NDSolve boundary condition relating derivatives in different variables

Question

I am trying to solve a wave equation (2nd order PDE) in z and t with absorbing boundary conditions, i.e. a boundary condition that relates the partial z and t derivatives. For this particular example, I believe the answer should be zero everywhere, but I'm interested ultimately in other, non-trivial, cases.

Here's my attempt:

NDSolveValue[{
  D[V[z, t], {z, 2}] == D[V[z, t], {t, 2}],
  DirichletCondition[V[z, t] == 0, t == 0],
  Derivative[0, 1][V][z, t] == NeumannValue[0, t == 0],
  Derivative[1, 0][V][0, t] == Derivative[0, 1][V][0, t],
  Derivative[1, 0][V][1, t] == -Derivative[0, 1][V][1, t]
  },
 V,
 {t, 0, 1}, {z, 0, 1}
 ]

which yields the error:

NDSolveValue::overdet: There are fewer dependent variables, {V[z,t]}, than equations, so the system is overdetermined.

I would appreciate suggestions for better ways to resolve this problem, or in general to approach the problem. (I realize the problem is solvable analytically--I'm trying it this way as a step along the way to a harder problem that will not be solvable analytically.)

The application here is modeling an electrical transmission line, and I've tried a few approaches so far, but haven't come across one that works.

I am using Mathematica 11.3.