Examine the following equations and test whether interference, beats or stationary waves will arise from the superposition of (i) 1 and 2 (ii) 1 and 3 (iii) 1 and 4 (iv) 2 and 3 (v) 2 and 4 (vi) 3 and 4 ?
`1. xi_(1)=acos2pi((t)/(T)+(x)/(lambda))`
`2. xi_(2)=acos2pi((t)/(T)-(x)/(lambda))`
`3. xi_(3)=acos2pi((t)/(T)+(x+P)/(lambda)+phi)`
`4. xi_(4)=acos2pi((t)/(T+alpha)+(x)/(lambda)),alphaltT`

Explanation

(i) 1 and 2 represent two waves of same amplitude and frequency travelling in opposite directions. Their superposition will produce stationary waves.
(ii) 1 and 3 represent two waves of same frequency and amplitude with a phase difference and travelling in the same direction . their supereposition will produce interference.
(iii) 1 and 4 represent two waves two waves of same amplitude travelling in same direction, but having slightly different frequencies. their superposition will produce beats.
(iv) Similarly, superposition of 2 and 3 will produce stationary waves.
(v) Superposition of 2 and 4 will produce interference.
(vi) Superposition of 3 and 4 will give rise to beats.