What is the equation of a partial standing wave which is formed by adding two opposite traveling waves with different amplitudes: $y(z,t) = A\cos(\omega t + bz) + B\cos(\omega t - bz)$ .
This is not a pure standing wave. Its magnitude must be a function of $z$ i.e. $f(z)$, but I do not know if it is of the form $y(z,t) = f(z) \cos (\omega t + bz)$ i.e like an Amplitude modulated traveling wave.
Assuming that $A_2 = A_1 + \Delta A > A_1$ we can always write \begin{align} y &= A_1 \cos(wt + k z) + A_2\cos(wt - k z) \\ &= A_1 \big[ \cos(wt + k z) + \cos(wt - k z) \big] + \Delta A \cos(wt - k z) \end{align} The first part with amplitude $A_1$ builds a standing wave, the second part with amplitude $\Delta A$ is simply superimposed.
The following image shows an incident wave (red) with amplitude $A_2=2$, which is only partially reflected on a mirror. The reflected wave (green) has amplitude $A_1=1$. The blue wave is the superposition. The horizontal axis is the position axis, which you called $z$ (I use $x$ instead).
The superposition does not look like a standing wave. Instead, it looks like a wave that is travelling with a periodically changing velocity.
