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I am having a lot of trouble understanding the method of characteristics to solve the wave equation.

In fact, I have a final exam tomorrow and I can't seem to get a question from a previous assignment. I know Math.SE isn't really meant for this kind of stuff but I am hoping someone would just briefly explain how my professor is getting the solution. I appreciate it.

Here is the problem: $$\frac{\partial^2 u}{\partial t^2} - 9 \frac{\partial^2 u}{\partial x^2} = 0$$ on the real axis (i.e., $-\infty < x < \infty$).

Here is the solution. It is a PDF to the professor's solution file.

The solution is on page 3 (question number 3).

I need to know how he's getting his solutions at different times.

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You have a error in the second term of your equation. – matt Dec 11 '11 at 5:09
It seems that the question is more than what you give; there is also a boundary condition $u(x,0)=0$ and $\frac{\partial u}{\partial t}(x,0) = 1$ if $|x|\leq 1$ and $0$ if $|x|\gt 1$. – Arturo Magidin Dec 11 '11 at 6:40
This is the third question you're asking on the method of characteristics. You never linked any of them to each other. (The other two are here and here.) In comments to each of the previous questions, I pointed you to the Wikipedia article and asked what part of that you don't understand. You never replied to any of those comments. I wonder why you think that others will take time to answer your questions if you don't bother to interact with their comments. – joriki Dec 11 '11 at 6:46
Regarding Wikipedia and your book, I can only repeat what I wrote in my previous comments: We don't have your book, but we have Wikipedia, and Wikipedia happens to have a fully worked out example that's almost identical to the one in one of your questions, so it would make a lot of sense if you told us what you don't understand in the Wikipedia article. (As a positive side-effect, that might enable us to improve the Wikipedia article, whereas we'll have a hard time improving your book.) – joriki Dec 11 '11 at 8:04
The solution your professor gave is via d'Alembert's integral formula. While it can be derived via the method of characteristics, one can also do it via a change of coordinates and many other methods. Is your question that you don't see how he arrived at the d'Alembert formula? – Willie Wong Dec 11 '11 at 14:07

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