Understand some points of Method of Characteristic and the solution of the Wave Equation.

by Greg   Last Updated September 22, 2018 15:20 PM

I started my first pde course. I don't have much experience with this yet.

I found an interesting question: Solution to one-dimensional Wave Equation with Method of Characteristics, but I didn't understand four points. I wish someone could help me.

  • First: In the Sy 1, he fixed the vectors tangents to the curve $\Gamma$?

  • Second: Can $\Gamma$ be any smooth curve?

  • Third: why $F(c_1,c_2) = 0$? who's $F$? I have no idea

  • Fourth: why $u = g(y-c_{0}x)$? Here, I tried to use the implicit function theorem, but I don't know what it would be like to calculate $F_u$.

The points first, three and four are the main problems.

For second, I think it can really be any smooth curve. I didn't seem to use properties of any particular curve.

Now, my question:

After, using the same ideia I will to got $u=f(y + c_{0}x)$. We know that the solution is $u(x,y) = f(y + c_{0}x) + g(y - c_{0}x)$... why the solution is the sum of the solutions to two one-dimensional equations?

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