Derive the expression for resultant displacement and amplitude when two waves having same amplitude and a phase difference `0/` superpose.

Explanation

Consider two waves having the same amplitude a and a phase difference of `phi`.
Let the displacement produced by source `S_1` be by `y_1` = a cos wt and by `S_2` be the
`y_2=a cos (wt + phi)`
The resultant displacement
`y=y_1+y_2`
`y=a cos wt + a cos (wt + phi)`
Using the formula , cos A + cos B = 2cos `((A+B)/2)cos ((A-B)/2)`
`y=a[cos wt + cos (wt + phi)]=2a cos ((wt+wt+phi)/2)cos ((wt-wt+phi)/2)`
`y=2a cos ((2wt+phi)/2) cos (phi/2)=2acos phi/2 cos ((2wt+phi)/2)`
The amplitude of the resultant displacement is = `2a cos phi/2`