No, the amplitude of the overlapping of two waves with different wavelengths and a path difference cannot be zero. When two waves with different wavelengths and a path difference interfere, they create interference patterns.
Interference occurs when two or more waves meet at a point in space. The resulting amplitude at that point is the sum of the individual amplitudes of the waves. Depending on the phase relationship between the waves, constructive interference or destructive interference can occur.
Constructive interference happens when the waves are in phase (peaks align with peaks and troughs align with troughs), and the amplitudes add up, resulting in a larger amplitude. Destructive interference occurs when the waves are out of phase (peaks align with troughs), and the amplitudes partially or completely cancel each other out, resulting in a smaller or zero amplitude.
The interference pattern is determined by the wavelength, path difference, and phase relationship between the waves. Even if the amplitudes of the individual waves are different, they can still interfere and create an interference pattern.
It's important to note that interference is a complex phenomenon that depends on various factors, including wavelength, path difference, amplitude, and phase. The resulting interference pattern can be constructive, destructive, or somewhere in between, depending on these factors.