Is it possible to reduce the speed and frequency of a light wave to zero in a liquid medium? Assume that two mirrors are located in a huge liquid medium – say, deep down on an ocean floor – with a refractive index of $n'$ as measured by an observer $A$ standing on the beach platform at rest WRT the liquid. One of the mirrors is at rest WRT $A$, and the other mirror recedes from $A$ at $v$. Remember that the mirrors are parallel to each other both perpendicular to the velocity vector.
Now, suppose that a tiny laser diode attached to the lowest part of the first mirror sends a photon towards the moving one at an angle nearly perpendicular to the surfaces of the mirrors. The photon, after being bounced between the mirrors for $i$ times, is received by a detector set on an upper side of the same mirror. [See the Figure.] Each time the photon hits the moving mirror, its speed reduces as measured by observer $A$. 
Can we say that, from the perspective of $A$, the velocity of the photon, complying with the relativistic velocity addition, reduces each time it hits the moving mirror till the speed, possibly along with the light frequency, approaches zero if the number of bounces is great enough? Is it correct to use the relativistic velocity addition formula in this example?
According to the figure, $c'_1$ is calculated to be:
$$c'_1=\frac{c}{n'} \tag{1}$$
Using the relativistic velocity addition formula, for $c'_2$ we can write:
$$c'_2=\frac{c'_1-v}{1-\frac{vc'_1}{c^2}}=\frac{\frac{c}{n'}-v}{1-\frac{v}{n'c}} \tag{2}$$ 
Similarly, for $c'_3$, $c'_4$, and $c'_i$, we have:
$$c'_3=\frac{c'_2-v}{1-\frac{vc'_2}{c^2}} \tag{3}$$
$$c'_4=\frac{c'_3-v}{1-\frac{vc'_3}{c^2}} \tag{4}$$
$$c'_i=\frac{c'_{i-1}-v}{1-\frac{vc'_{i-1}}{c^2}} \tag{5}$$
As we see, we can arbitrarily reduce the speed of light if we repeat the calculation enough. Does this mean that the speed of light can be as slow as a turtle in a medium by this means? Do the accurate calculations become more complicated than those done here by considering the fact that the refractive index varies as a function of frequency as well as velocity $v$?

 A: In other words, does speed of light (after reflection) depend on speed of mirrors as well as speed of water or not? Clearly the answer is NO. 
Let's redo this experiment, but this time in space not water. Simply enough we can see that speed of light will be c no matter what. Wavelength is the only thing which is affected by velocity of mirrors and that's because of Doppler effect.
Water is no exception(why would it be?). Light just doesn't care about what it sees at boundaries. Its speed from A perspective will be $c/n$ before and after reflections. From perspective of moving mirror however, light's speed will oscillate between $\frac{c/n-v}{1-\frac{cv}{nc^2}}$ (before reflection) and $\frac{c/n+v}{1+\frac{cv}{nc^2}}$ (after reflection) where $v$ is speed of water.
A: Consider Maxwell's equations in a linear transparent medium without free currents or charges:
$$\nabla \cdot \vec E = 0$$
$$\nabla \cdot \vec B = 0$$
$$\nabla \times \vec E = -\frac{\partial}{\partial t} \vec B$$
$$\nabla \times \vec B = \mu \epsilon \frac{\partial}{\partial t} \vec E$$
Which can be reduced to:
$$\left(\frac{1}{\mu \epsilon} \nabla^2 - \frac{\partial^2}{\partial t^2}\right)\vec E = 0$$
Since $\mu \epsilon = \mu_r \epsilon_r \mu_0 \epsilon_0 = n^2/c^2$ this represents a wave moving at a speed $v=c/n$.
In other words, from Maxwell's equations the speed of the wave is determined entirely by the electromagnetic properties of the medium, as summarized by the refractive index $n$. Neither the speed of the source nor the speed of any intervening reflective surface are involved. As long as Maxwell's equations are valid in a medium, the speed of the wave depends only on the electromagnetic properties of the medium itself.
Generally, the relative permittivity and permeability of a material are measured in the rest frame of the material, and thus the refractive index describes the $v=c/n$ of the wave in the rest frame of the material. Therefore, to obtain the velocity in another frame one simply applies relativistic velocity addition using the known speed of the light in the rest frame of the medium and the speed of the medium in another frame to obtain the speed of light in the other frame. This is consistent with the famous Fizeau experiment which predated relativity but provided probably the first evidence supporting relativistic velocity addition.
Thus it is incorrect that "Each time the photon hits the moving mirror, its speed reduces as measured by observer A". Upon recognizing this the rest of the problem disappears.

Do the accurate calculations become more complicated than those done here by considering the fact that the refractive index varies as a function of frequency as well as velocity v?

Yes, the light will Doppler shift (in the stationary mirror's frame) upon hitting the moving mirror. As a result, if $n$ is a function of frequency then the speed will change. As the amount of Doppler shift depends on the speed of the mirror, and the frequency depends on the Doppler shift, and the refractive index depends on the frequency, and the speed of the light wave depends on the refractive index, then it is true that (indirectly) the speed of the light wave depends on the speed of the mirror. However the relationship is not as simple as velocity addition, but rather follows the causal chain outlined above. In the absence of a frequency dependence of $n$ there is no dependence of the speed of light on the velocity of the mirror.
