The distance at any time due to any value of pulse meeting simultaneously at a position in a medium is the vector addition of the individual distances due each one of the ray at that point at the similar time.
If 
are the distances at a particular time at a particular point, due to individual rays, then the resultant distance. 
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When two waves of same amplitude on a string approach each other, then on collision, they superimpose to give a resultant pulse of zero amplitude. After crossing, the two waves travel independently as given in as if nothing had occurred.
Important applications of superposition rule :
(a) Interference of waves
(b) Stationary waves
(c) Beats
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