Hey I'm trying to understand what kind of boundary data I can pose for the wave equation. Let's work in one dimension for now. It appears I should be able to pose any Neumann, Dirichlet or Robin boundary conditions.

I've heard that you need your boundary data to be 'consistent with the Cauchy data'. I would like to understand this better.

If I think about the infinite string problem so solve $u_{tt} - u_{xx} = 0$ on $[0,\infty)$ and I consider a characteristic $x+t = x_0$ I must have $(u_t(x_0 - t, t) - u_x(x_0-t,t))$ is constant (I'm thinking of a point (x=0,t) on the boundary x=0 and a point from my the x-axis (x_0,0) with a characteristic connecting them). This tells me that I can only choose one of $u(x=0,t)$ or $u_x(x=0,t)$ but not both since the must agree on the bounary.

Is this what it means for "cauchy data to be consistent with boundary data"? Is there an analagous statement for the heat equation?

Initial-boundary value problems and the Navier-Stokes equationby H.-O. Kreiss and J. Lorenz has a treatment for the heat equation case in Chapter 7. – Willie Wong Aug 24 '10 at 15:14