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?



Can I pose any bounary data for the wave equation on $[0,\infty)$ for given Cauchy data?
I am just going to answer here the problem actually posed in the title. 


The one dimensional wave equation can be re-written as $\partial\_u\partial\_v \phi = 0$, where $u$ and $v$ are the null variables $x + t$ and $x - t$. The initial data then is prescribed on $u+v \geq 0, u-v = 0$, while the boundary is $u+v = 0, u-v \geq 0$. 


So we see that the wave equation implies that the function $\psi(x,t) = \partial\_u \phi(x,t)$ solve a transport equation with negative velocity (cf. my second comment above). Thus if your boundary condition is given such that $\partial\_u \phi(x,t)$ is well-determined along the boundary by just the data given there, you will reach an inconsistency. This is one of the ways of seeing why you cannot prescribe simultaneously Dirichlet and Neumann conditions at the same time. 


(On the other hand, to make the IBVP well-posed, you *need* to specify $\partial\_v \phi(x,t)$ along the boundary, since it satisfies a transport equation with positive velocity. That just one of Dirichlet or Neumann conditions suffice follows from the fact that $\partial\_v \phi(0,t)$ can be solved from $\partial\_u \phi(0,t)$ [transported from Cauchy data] and the boundary data.)


