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SetThe scalar conservation law is $u_t+a(t)f^{\prime }(u)u_x=0$. Its characteristic curve issuing from $\xi$ satisfiesThe Jacobian of the one-dimensional characteristic map isBefore characteristic crossing, differentiation with respect to $\xi$ givesBecause $u_0$ has compact support, $m$ is continuous and vanishes outside a compact set. It therefore attains its minimumby the hypothesis. Since $a(t)>\epsilon$, the function $A$ is strictly increasing and tends to infinity. There is consequently a unique first time $T_{\ast }>0$ satisfyingAt a minimizer of $m$, the numerator $u_0^{\prime }$ is nonzero because $f^{\prime \prime }$ is nonzero, while the denominator tends to zero as $t\uparrow T_{\ast }$. Hence the classical solution has gradient blow-up:

A rectangular lock of length ${\ell }_0$ and depth $h_0$ has rarefaction head speed $-c_0$, where $c_0=\sqrt{g^{\prime }{h}_0}$. It reaches a closed rear wall at $t_r={\ell }_0/c_0$. The first reflected positive characteristic curve follows $x={\ell }_0+2c_{0}t-3c_0{t}_r^{2/3}{t}^{1/3}$ inside the undisturbed rarefaction, then crosses the uniform shelf and reaches the front at the displayed time. This marks the end of the initial constant-speed front stage in the ideal fixed-front-Froude model.

A shallow gravity current is supercritical when its Froude number exceeds one: both gravity-wave characteristic curves move downstream relative to the bed.

A surface gravity wave breaks when its smooth free-surface motion becomes unstable and overturns or forms a strongly dissipative front. In the nondispersive shallow-water approximation, intersecting compressive characteristic curves signal failure of a smooth simple wave; a hydraulic bore replaces it at the integral-model level. Dispersion can instead produce an oscillatory bore, while physical breaking involves turbulence and energy loss beyond the inviscid equations.

A transport equation carries a quantity along the flow of a velocity field $b$. For a smooth solution of $u_t+b\cdot \nabla u=0$, the value of $u$ is constant along each characteristic curve. With constant speed $c$, the chain rule gives $\frac{d}{dt}u(x_0+ct,t)=u_t+c{u}_x=0$, hence $u(x,t)=g(x-ct)$. This is exact translation: it preserves the shape rather than smoothing it, as the periodic heat and transport comparison shows.

For a one-dimensional wave equation whose principal symbol is ${\xi }_t^2-c(y{)}^2{\xi }_y^2$, a travel-time coordinate isIts two families of characteristic curves are $s(y)-t=\text{constant}$ and $s(y)+t=\text{constant}$ wherever the wave speed $c$ is positive.

In a channel with width $2\lambda xz$ at height $z$, a well-mixed particle-laden gravity current of depth $h$ occupies area $\lambda x{h}^2$. Its volume conservation, momentum and particle deposition flux balances givewhere the reduced gravity is $g^{\prime }=d\varphi$. The $h/3$ coefficient is the cross-section-weighted mean of $h-z$ in the hydrostatic pressure force. The characteristic curves have speeds $u$ and $u\pm \sqrt{g^{\prime }h/2}$.

A vertically mixed gravity current with horizontally varying reduced gravity $b$ obeys $h_t+(hu{)}_x=0$ and $(hu{)}_t+(h{u}^2+b{h}^2/2{)}_x=0$. Consequently the material acceleration is $Du/Dt=-b{h}_x-h{b}_x/2$. Advected density variables add repeated contact characteristic curves of speed $u$ to the gravity-wave characteristic curves of speeds $u\pm \sqrt{bh}$.

For a rectangular channel with stationary width $b(x)$ and constant reduced gravity $g^{\prime }$, volume conservation and momentum conservation give $h_t+u{h}_x+h{u}_x=-hu{b}^{\prime }/b$ and $u_t+u{u}_x+g^{\prime }{h}_x=0$. With $c=\sqrt{g^{\prime }h}$ the characteristic curves have speeds $u\pm c$. The combinations $u\pm 2c$ obey the displayed compatibility equations; they are Riemann invariants only when the width source term vanishes.