In the context of wave interference, the phase difference is a critical factor that influences the resultant wave’s amplitude. Specifically, when the phase difference between two waves is equal to $\pi$ (or 180 degrees), the waves are considered to be in anti-phase. In this configuration, the crest of one wave aligns with the trough of the other wave, resulting in a unique interference pattern.

When this anti-phase condition occurs, destructive interference takes place. In this scenario, the amplitudes of the two waves effectively cancel each other out, which can lead to a minimum intensity or even zero intensity at the point of interference. This cancellation happens because the positive displacement from one wave is precisely offset by the negative displacement from the other wave, resulting in a point of minimal disturbance known as a “node.”

This principle is essential for understanding various physical phenomena, such as the interference patterns observed in a double-slit experiment, the noise-cancelling capabilities of certain headphones, and the formation of standing waves in musical instruments. In each of these instances, the principle of superposition is at work, and the phase difference between the waves determines whether the interference is constructive (amplitudes add) or destructive (amplitudes cancel).

To summarize, a phase difference of $\pi$ leads to destructive interference, where the amplitudes of the two waves subtract from each other. This results in minimal or zero intensity at the interference point, creating a node. Grasping this concept is vital for comprehending various wave-related phenomena in physics.