Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 79 / 2

At a fixed time, the complete spatial derivative isThe first term describes geometric variation along the channel at a fixed surface elevation. The second describes the extra wetted area produced by a varying water level; $S_h$ is the surface width.

Assume a fixed channel, negligible lateral inflow, uniform cross-sectional velocity, incompressible water and the hydrostatic long-wave approximation. Integrating volume conservation over a short channel segment givesInviscid horizontal momentum, with $h$ measured from the fixed datum, is $u_t+u{u}_x+g{h}_x=0$. The coefficient matrix acting on $(h_x,u_x)$ has eigenvaluesThese are the variable-section channel characteristics. Put $D_{\pm }={\partial }_t+(u\pm c){\partial }_x$. Adding the corresponding linear combinations of the two equations givesThe opposite signs in the two combinations cancel both derivative terms in the source calculation.

If $c=c(h)$, then $D_{\pm }c=c_h{D}_{\pm }h$ and $d{c}^2/dh=2c{c}_h$. Taking $\xi =t$ along a characteristic gives the requested formA sufficient geometry assumption is $S(x,h)=b(x)A(h)$, for which $S/S_h=A/A^{\prime }$ is independent of $x$ at fixed height.

For a completely general $S(x,h)$, the printed form omits an explicit wave-speed geometry term. The correct conversion is insteadFor example, a lake at rest with constant $h$ but position-dependent $c$ has $D_{\pm }c=\pm c{c}_x$. The printed left side is then nonzero while its right side is zero. The $h$-based compatibility equation remains valid, including when ${\partial }_h(c^2)=0$ makes the speed-based expression singular.

Linearize about water at rest with uniform surface elevation $h_0$. Write $h=h_0+\eta$, $u=v$, $S_0=S(x,h_0)$ and $T_0=S_h(x,h_0)$. To first order the shallow water equations becomeDifferentiate continuity in time and eliminate $v_t$:For a time-harmonic standing disturbance, use $\eta (x,t)=\mathrm{Re}[\eta (x)e^{-i\omega t}]$. ThenDividing by $S_0$ provesThis identifies the coefficient as the local long-wave wavenumber. It is constant under the fixed-shape geometry used in part (a), when $c_0$ is independent of $x$; that assumption is needed for the constant-coefficient mode calculation in part (c).

Assume $k$ is constant, as for constant resting long-wave speed. With $S_0^{\prime }/S_0=\beta$, put $\eta =e^{-\beta x/2}Y$. The amplitude equation becomesIf $k^2>{\beta }^2/4$, write $\mu =\sqrt{k^2-{\beta }^2/4}$: the solution is an oscillatory sine-cosine combination multiplied by $e^{-\beta x/2}$. If $k^2={\beta }^2/4$, it is $e^{-\beta x/2}(A+Bx)$. If $k^2<{\beta }^2/4$, it is a combination of real exponentials, $e^{-\beta x/2}(A{e}^{\gamma x}+B{e}^{-\gamma x})$, where $\gamma =\sqrt{{\beta }^2/4-k^2}$. These classify the local spatial behavior.

For a nonzero-frequency mode, zero velocity at both dams and $v_t=-g{\eta }_x$ impose ${\eta }^{\prime }(0)={\eta }^{\prime }(L)=0$. In the oscillatory case, the first condition givesIts derivative at $L$ is $-A{k}^2{e}^{-\beta L/2}\sin (\mu L)/\mu$. Hence the standing modes in an exponentially tapered channel haveThe hyperbolic and repeated-root cases provide no additional nonzero-frequency mode satisfying both end conditions. There is a separate zero-frequency uniform-level mode $\eta =\mathrm{constant}$, corresponding to a change of the basin's conserved water volume.

A failed dam launches a rarefaction wave back into the reservoir and an advancing current into the initially dry downstream channel. Take the failed dam as $x=0$, initial depth $h_0$ and zero velocity for $x<0$, and a dry bed for $x>0$. Locally assume a flat, prismatic triangular section, $S=C{h}^2$. The early flow is governed byIts Riemann invariants are $u\pm 4c$, since $\int (g/c)dh=4c$. In the left-running simple wave, $u+4c=4c_0$, where $c_0=\sqrt{g{h}_0/2}$, and the expanding fan has $x/t=u-c$. Solving givesThus the triangular-channel dry-bed dam break profile isThe reservoir-side edge moves at $-c_0$, and the dry front moves atVelocity is zero in the undisturbed reservoir and has the fan value above where water is present. This approximation applies before the rarefaction reaches the surviving dam and before channel variation, friction or finite-volume reflections substantially modify the flow. If water is already present downstream, a bore replaces the dry-bed front and requires a different matching problem.