The Two Towers

Two radio towers, A and B, located at a distance $d$ apart from one another transmit radio waves of wavelength $\lambda$. The emitted waves have exactly the same phase. The waves are observed by a receiver which is located far away from the towers, see Fig. 3. When the transmitter in the tower B is turned off, the intensity of the received signal is $I_A$; when the transmitter in the tower A is turned off, the intensity becomes $I_B$. Now, consider the situation when both transmitters are on.

(a)

Find the expression for the intensity of the received signal $I$ as a function of the angle $\theta$ (Simplify your expression so that the final answer involves no complex numbers!).

(b)

What is the maximal value of the intensity $I$? At which values of $\theta$ do the maxima occur?

(c)

What is the minimal value of the intensity $I$? At which values of $\theta$ do the minima occur?

(d)

If the waves emitted by the towers were out of phase, would the minimal/maximal values of $I$ change? What about the angles at which the minima/maxima occur?

Figure 3: The two towers.