I thought these problems were pretty routine, and then I came across one that I am not able to do.

Find the admissible solution to the equation for subject to the following initial conditions

My attempt: We see that and so

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Thus, initially we have two shock waves that should then combine to create a single shock wave. Using the Rankine-Hugoniot condition we have

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This doesn't make sense to me, because I expected the second shock to be in the opposite direction. I must be doing something wrong.

EDIT: so there is actually a rarefaction wave at . I am just having a hard time picturing this in my head now.

EDIT2: LOL, now I feel stupid!

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