Two loudspeakers `S_(1)` and `S_(2)` each emit sounds of frequency `220 H_(Z)` uniformly in all directions. `S_(1)` has an acoustic output of `1.2 xx 10^(-3) W` and `S_(2)` has `1.8 xx 10 ^(-3) W` . `S_(1)` and `S_(2)` vibrate in phase.Consider a point `P` such that `S_(1) P = 0.75 m` and `S_(2) p = 3 m` . How are the phases arriving at `P` related? What is the intensity at `P` when both `S_(1)` and `S_(2)` are on ? Speed of sound in air is `330 m//s`.

Explanation

`lambda = (nu)/(f) = (330)/(220) = 1.5m`
`Delta phi_(1) = ((2 pi)/(lambda)) (Deltax_(1)) = ((2 pi)/(1.5)) (0.75)` = `pi`
`Delta phi_(2) = ((2 pi)/(lambda)) (Deltax_(2)) = ((2 pi)/(1.5)) (3) = 4pi`
Since, `phi = Delta phi_(2) - Delta phi_(1) = 3 pi`
They will interfere destructively.
`:. I_(R) = (sqrt(I_(1)) - sqrt(I_(2)))^(2)`
where,
`I_(1) = p_(1)/(4 pi r_(1)^(2)) = (1.2 xx 10^(-3))/((4pi) (0.75)^(2))`
`= 16.97 xx10^(-5) W//m^(2)`
`I_(2) = p_(2)/(4 pi r_(2)^(2)) = (1.8 xx 10^(-3))/((4pi) (3)^(2))`
`= 1.59 xx 10^(-5) W//m^(2)`
Subsituting in Eq. (i), we get
`I_(R) = 8.2 xx 10^(-5) W//m^(2)`