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Wave interference is a fundamental concept in physics that describes how two or more waves superpose to form a resultant wave. This phenomenon occurs when waves overlap, and it can lead to an increase (constructive interference) or decrease (destructive interference) in wave amplitude. In the context of light, a wave-like form of energy, interference results in patterns of bright and dark fringes.

This can be observed in experiments such as the double-slit experiment, where light waves pass through two narrow slits and interfere with one another on the other side, creating an interference pattern on a screen. The areas where the waves reinforce each other appear as bright bands, and the areas where they cancel out appear as dark bands. Understanding this concept is crucial for analyzing diffraction and interference patterns in exercises like the one provided.

The calculation of fringe patterns is an integral part of understanding wave interference in double-slit experiments. To determine the distance between fringes, one must know the formula that relates the slit separation, the wavelength of light, and the distance from the slits to the screen. For dark fringes, the formula is given by \[\begin{equation}d \sin(\Theta) = m\lambda\end{equation}\]where \(d\) is the distance between slits, \(\Theta\) is the angular position, \(m\) is the order of dark fringe (integer values, where \(m = 1, 2, 3, ...\) represent the first, second, third, etc., dark fringes), and \(\lambda\) is the wavelength of light.

When analyzing the problem provided, we first calculate the angular positions of the first and second dark fringes using this formula and then use the small-angle approximation (\(\tan(\Theta) \approx \sin(\Theta)\)) to convert these angles into distances on the screen. By finding the distances for both the first and second dark fringes and taking the difference, we obtain the distance between the two fringes.

Diffraction and interference are two phenomena that often occur together, especially in the study of light and wave behavior. Diffraction refers to the bending and spreading out of waves as they pass around an obstacle or through an opening. It's what allows waves to 'turn corners' and propagate beyond the geometrical shadow of the obstruction.

Interference, on the other hand, relates to the combination of two or more waves overlapping and forming a new wave pattern. In the double-slit experiment, diffraction occurs as the light waves pass through the slits and subsequently interfere with one another to produce bright and dark fringes on a screen.

Since these phenomena are closely related, their combined effect must be considered to accurately predict the resulting light pattern observed in such experiments, as exemplified in the problem at hand where understanding both has enabled us to calculate the distance between dark fringes on the screen.