The requirement for a small path difference between waves from two sources is often related to interference phenomena, such as constructive or destructive interference. Interference occurs when two or more waves superpose, meaning they combine or interact with each other.
To understand why a small path difference is necessary for certain interference patterns, let's consider two coherent sources emitting waves of the same frequency. When these waves overlap, they can either reinforce or cancel each other out, depending on the phase relationship between them.
Constructive interference occurs when the peaks of one wave align with the peaks of the other wave, or when the troughs align with the troughs. In this case, the amplitudes of the waves add up, resulting in an overall larger amplitude at the points of overlap. The constructive interference pattern will be most pronounced when the path difference between the two waves is an integer multiple of their wavelength (λ).
Destructive interference, on the other hand, occurs when the peaks of one wave align with the troughs of the other wave. In this case, the amplitudes of the waves subtract from each other, resulting in an overall smaller amplitude at the points of overlap. Destructive interference is most significant when the path difference is a half-integer multiple of the wavelength (λ/2).
Therefore, for constructive or destructive interference to occur and create distinct interference patterns, it is necessary for the path difference between the waves to be small compared to the wavelength. If the path difference becomes too large, the interference effects become less noticeable or even negligible.
It's worth noting that this requirement applies to specific interference phenomena, such as Young's double-slit experiment or the interference patterns observed with thin films. In other contexts, such as diffraction or general wave propagation, the path difference can be large and still contribute to observable effects. The importance of the path difference depends on the specific phenomenon under consideration.