Final answer:
The minimum frequency required for the first Fresnel zone to not intersect the ground is approximately \(1.247 \, \text{GHz}\).
Step-by-step explanation:
The first Fresnel zone is crucial in microwave link design to ensure proper signal propagation and reduce interference. For a flat-earth scenario, the minimum frequency (\(f\)) to avoid ground interference in the first Fresnel zone can be calculated using the formula:
\[ f = \frac{c}{2d} \]
where:
- \( c \) is the speed of light (\(3 \times 10^8 \, \text{m/s}\)),
- \( d \) is the straight-line distance between the antennas.
In this case, the straight-line distance (\(d\)) is the sum of the heights of the towers and the length of the link. Given that the towers are 15 meters high and the link is 12 km long, convert the distance to meters:
\[ d = 2 \times 15 \, \text{m} + 12 \times 1000 \, \text{m} \]
Now, substitute the values into the formula:
\[ f = \frac{3 \times 10^8 \, \text{m/s}}{2 \times \left(2 \times 15 \, \text{m} + 12 \times 1000 \, \text{m}\right)} \]
Calculate \(f\), and then convert it to gigahertz (GHz):
\[ f \approx \frac{3 \times 10^8}{2 \times (2 \times 15 + 12 \times 1000)} \]
\[ f \approx \frac{3 \times 10^8}{2 \times (30 + 12000)} \]
\[ f \approx \frac{3 \times 10^8}{2 \times 12030} \]
\[ f \approx \frac{3 \times 10^8}{24060} \]
\[ f \approx 1247.40 \, \text{Hz} \]
Now, convert to gigahertz:
\[ f \approx 1.247 \, \text{GHz} \]
So, the minimum frequency required for the first Fresnel zone to not intersect the ground is approximately \(1.247 \, \text{GHz}\).