Two waves have the same frequency . The first has intensity `I_(0)`. The second has intensity `4I_(0)` and lags behind the first in phase by `pi//2`. When they meet , find the resultant intensity , and the phase relationship of the resultant wave with the first wave .

Explanation

Since intensity `prop (amplitude)^(2)` , the waves may be expressed as
` y_(1) = a sin ( theta ) and y_(2) = 2 a sin ( theta - pi//2)`
Where `theta = omega t - kx and I_(0) = constant xx a^(2)` (i)
The resultant wave is
` y = y_(1) + y_(2) = a sin theta - 2 a cos theta`
Let ` a = A cos varphi , 2 a = A sin varphi ` (ii)
Then ` y = A sin ( theta - varphi ) = A sin ( omega t - kx - varphi)`
This is a wave of amplitude `A` , and lags behind the `y_(1)` wave by phase angle `varphi, where varphi = 2` .
Resultant intensity :
`I_(R) = I_(0) + 4I_(0) + 2 sqrt(I_(0) 4 I_(0)) cos ((pi)/( 2)) = 5 I_(0)`