Explain the superposition principle for waves forming in calm water.

(N/A) The figure $(a)$ shows two needles oscillating with equal phase,representing two coherent sources. Consider two needles moving periodically up and down in an identical fashion in a trough of water as shown in the figure; they produce two water waves.
At any particular point,the phase difference between the displacements produced by each of the waves does not change with time,so both sources are called coherent sources.
Figure $(b)$ shows the position of crests (solid circles) and troughs (dashed circles) at a given instant of time.
Now consider a point $P$ as shown in figure $(a)$ for which $S_{1}P = S_{2}P$.
Since waves from $S_{1}$ and $S_{2}$ will take the same time to travel to point $P$,they arrive with the same phase.
The displacement produced by the source $S_{1}$ at point $P$ is given by $y_{1} = a \cos \omega t$,and the displacement produced by the source $S_{2}$ at point $P$ is given by $y_{2} = a \cos \omega t$,where $a$ is the amplitude.
According to the superposition principle,the resultant displacement at $P$ is:
$y = y_{1} + y_{2} = a \cos \omega t + a \cos \omega t$
$\therefore y = 2a \cos \omega t$