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The flood wave in a river follows the conservation law
$$ A_t + (A^{3/2})_x = 0 $$
where $A(x,t)$ is the cross sectional area of the water at the location $x$ and time $t$. A sudden heavy rain creates a flood with river cross sectional area along its path as follows (set it as $t=0$)
$$ A(x,0) = leftlbracebegin{aligned}
&1 && xleq 0 \
&4 && 0 < x leq 10\
&1 && x> 10
end{aligned}right. . $$

(a) Find and draw the characteristics of the equation.

(b) Find the solution of cross sectional area along the river at $t=1$.

(c) Assume a town is located at $x=31$, when will the flood crest reach it?

TRY:

First, we write the PDE as $A_t + frac{3}{2} A^{1/2} A_x = 0 $

Now, solving this equation using method of characteristiscs, we obtain

$$ x = frac{3}{2} sqrt{ A(r,0) } t + r $$

for characteristics equation and solution is implicity given by

$$ A(x,t) = phi ( x - 3/2 sqrt{ A } s ) $$

where $phi(x) = A(x,0) $. So, we have characteristic are described by

$$ x = begin{cases} 3/2 t + r,& r leq 0 \ 3t + r,& 0 < r leq 10 \ 3/2t+r,& r > 10 end{cases} $$

am I correct so far?

Here is a plot of the characteristic curves in the $x$-$t$ plane as deduced from the initial data:

The flux $A mapsto A^{3/2}$ is convex for positive cross sectional area $A$. Hence, the classical theory for weak entropy solutions of conservation laws applies.
One can observe that

• At $x=0$, characteristics separate. A rarefaction wave is created, which waveform $v(x/t)$ is given by the reciprocal of the function $A mapsto frac{3}{2}sqrt{A}$, i.e., $v(x/t) = frac{4}{9}(x/t)^2$.




• At $x=10$, characteristics intersect. According to the Lax entropy condition, a shock wave is created, which speed $s$ satisfies the Rankine-Hugoniot condition $s = {(4^{3/2}-1^{3/2})}/{(4-1)}= frac{7}{3}$.

Therefore, the entropy solution for small positive times reads
$$
A(x,t) = leftlbrace
begin{aligned}
& 1 && x leq tfrac{3}{2} t \
& left(tfrac{2x}{3t}right)^2 && tfrac{3}{2} t leq x leq 3 t\
& 4 && 3 t leq x leq 10 + tfrac{7}{3} t \
& 1 && x geq 10 + tfrac{7}{3} t
end{aligned}
right.
$$
which is valid at $t=1$. Indeed, the rarefaction wave does not catch the shock before the time $t^*$ such that $3 t^* = 10 + tfrac{7}{3} t^*$, i.e., $t^* = 15$. The shock wave will reach $x=31$ at the time $21/frac{7}{3} = 9$ at which the previous solution is still valid ($t < t^*$).