Obtain the resultant wave of more than two wave functions by representing the superposition principle mathematically.

Explanation

Let `y _(1) (x,t) and y _(2) (x,t)` be the displacement that any element of the string would experience if each wave travelled alone.
The displacement y(x,t) of an element of the string when the waves overlap is then given by,
`y (x,t) =y _(1) (x,t) +y_(2) (x,t)" "...(1)`
If we have two or more waves moving in the medium the resultant waveform is the sum of wave functions of individula waves.
`y _(1) =f _(1) (x-vt)`
`y _(2) =f_(2) (x-vt)`
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If `y _(n) = f _(n) (x-vt),` then resultant wave function, `y =f _(1) (x-vt) + f _(2) (x-vt)+,....f _(n) (x-vt)`
`therefore y = sum _(i =1) ^(n)f _(i) (x - vt) ` where ` i=1,2,3,...,n`
The phenomenon of superpostition of two waves of a same medium is called interfeference.