$3\mathrm{.}2($a$)$ Explain the advantages and disadvantages of the $2-$ray ground reflection model in the analysis of path loss.

$($b$)$ In the following cases, tell whether the $2-$ray model could be applied, and explain why or why not:

$h_t=35m.h_r=3m,d=250m$

$h_t=30m,h_r=1\mathrm{.}5m,d=450m$

$($c$)$ What insight does the $2-$ray model provide about large$-$scale path loss that was disregarded when cellular systems used very large cells?

$3\mathrm{.}3$ Prove that in the $2-$ray ground reflected model, $=d^{\text{'}\text{'}}-d^{\text{'}}$~~$2h_r\frac{h_r}{d}.$ Show when this holds as a good approximation. Hint: Use the geometry of Figure P$3\mathrm{.}3$ given below.

$3\mathrm{.}4$ In a $2-$ray ground reflected model, assume that ${}_{}$ must be kept below $6\mathrm{.}261$ radians for phase cancellation reasons. Assuming a receiver height of $2$ m $,$ and given a requirement that ${}_i$ be less than $5,$ what are the minimum allowable values for the T$-$R separation distance and the height of the transmitter antenna? The carrier frequency is $900$ MHz $.$ Refer to Figure P$3\mathrm{.}3.$

$3\mathrm{.}5$ In the $2-$ray path loss model with $=-1,$ derive an appropriate expression for the location of the signal nulls at the receiver.