Final answer:
Destructive interference occurs when two light waves combine to create a region of darkness, and this requires the waves to be a half-wavelength out of phase.
Step-by-step explanation:
Destructive interference in the context of light waves occurs when two waves combine to produce a region of darkness. This optical phenomenon arises from a specific condition: the waves involved must be a half-wavelength out of phase. In essence, the explanation for the occurrence of a dark region lies in the alignment of the crest of one wave with the trough of another, resulting in the cancellation of their respective amplitudes.
To elaborate, destructive interference is a consequence of the superposition of waves. When the waves are precisely half a wavelength out of phase, the positive and negative amplitudes coincide spatially. This alignment leads to the constructive cancellation of the wave's peaks and troughs, resulting in a net amplitude of zero at that particular location. The cancellation of the wave's intensity creates a region of darkness in the combined wave pattern.
The key condition for destructive interference is that the path length difference between the two waves must be any half-integral number of wavelengths, such as ½λ, ¾λ, 5/2λ, and so forth. This condition ensures that the waves, when combined, undergo phase shifts that bring their crests and troughs into perfect alignment, reinforcing the destructive interference effect.
In summary, the statement that the waves are a half-wavelength out of phase is indeed accurate and essential for the observed destructive interference. This alignment of wave phases results in the cancellation of amplitudes, leading to the creation of a region of darkness in the combined wave pattern, a phenomenon fundamental to the understanding of wave behavior in optics.