Two point sound sources `A` and `B` each to power `25pi w` and frequency `850 Hz` are `1 m` apart. The sources are in phase

(a) Determine the phase difference between the waves emitting from `A` and `B` received by detector `D` as shown in figure.
(b) Also determine the intensity of the resultant sound wave as recorded by detector `D`. Velocity of sound `= 340 m//s`.

Answer

`pi`

Explanation

Path difference is
`Delta = sqrt((2.4)^(2) + (1)^(2) ) - 2.4`
`A = (v)/(n) = (340)/(850) = 0.4m`
`phi = (2pi)/(lambda) xx Delta`
`(2pi)/(0.4)xx0.2 = pi`
Intensity of a point source is
`I = (P)/(4pix^(2))`
Intensity of wave from A at D is
`I_(1) = (25pi)/(4pi(2.4)^(2))`
Intensity of wave from B at D is
`I = (25pi)/(4pi(2.6)^(2))`
Thus resulting intensity at D is
`I = (sqrt(I_(1) ) - sqrt(I_2))^(2)`
= ` (sqrt((25)/(4(2.4)^(2)) ) - sqrt((25)/(4(2.6)^(2))))^(2)`
`((5)/(4.8) - (5)/(5.2))^(2) W//m^(2)`
`(1.0416 - 0.9615)^(2) W//m^(2)`
= `0.0064W//m^(2)`