatic triangular-channel shallow water equations, $c^2=G\varphi h/2$. The characteristic speeds are $u,u\pm c$. Along $dx/dt=u$, $d\varphi /dt=-2w\varphi /h$. Along $dx/dt=u\pm c$, the hyperbolic system has compatibility relationThe concentration differential cannot generally be discarded to obtain Riemann invariants $u\pm 4c$. Positive depth and concentration are required for this nondegenerate form.

A simple wave is a region of a hyperbolic flow in which one Riemann invariant is constant throughout, leaving the other invariant to determine all fluid variables.