Show that two waves interfere constructively when the path difference them is an integral multiple of wave length.

Explanation

Let , `Y_1=A_1 sin (omega t- phi_1) and Y_2 =A_2 sin (omega t-phi_2)` when these waves interfere, the resultant displacement will be `Y=Y_1+Y_2`
i.e. `Y=A_1 sin (omega t-phi_1)+A_2 sin (omega t-phi_2)`.
i.e., `Y=A sin (omega t-phi)`
where, `A=sqrt(A_1^2+A_2^2+2A_1 A_2 cos (phi_1-phi_2))`
and `phi=tan^(-1){(A_1 sin phi_1+A_2 sin phi_2)/(A_ cos phi_1+A_2 cos phi_2)}`
For a constructive interference , `A=A_("max")`.
This implies that
`cos (phi_1-phi_2)=+1,` So that `A=A_1+A_2`
i.e. `phi_1-phi_2=0,2 pi, 4pi..........2npi`
or `(phi_1-phi_2)=2npi` where `n=0,1,2...........`
For a constructive interference, path difference `delta=nlambda`