$1.(10$ points$)$ Write down the Rankine$-$Hugoniot relations for a two$-$dimensional shock wave with a constant radius of curvature in one direction.

$2.(25$ points$)$ Answer the following questions:

$($a$)$ Find the characteristic line equation$($s$)$ of the one$-$dimensional wave equation.

$($b$)$ Given the coordinates $($with the origin at the middle of the throat$),$ Mach numbers, and wave angles of two interior points inside the divergent section of a choked nozzle:

Find the location of point $3$ and the corresponding wave angle $\backslash ((\backslash$theta$)\backslash )$ and Mach number $($M$).$

$($c$)$ The Mach number and the wave angle at the point $\backslash ((2\mathrm{.}0,2\mathrm{.}0)\backslash )$ are $2\mathrm{.}0$ and $6$ degrees, respectively. Find the location of the point on the wall, which is the intersection of the characteristic line through the point and the wall. Also, find the Mach number and the wave angle at that point.

The function of the geometry can be given as follows:

$\backslash [$

y$=$x$^\{0\mathrm{.}5\}$

$\backslash ]$

Hint: Assume a point on the wall, and iterate the location with the input of the slope of the characteristic line.