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

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?

• These notes are a useful start and answer the third and fourth points you asked about. I suggest reading the whole set of notes to understand what is going on first. – Mattos Sep 23 '18 at 12:27