When two waves have equal amplitudes but opposite phases, it means that the peaks and troughs of one wave align with the troughs and peaks of the other wave, respectively. In other words, they are completely out of phase with each other.
When two waves are in phase, their peaks and troughs align, resulting in constructive interference, where the amplitudes of the waves add up. However, when two waves are exactly out of phase, their peaks and troughs align in opposite directions, causing destructive interference, where the amplitudes of the waves cancel each other out.
In the case of waves with equal amplitudes but opposite phases, the resulting interference leads to complete destructive interference. As a result, the waves will effectively cancel each other out, resulting in a net amplitude of zero at any given point where the waves overlap. This phenomenon is called "phase cancellation."
Physically, phase cancellation can be observed in various contexts. For example, in acoustics, if two identical sound waves with equal amplitudes but opposite phases are played simultaneously, they can interfere destructively, causing a reduction in the overall perceived sound level. This effect is sometimes used in noise-canceling technology to reduce unwanted ambient noise.
Mathematically, when two waves have equal amplitudes but opposite phases, they can be represented as:
Wave 1: A * sin(ωt) Wave 2: -A * sin(ωt)
Here, A represents the amplitude of the waves, ω is the angular frequency, and t is the time. The negative sign in front of Wave 2 indicates the opposite phase.
When these two waves are added together, the result is:
Resultant Wave: A * sin(ωt) - A * sin(ωt) = 0
Thus, the resulting wave has an amplitude of zero, indicating complete cancellation.