Oscillation of Two Fluid Interface in a U-tube: The Story before Statics 

Attached above is an example problem from College Physics by Nicholas Giordano.
Of course, the problem is extremely simple - and I solved it too; however, I've a conceptual doubt. 
Here's a part of the explanation given in the book:

Consider the pressure at point B. This point is inside liquid 1 and is just below the boundary that separates the two liquids. Because we consider point B to be at the same depth as point A (also located in liquid 1, but on the other side of the U-tube), the pressures at points A and B are the same.
  We next consider point C; this point is in liquid 2, and just above the boundary
  between the liquids. Because points B and C are extremely close together, the pressures at these two locations must be the same. You can see how that is true by imagining a thin (massless) plastic sheet stretched along the boundary between liquids 1 and 2. Because the system is at rest, this sheet is in translational equilibrium; hence, the total force on the sheet must be zero. That can only be true if the pressure is the same on the top and bottom surfaces of the sheet. We have thus concluded that the pressure is the same at points A, B, and C.

Now, for a very similiar problem in Physics by Resnick, Halliday & Krane (abbreviated RHK hereafter), something else is added as well- 
For concluding that the pressure is same at B and C, RHK uses the same argument, that of translational equilibrium of the interface. Just after, it also mentions the following:

When we first pour oil in the tube, however, there may be a difference in pressure and an unbalanced force that would cause the system to move, until it reached the static situation as shown in the figure.

I'm not able to picture this phenomenon clearly. What kind of motion is RHK talking about? My instructor said there would be oscillations of the interface, which would eventually die down due to energy loss to the vessel and surroundings.


*

*Will these oscillations be simple harmonic? (Damped, I guess?)

*Under what assumptions, if any, would these oscillations be simple harmonic?

*What effect does the viscosity of the two fluids have on these oscillations?

*Is it necessary to include viscosity in our analysis? I ask this because such problems usually start with the assumption of ideal fluids, which are inviscid by definition. 

*Can we talk about these oscillations in terms of an equation of motion, for the interface? A detailed analysis would help me understand the concept better, and it will also clear up doubts of many (my classmates, at least) in such grey areas of fluid mechanics, which no text I've come across, addresses.


Thanks a lot.
 A: First of all, as mentioned by Sammy, the answers of this post should be somewhat helpful:
Oscillations in U-tube
I will take the approach of using energy conservation to answer your question. First, I denote point $B$ of the equilibrium position to be of height $H$, and I will define $h_1=y_L$ and $h_2=y_R$ as given by the problem.
Furthermore, I will denote the deviation from the equilibrium position as $h$, oil density as $\rho_o$, water density as $\rho_w$, and the area of the cross section as $A$. Then the potential energy can be written as:
$$ V=\frac{\rho_wg(H+h_1-h)^2S}{2}+\frac{\rho_wg(H+h)^2S}{2}+\rho_ogh_2(H+\frac{h_2}{2}+h)S $$
The first term is due the potential energy from the water in the "left side" of the tube. The second term is due to the water in the "right side" of the tube. The third term is due to the oil.
And the kinetic energy can be written as:
$$ T=\frac{1}{2}\rho_wLS(\dot{h})^2+\frac{1}{2}\rho_oh_2S(\dot{h})^2 $$
The sum of the two has to equal to some constant energy:
$$ T+V=S(\frac{\rho_wg(H+h_1-h)^2}{2}+\frac{\rho_wg(H+h)^2}{2}+\rho_ogh_2(H+\frac{h_2}{2}+h)+\frac{1}{2}\rho_wL(\dot{h})^2+\frac{1}{2}\rho_oh_2(\dot{h})^2)=E $$
If you take the time derivative of both sides, you find:
$$
\dot{h}[2\rho_wgh+(\rho_wL+\rho_0h_2)\ddot{h}]=0
$$
where I've used the fact that $\rho_0h_2=\rho_wh_1$. So in fact, this is a simple harmonic oscillator assuming ideal liquids.
As for the remainder of your questions. I think the best way to intuitively grasp them is to think of viscosity as a source of energy dissipation. And whenever you have energy dissipation, the oscillation will be damped in some form.
