A radar array consists of five vertical antennae arranged in a line, each emitting radio waves of frequency 120MHz in phase with one another uniformly in all directions. The spacing between each of the antennae is 8.0m.
Apart from the normal direction perpendicular to the line joining the antennae, at what other angles will the wave fronts from the 5 antennae arrive in phase at a distant location to produce a strong radar signal?
To determine the angles at which the wave fronts from the five antennae arrive in phase at a distant location, we can use the concept of phase difference and interference.
In this setup, the wave fronts from each antenna will reach the distant location with a phase difference. For interference to occur and produce a strong radar signal, the phase difference between any two adjacent antennas must be an integral multiple of the wavelength.
First, let's calculate the wavelength (λ) corresponding to the given frequency (f) of 120MHz:
Speed of light (c) = 3x10^8 m/s (approximately)
λ = c/f = (3x10^8 m/s) / (120x10^6 Hz) = 2.5 meters
Now, let's consider the first antenna, A1, as the reference point. The phase difference between antenna A1 and antenna A2 is given by:
Δφ = 2π * (distance between A1 and A2) / λ = 2π * 8.0 m / 2.5 m = 16π / 2.5 radians
For interference to occur, the phase difference Δφ should be an integral multiple of 2π (360 degrees). So, we can write:
Δφ = 2πn, where n is an integer
Therefore, the angles at which the wave fronts from the five antennae arrive in phase at a distant location can be calculated as follows:
Angle = sin^(-1)(λ / d), where d is the distance between the antennae.
For A2 to be in phase with A1:
Angle1 = sin^(-1)(2.5 m / 8.0 m) = 18.19 degrees
For A3 to be in phase with A1:
Angle2 = sin^(-1)(2.5 m / 16.0 m) = 9.58 degrees
For A4 to be in phase with A1:
Angle3 = sin^(-1)(2.5 m / 24.0 m) = 6.14 degrees
For A5 to be in phase with A1:
Angle4 = sin^(-1)(2.5 m / 32.0 m) = 4.55 degrees
Therefore, apart from the normal direction perpendicular to the line of the antennae, the wave fronts from the five antennae will arrive in phase at the distant location at angles approximate to:
18.19 degrees, 9.58 degrees, 6.14 degrees, and 4.55 degrees.