Two sinusoidal wave of the same frequency are to be sent in the same direction along a taut string. One wave has an amplitude of `5.0 mm`, the other `8.0 mm`. (a) What phase difference `phi_(1)` between the two waves results in the smallest amplitude of the resultant wave? (b) What is that smallest amplitude? (c) What phase difference `phi_(2)` results in the largest amplitude of the resultant wave ? (d) What is that largest amplitude? (e) What is resultant amplitude if the phase angle is `(phi_(1) - phi_(2))//2`?

Answer

(a) `pi` rad; , (b) `3.0 mm;` , (c) `0` rad; , (d) `13 mm;` , (e) `9.4 mm`

Explanation

(a) For smallest amplitude ,
`A_(R) = |A_(1) - A_(2)|` and that is possible when `phi_(1) = pi` between `A_(1)` and `A_(2)`
(b) `A_(R) = |A_(1) - A_(2)| = 3 mm`
(c) For largest amplitude ,
`A_(R) = |A_(1) + A_(2)|` and that is possible when `phi_(2) = 0` between `A_(1)` and `A_(2)`
(d) `A_(R) = |A_(1) + A_(2)| = 13 mm`
(e) when `phi = (phi_(1) - phi_(2))/(2) = (pi - o)/(2) = (pi)/(2)`
`:. A_(R) = [A_(1^(2)) + A_(2^(2)) + 2A_(1) A_(2)cos (pi)/(2)]^(1//2)`
`= 9.4 mm`