When a number of similar waves pass through a medium simultaneously, each wave travels through the medium as though the others were not present. However, at any point in the medium, the net effect of the waves reaching that point at any instant, is produced. The principle of superposition of waves states that 'if two or more waves of the same nature travel past a point of the medium, then the resultant displacement of the medium at that point is given by the vector sum of the individual displacements due to the waves.'
If are the individual displacements
caused by two waves reaching a point, then the resultant displacement is given by
This principle is applicable when the equations of the waves under consideration are linear. It holds good in the case of waves in an elastic medium and electromagnetic waves, but not in the case of shock waves produced by large explosions.
Superposition of waves can be studied in three different cases
Superposition of two waves of the same frequency
traveling in the same direction, giving rise to a sustained interference pattern.
Superposition of two waves of slightly different frequencies
traveling in the same direction, giving rise to the phenomenon of beats.
Superposition of two waves of same frequency and same amplitude moving in opposite directions, giving rise to the formation of standing waves or stationary waves.
are the individual displacements
caused by two waves reaching a point, then the resultant displacement is given by 
Superposition of two waves of the same frequency
traveling in the same direction, giving rise to a sustained interference pattern.
Superposition of two waves of slightly different frequencies
traveling in the same direction, giving rise to the phenomenon of beats.
Superposition of two waves of same frequency and same amplitude moving in opposite directions, giving rise to the formation of standing waves or stationary waves.