A source `S` and a detector `D` are placed at a distance `d` apart. A big cardboard is placed at a distance `sqrt(2) d` from the source and the detector as shown in figure. The source emits a wave of wavelength `= d//2` which is received by the detector after reflection from the cardboard. It is found to be in phase with the direct wave received from the source. By what minimum distance should the cardboard be shifted away so that the reflected wave becomes out of phase with the direct wave?

Explanation

`Delta x_(1) = SPD - SD = 2 SP - SD = n lambda` (i)
`Delta x_(2) = SP'D - SD = 2SP' - SD = (2n + 1) (lambda)/(2)` (ii)
`Delta x_(2) - Delta x_(1) = 2(SP' - SP] = (lambda)/(2)`
`SP' - SP = (lambda)/(4) = (d//2)/(4) = (d)/(8)`
`sqrt((sqrt(2)d + x)^(2) + ((d)/2)^(2)) - sqrt((sqrt 2d) ^(2) + ((d)/(2))^(2)) = (d)/(8)`
`sqrt ((sqrt2d + x)^(2) + d^(2)/(4)) - (3d)/(2) = (d)/(8)`
`sqrt((sqrt2d + x)^(2) + d^(2)/(4)) = (13d)/(8)`
Squaring both sides
`(sqrt2d + x)^(2) + d^(2)/(4) = (169d^(2))/(64)`
`(sqrt2d + x)^(2) = (169/(64) - (1)/(4)) d^(2) = (153d^(2))/(64)`
`(sqrt2d + x) = sqrt(153d)/(8)`
`x = sqrt(153d)/(8) - sqrt2d = 0.13d`