In destructive interference, waves can cancel each other out when their crests align with the troughs or their troughs align with the crests. This occurs when two or more waves of the same frequency and amplitude are superimposed or combined.
When waves meet, they undergo superposition, which means their amplitudes (peak heights) are added together at each point. If two waves with equal amplitude, but opposite phase (180 degrees out of phase), meet at a point, the positive displacement (crest) of one wave coincides with the negative displacement (trough) of the other wave. As a result, their amplitudes cancel each other out, resulting in a net displacement of zero at that point.
Mathematically, if we represent the waves as functions, such as f₁(x) and f₂(x), their superposition would be f(x) = f₁(x) + f₂(x). If the waves have opposite phases, the sum of their amplitudes will be zero at certain points, leading to destructive interference.
It's important to note that destructive interference occurs when the waves have the same frequency and amplitude, as well as a constant phase difference. If any of these factors vary, the interference pattern will be different.
Destructive interference is a fundamental concept in wave phenomena and is observed in various contexts, including the double-slit experiment, where it produces dark fringes in the interference pattern, indicating regions where waves have canceled each other out.