@Rouben Rostamian  Well in the first case we get a rarefaction wave that looks like the diagram below using your code.

However, I am having trouble plotting shock waves for the second case since I'm not sure how to implement it using the code that you have given.

@Rouben Rostamian  Of course, did not see the comma there, thanks. One more question, if I wanted to show shock waves on each plot how would I go about implementing this?

@Rouben Rostamian  Thanks for your reply. I was also wondering if there is a way to have two sets of graphs; 1 set containing only the characteristics and another set containing both characteristics and rarefaction waves as you have above. I tried using your code to only get the characteristics and leave out the rarefaction but it didn't work. any advise?

@Rouben Rostamian  I presume the lines represented by cyan and magenta colours are the rarefaction waves?

@tomleslie What you have described and said is exactly what I'm trying to solve. To make it easier to understand, I have attached the problem sheet. The reason why I included the $ signs were because I thought it would read it as TeX does, but have realised that the site doesn't.