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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.2011-12-11
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    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 $|x|\gt 1$.2011-12-11
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    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](http://math.stackexchange.com/questions/88443) and [here](http://math.stackexchange.com/questions/88825).) 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.2011-12-11
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    Sorry I thought I mentioned it but the wikipedia isn't really useful. Not for me anyways. All the information thats on wikipedia is in my book (ie, PDE by haberman and strauss). Wikipedia and other resource sites are my first priority. The reason i've asked three times now is because I am not understanding it. In the first question I didn't get an answer. No answer in the second one either.2011-12-11
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    I can only speak for myself, but I suspect that others may have felt the same and that this explains why you didn't get an answer on a site where people are usually rather quick and keen to answer and there is probably no shortage of people who understand the method of characteristics. You didn't get an answer (at least from me) because you ignored the comments. The spirit of this site is very interactive, and if you don't respond to people, it looks as if you're just trying to grab answers from them without bothering to interact with them.2011-12-11
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    Also, when people see that someone (in the case of your second question, two different people, both with high reputation) has responded to your question, they'll think that the question is already being dealt with and will probably prefer to spend their time on other questions that haven't gotten a response yet. That makes it even more important to interact with the people who have already taken an interest in your question. My comments implied that I was willing to help if you pointed out what you didn't understand, and you didn't take up that offer.2011-12-11
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    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.)2011-12-11
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    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](http://williewong.wordpress.com/2011/05/12/decay-of-waves-iia-minkowski-background-homogeneous-case/) and many other methods. Is your question that you don't see how he arrived at the d'Alembert formula?2011-12-11

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