Suppose we are given the wave equation and some initial condition. About the boundary conditions... why have we to prescribe as many boundary conditions as the number of characteristic lines which enter in the domain? Can anyone help me?
Thanks in advance
When solving the wave equation (WE) you must consider the given domain or boundaries. For example, when considering the problem of a vibrating string, the domain is finite and the PDE, i.e.:
$$u_{tt} - u_{xx} = 0,$$ (this should be the non-dimensional form of the WE) has to be solved with two boundary conditions (for example, the ends are fixed: $u(0,t)=0$ and $u(L,t) = 0$ if $L$ is the length of the string) and a given initial condition for position, $u(x,0)$, and velocity, $u_t(x,0)$. This often leads to a well posed problem for solving the PDE in this finite domain. You can solved this using the Sturm-Liouville theory and applying separation of variables.
Things change if you consider infinite (or semi-inifinite) domains, for example, the axial evolution of pressure disturbations of a perfect gas through a finite pipe. In this case, two initial conditions can be prescribed (initial velocity and pressure gradient). Anyway, this yields to the well-known D'Alembert's formula, which states that the initial condition propagates through the characteristics and, thus, the solution is constant through them.
So, in resume, you can solve two very different problems depending on the domain you consider.
I hope this may be helpful to you.