Superposition of waves 
Let two waves moving in the direction from $D$ to $B$ and $C$ to $A$. These waves are out of phase at $O$. Then will the particles along $OA$ and $OB$ oscillate? If they do oscillate,then can you please explain why they will do so intuitively?
Both of the waves are of same frequency and amplitude.
The reason behind this question is that suppose the particles involved in the wave OC is pulling the particle O upwards whereas the wave OD is pulling it downwards,then the particle O would come to standstill.then how would the both waves propagate then? Will it get stopped there since these are cancelling each other?
 A: Discrete medium
Let's analyze the setup by placing masses at fixed intervals along the diagonals $\overline{AC}$ and $\overline{BD}$, such that one mass lands exactly on $O$. To make it easier to visualize, imagine the masses being mounted on guides such that they can only move up and down, so we consider transversal waves. Now connect all the masses to their nearest neighbors with springs. Every mass will be connected to two others, except at the points $A$, $B$, $C$, and $D$, which have only one neighbor, and $O$, which as four.
In this setup, it is quite obvious that if two waves are coming from $D$ and $C$, respecitvely, perfectly out of phase, one of the neighbors of $O$ (in the direction of $D$) will try to pull it up, while the other (from $C$) wants to pull it down with the same force. They cancel out and the mass at $O$ does not move. Because $O$ does not move, and all masses between $A$ and $O$, and between $B$ and $O$ are only connected to all other masses via $O$, none of these masses move.
This is one of the rare situations where the intuitive solution is correct, and one must think a bit more carefully about the mathematical reason. One might expect that, because of the principle of superposition for waves, that they might just pass through each other without interference.
The reason is that if there is only a wave travelling from $D$ towards $B$, it will split up at $O$ and travel to $A$, $B$, and $C$ with equal amplitudes. The same goes for a wave starting at $C$. If there are waves starting and $C$ and $D$, but out of phase, the waves coming from $D$ and the waves coming from $C$ completely cancel between $A$, $O$, and $B$ (superposition out of phase, or destructive interference).
Continuous media
The above conclusions do not straightforwardly translate to electromagnetic waves. If we imagine the lines as waveguides, as long as they have some finite width, the crossover region at $O$ is also of finite size, and the amplitude in that region will no longer exactly cancel, allowing waves to propagate through to $A$ and $B$. The very same argument also holds for water waves or any other setup where the structure of the medium is much smaller than the wavelength.
Only if we make the waveguides infinitely narrow, will we recover the above situation. This is consistent with the setup of masses connected by springs, because for infinitely small slits, the diffracted intensity becomes direction independent, and we again have cases of destructive interference on the sections $\overline{OA}$ and $\overline{OB}$.
A: If you are talking about sound waves of the same frequency from speakers at D and C, When out of phase the molecules at point O would be moving in a circle (vector sum of two longitudinal displacements changing with time). At other points within the square area the vector sum would be more complex as the amplitudes and phase differences change.
