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The one-dimensional open-channel shallow water equations express conservation of water volume and depth-averaged momentum. Here $h$ is water depth, $u$ the mean velocity, $\eta$ the free-surface elevation and ${\tau }_b$ the resisting bed shear stress. The elevation includes the bed profile, so its gradient already includes gravitational driving by bed slope. A constant volume flux in space and time forces $h_t=0$ through the first equation; an unsteady reduction that imposes that flux while retaining momentum relaxation is consequently an approximation to this full system.

Shallow-water potential vorticity is materially conserved by inviscid, unforced shallow water equations. Linearizing about rest with depth $H_0$ gives an anomaly proportional to

For the dimensionless shallow water equations with gravity coefficient one, let the left state be $(h,u)$, $h>0$, and the right state $(1,0)$. The Rankine-Hugoniot conditions are $s(h-1)=hu$ and $shu=h{u}^2+(h^2-1)/2$. For $h\ne 1$ and $s>0$, eliminating $u$ gives the displayed formulas. The characteristic speeds are $u\pm \sqrt{h}$ and $\pm 1$. Because $u-\sqrt{h}<s$ always, the positive branch can only be an admissible family-plus shock. Its remaining Lax shock admissibility conditions, $1<s<u+\sqrt{h}$, both reduce to $h>1$. At $h=1$ there is no jump and the labelled interface has no determined shock speed.

For frictional wide-channel shallow water equations, a smooth increasing wave connecting $(h,u)=(1,1)$ to water depth $H>1$ and moving upstream has $c=(H^{3/2}+1)/(H-1)$ and $h^{\prime }=[h^3-(1+c-ch{)}^2]/[h^3-(1+c{)}^2{F}^2]$. Positivity of the denominator at the upstream endpoint requires $F<1/(1+c)$. For the numerator put $r=\sqrt{h}$, $R=\sqrt{H}$: its nontrivial factor is $(r+1)(r-R)(r-R/(R-1))$. The endpoint at $H$ is the first positive downstream zero exactly when $R\le 2$, giving the amplitude restriction. At $H=4$ that zero is double; otherwise the profile cannot cross an intervening equilibrium.

An incoming tide can form a moving surge that propagates upstream against a river's current. A discontinuous idealization is a shock wave of the shallow water equations, while weaker or dispersive bores can have an undular wave train. A boundary overtaking upstream-propagating characteristic curves warns that a smoothly attached quasi-steady description may fail; the appropriate discontinuity must satisfy the conservative jump conditions of the chosen model.

For inviscid nonrotating shallow water equations, columnar horizontal flow has material vorticity equation $D\zeta /Dt=-\zeta {\nabla }_h\cdot {\mathbf{u}}_h$ and depth equation $Dh/Dt=-h{\nabla }_h\cdot {\mathbf{u}}_h$. Thus $\zeta /h$ is conserved, not $\zeta$ itself. Squashing a column to much smaller depth decreases its vertical vorticity. This permits a late draining flow to approximate an irrotational $u_{\theta }=C/r$ profile outside the drain, despite nonzero initial solid-body vorticity. At finite nonzero depth the residual vorticity need not be exactly zero. In a rotating frame, replace $\zeta$ by absolute vorticity $\zeta +f$ in the conserved ratio. Kelvin's circulation theorem remains compatible with these changes because it follows a material loop, not a fixed spatial circle.

A flat pool of depth $h_0$ between a wall at $-a$ and a removed dam at zero initially develops the shallow water equations rarefaction $u=2(c_0+x/t)/3$, $c=(2c_0-x/t)/3$, where $c_0=\sqrt{g{h}_0}$. It has $Q_{+}=u+2c=2c_0$ and front $x=2c_{0}t$. Its right-running characteristics satisfy $x_{+}=2c_{0}t+C{t}^{1/3}$, obtained by integrating $x_{+}^{\prime }-x_{+}/(3t)=4c_0/3$.

The first rarefaction signal reaches the wall at $(-a,a/c_0)$. The right-running characteristic through that event is the displayed curve. To its right the original fan remains valid; to its left reflected wall data alter the invariant. Continuing the original fan to the wall after that time would give nonzero wall velocity, contradicting impermeability. This is a concrete domain of dependence calculation for a nonlinear hyperbolic system.

Two smooth differential systems related by a nonlinear change of dependent variables need not define the same weak solutions. If $U_t+F(U{)}_x=0$, a smooth function $V=g(U)$ satisfies $V_t+g^{\prime }(U)F^{\prime }(U)U_x=0$ for smooth $U$; even when this can be written as another divergence, its Rankine-Hugoniot conditions need not be equivalent to those in $U$. The smooth chain rule does not justify multiplying jump distributions by discontinuous coefficients.

For example the shallow water equations can be written smoothly as $h_t+(hv{)}_x=0$ and $v_t+(h+v^2/2{)}_x=0$. Mechanical mass and momentum conservation instead select $h_t+(hv{)}_x=0$ and $(hv{)}_t+(h{v}^2+h^2/2{)}_x=0$. These systems agree for smooth positive $h$, but their jump laws differ. The states $(h,v)=(2,\sqrt{3}/2)$ and $(1,0)$ satisfy the mass/momentum jump laws with speed $\sqrt{3}$, while the velocity-form jump equation would demand $11/(4\sqrt{3})$. A model for discontinuities must specify the conservative variables or an equivalent admissible regularization, rather than using smooth equivalence alone.