# Phase Differences and Wave Equation in ULA Antennas with Far-Field Assumption

I am analyzing a signal received by two antennas, where the second antenna is positioned farther than the first in the direction of wave propagation. I am encountering some confusion when it comes to the phase difference and gain between these two antennas. The wave equation of the signal is given by

$$s(x, t) = \alpha \exp(i(\omega_0 t - kx)) = \alpha \exp(2\pi i(f_0 t - \frac{1}{\lambda}x)),$$

where $$\f_0\$$ ($$\\omega_0\$$) is the (angular) frequency of the wave.

Considering a scenario where the second antenna is at a distance $$\\Delta x\$$ farther from the transmitter than the first antenna ($$\\Delta x = d \sin \theta\$$ or $$\\Delta x = d \cos \theta\$$), the signal is received at the second antenna $$\\frac{\Delta x}{c}\$$ seconds later. This leads to the signal expression

$$s(x + \Delta x, t + \frac{\Delta x}{c}) = \alpha \exp(2\pi i(f_0 (t + \frac{\Delta x}{c}) - \frac{1}{\lambda}(x + \Delta x))) = \alpha \exp(2\pi i(f_0 t - \frac{1}{\lambda}x)) = s(x, t),$$

due to the fact that $$\\frac{f_0}{c} = \frac{1}{\lambda} \Rightarrow \frac{f_0}{c}\Delta x = \frac{1}{\lambda} \Delta x\$$. It appears that the phase difference arising from the delay difference is canceled out by the phase difference due to the path length difference.

However, my confusion arises when considering Uniform Linear Array (ULA) antennas operating under the far-field assumption. These antennas are known to exhibit phase differences. Could someone elucidate where this phase difference is coming from?

Additionally, I am looking to understand the relation between the gains of the two antennas. If the first antenna has a gain of $$\\beta\$$, would the gain of the second antenna, which is farther than the first, be $$\\exp(\frac{2\pi i}{\lambda}\Delta x)\$$ or $$\\exp(-\frac{2\pi i}{\lambda}\Delta x)\$$ given that $$\\Delta x > 0\$$?

Any insights or clarifications on the phase and gain differences between the two antennas would be greatly appreciated.

• The difference in arrival time of the signal between the two antennas can be expressed in time (ps, ns, etc), or in degrees (phase). The difference is that the former is just based on the geometry - path length difference between the two antennas - while the later also depends on the wavelength, or frequency of the signal. Commented Jan 29 at 17:17