Two waves
` y_(1) =A_(1) sin (omega t - beta _(1)), y_(2)=A_(2) sin (omega t - beta_(2)`
Superimpose to form a resultant wave whose amplitude is

Options

A. `sqrt(A_(1)^(2)+A_(2)^(2)+2A_(1)A_(2) cos (beta_(1)-beta_(2)))`
B. `sqrt(A_(1)^(2)+A_(2)^(2)+2A_(1)A_(2) sin (beta_(1)-beta_(2)))`
C. `A_(1)+A_(2)`
D. `|A_(1)+A_(2)|`

Answer

A

Explanation

`:' y_(1) = A_(1) sin (omegat - beta_(1))`
and `y_(2) = A_(2) sin (omegat- beta_(2))`
`:.` Phase difference, `Deltaphi = beta_(1) - beta_(2)`
`:' A = sqrt(A_(1)^(2) + A_(2)^(2) + 2A_(1)A_(2).cos Deltaphi)`
`= sqrt(A_(1)^(2) + A_(2)^(2) + 2A_(1)A_(2).cos (beta_(1) - beta_(2)))`