Question

A source S and a detector D of high frequency waves are a distance d apart on the ground. The direct wave from S is found to be in phase at D with the wave from S that is reflected rays make the same angle with the reflecting layer. When the layer rises a distance h, no signal is detected at D. Neglect absorption in the atmosphere and find the relation between d, h, H and the wavelength $\lambda$ of the waves.

Answer 1

When a ray is reflected from a height of $H$ , and other comes direct to $D$ , they are in phase i.e. constructive interference is occured at $D$ ,

path difference for destructive interference is given by ,

$\Delta x=n\lambda$ , n=0,1,2,3....

let we have a point $P$ at height $H$ ,

now , $SD=d$ ,

$SP=\sqrt{H^2+(d/2{)}^2}$ ,

and $PD=\sqrt{H^2+(d/2{)}^2}$ ,

$SP+PD=2\left(\sqrt{H^2+(d/2{)}^2}\right)=\sqrt{4H^2+d^2}$ ,

hence path difference ,

$\Delta x=SP-SD=\sqrt{4H^2+d^2}-d$ ,

or $n\lambda =SP-SD=\sqrt{4H^2+d^2}-d$ ,....................eq1

when is ray is reflected from a height of $H+h$ , and other comes direct to $D$ , there is no signal found at $D$ i.e. desstructive interference is occurred at $D$ ,

path difference for destructive interference is given by ,

$\Delta x=(n+1/2)\lambda$ , n=0,1,2,3....

let we have a point $Q$ at height $H+h$ ,

now , $SD=d$ ,

$SQ=\sqrt{(H+h{)}^2+(d/2{)}^2}$ ,

and $QD=\sqrt{(H+h{)}^2+(d/2{)}^2}$ ,

$SQ+QD=2\left(\sqrt{(H+h{)}^2+(d/2{)}^2}\right)=\sqrt{4(H+h{)}^2+d^2}$ ,

hence path difference ,

$\Delta x^{\prime }=SQ-SD=\sqrt{4(H+h{)}^2+d^2}-d$ ,

or $(n+1/2)\lambda =SQ-SD=\sqrt{4(H+h{)}^2+d^2}-d$ ,...................eq2 ,

now subtracting eq1 by eq2 ,

$\lambda /2=\sqrt{4(H+h{)}^2+d^2}-\sqrt{4H^2+d^2}$ ,

or $\lambda =2\sqrt{4(H+h{)}^2+d^2}-2\sqrt{4H^2+d^2}$