Say we are looking for a solution to the Helmholtz equation $$(\Delta + k^2) u = 0,$$ in in the upper half space ($y > 0$) in 2D with a Dirichlet boundary condition on the $x$-axis, that is, $u(x, 0) = 0$. Also, we have an incident wave $u^{inc} = e^{-iky}$ which is orthogonal to the the boundary.
Then the solution can be obtained by the method of images as $$u(x, y) = u^{inc}(x, y) -u^{inc}(x, -y).$$
Now the second term on the RHS represents the wave that reflects off the boundary. My question is, how is there any reflection when we have a Dirichlet boundary condition? I thought Neumann boundary conditions are required for a wave to reflect at a boundary? Dirichlet conditions transmit the wave, not reflect it?
So what am I misunderstanding here...can you have a problem with a Dirichlet boundary condition yet the waves also reflect off the boundary?