Answer:
Space Wave Propagation (LOS) It is a mode of wave propagation in which the radio waves emitted from the transmitter antenna reach the receiving antenna through space. These radio waves are called space waves. This mode of propagation is used for television broadcast and microwave link. Maximum distance between the transmitting and receiving antenna in line of sight communication is \[{{d}_{M}}=\sqrt{2R{{h}_{T}}}+\sqrt{2R{{h}_{R}}}\] ? (i) Given, \[{{h}_{T}}+{{h}_{R}}=h\] or \[{{h}_{R}}=h-{{h}_{T}}\] Substituting \[{{h}_{R}}\] in Eq. (i), we get \[{{d}_{M}}=\sqrt{2R{{h}_{T}}}+\sqrt{2R\,(h-{{h}_{T}})}\] Differentiating above equation w.r.t. \[{{h}_{T}},\] we get \[\frac{d\,({{d}_{M}})}{d{{h}_{T}}}=\sqrt{2R}\times \frac{1}{2\sqrt{{{h}_{T}}}}-\sqrt{2R}\times \frac{1}{2\sqrt{h-{{h}_{T}}}}\] For maximum range, we have \[\frac{d\,({{d}_{M}})}{d{{h}_{T}}}=0\] or \[\frac{\sqrt{2R}}{2}\left[ \frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}} \right]=0\] or \[\frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}}=0\] or \[\frac{1}{\sqrt{{{h}_{T}}}}-\frac{1}{\sqrt{h-{{h}_{T}}}}\] or \[h-{{h}_{T}}={{h}_{T}}\] \[\therefore \] \[{{h}_{T}}=\frac{h}{2}={{h}_{R}}\]
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