Behavior of strings with displacement in the middle

[snickersnee]

Homework Statement

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I have a string that's infinite on one side, as well as a string that's infinite on both sides. The displacement at the center (x=0) is the same for both, as shown in the pictures.

Homework Equations

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I think the G term goes to 0 because there's no incoming wave but I'm not sure

The Attempt at a Solution

Here's my solution for the string that's infinite on one side, would this be any different if the string was infinite on both sides? Maybe not because multiplying by H(t) makes the function zero at negative values of x. I'd appreciate your comments.

I also need to graph the strings at a few representative times, and draw the characteristics diagram, so if someone has any suggestions regarding those steps it would be well appreciated. In particular, what times would be representative? And how would I go about drawing the characteristics diagram? Thanks in advance.

 

[Simon Bridge]

Try using words to describe what you expect would happen in each case.