$u_{tt} = au_{xx}, x \in [0,L]$, and $t>0$
$u(0,t) = g(x)$
$u_x(L,t) = \mu u_t(L,t)$
$u(x,0) = u_0(x)$
$u'(x,0) = u_1(x)$
This type of boundary feedback in 1D wave equation is known to be stable. The first is a dirichlet BC, whats the second one?. Anyone sources where I can read more about it? I can't seem to find more about it because, I do not know what are they are called.
I doubt that it has a well agreed-upon name. From the physics point of view there are many interpretations of this boundary condition, and probably many names, depending on which area of physics you consider the wave equation to represent: electromagnetics, mechanical, acoustics, or something else.
For example, from the mechanical side $v=u_t$ is a velocity and $\frac{1}{\mu}u_x$ is a force, force distribution, pressure, or flux. So, this boundary condition relates the velocity of the particle at $L$ to the pressure at the same point (pressure or flux coming from outside the modeling domain $[0,L]$). In a wave equation context, a first order derivative in time represents damping. This means that this boundary condition gives a damping that is proportional to the pressure/flux. This has a stabilizing effect on the equation.
Yet another perspective on this equation would be to view it in the frequency domain replacing $u_t$ with $j\omega u$, assuming things are linear. The boundary condition then reads $\frac{1}{\mu}u_x-j\omega u=0$ and is a form of Robin condition, but now in the frequency domain. Again, whenever you see $j\omega u$ in a frequency domain expression you know there is damping in the system.