Superposition - broadcast antenna question

[rpa3010]

The broadcast antenna of an Am radio station is located at the edge of town. The station owners would like to beam all of the energy into town and none into the countryside, but a single antenna radiates energy equally in all directions. There are two antennas separated by a distance L. Both antennas broadcast a signal at wavelength lambda, but antenna 2 can delay its broadcast relative to antenna 1 by a time interval delta t in order to create a phase difference delta phi_0 between the sources. Your task is to find values of L and delta t such that the waves interfere constructively on the town side and destructively on the country side.

We are given this additional information:
Let antenna 1 be at x = 0. The wave that travels to the right is asin[2pi(x/lambda - t/T)]. The wave that travels to the left is asin[2pi(-x/lambda - t/T)]. Antenna 2 is at x = L. It broadcasts waves asin[2pi((x-L)/Lambda - t/T) + phi_20] to the right and asin[2pi(-(x-L)/lambda - t/T) + phi_20] to the left.

a) What is the smallest value of L for which you can create perfect constructive interference on the town side and perfect destructive interference on the country side? the answer should be a multiple or fraction of the wavelength lambda.

b) What phase constant phi_20 of antenna 2 is needed?

 

[Nate810]

a) The smallest value of L for which we can create perfect constructive interference on the town side and perfect destructive interference on the country side is one-half of the wavelength (λ/2).b) The phase constant phi_20 of antenna 2 needed is π.

 

[m2003]

I would approach this problem by using the principle of superposition, which states that when two or more waves overlap, the resulting wave is the sum of the individual waves. In this case, we have two antennas broadcasting waves that can interfere with each other.

To achieve perfect constructive interference on the town side and destructive interference on the country side, we need to ensure that the waves from both antennas are in phase on the town side and out of phase on the country side. This can be achieved by adjusting the distance L between the antennas and the phase difference delta phi_0 between the sources.

a) To find the smallest value of L for perfect constructive interference on the town side, we need to make sure that the waves from both antennas have the same phase when they reach the town. This can be achieved when the distance traveled by the wave from antenna 1 to the town is equal to the distance traveled by the wave from antenna 2 to the town. This can be represented as:

lambda = (x/lambda - t/T) - ((x-L)/lambda - t/T)

Solving for L, we get L = lambda/2. This means that the distance between the two antennas must be half the wavelength for perfect constructive interference on the town side.

Similarly, for perfect destructive interference on the country side, we need to make sure that the waves from both antennas have a phase difference of pi when they reach the country side. This can be achieved when the distance traveled by the wave from antenna 1 to the country side is equal to the distance traveled by the wave from antenna 2 to the country side. This can be represented as:

lambda = (x/lambda - t/T) - ((x-L)/lambda - t/T) + pi

Solving for L, we get L = 3lambda/2. This means that the distance between the two antennas must be three times the wavelength for perfect destructive interference on the country side.

b) To determine the phase constant phi_20 of antenna 2, we can use the equation:

phi_20 = delta phi_0 - 2piL/lambda

Substituting the values of L = lambda/2 and L = 3lambda/2, we get phi_20 = -pi and phi_20 = 0, respectively. This means that for perfect constructive interference on the town side, the phase difference between the two antennas must be -pi, and for