Interference & Superposition of Waves

Superposition is the physical process that occurs when two or more waves travel through the same medium simultaneously. At any point where the waves overlap, they combine to form a single resultant wave pulse.

The Principle of Superposition states that the resultant displacement at any point is the vector sum of the individual displacements of the overlapping waves. This means that if one wave has a displacement of $+y_1$ and another has $+y_2$, the total displacement is $y_{total} = y_1 + y_2$.

Interference is the specific term used to describe the pattern or effect produced by the superposition of waves. It results in a new wave amplitude that can be larger, smaller, or even zero compared to the original waves.

The type of interference at any specific point in space is determined by the path difference, which is the difference in the distance traveled by waves from their respective sources to that point.

Constructive Interference occurs when the path difference is a whole number of wavelengths, expressed as $\Delta L = n\lambda$, where $n = 0, 1, 2, ...$. In this state, the waves arrive in phase (peaks align with peaks).

Destructive Interference occurs when the path difference is an odd number of half-wavelengths, expressed as $\Delta L = (n + \frac{1}{2})\lambda$. In this state, the waves arrive in antiphase (peaks align with troughs), leading to cancellation.

Identify the Phase: When solving problems, always check if the waves are meeting 'peak-to-peak' or 'peak-to-trough'. If they meet at the same point in their cycle, the interference is constructive; otherwise, it is likely destructive.

Check for Coherence: If a question asks why an interference pattern is not visible, the most common answer is that the sources are not coherent (they lack a constant phase relationship).

Sanity Check for Path Difference: Always express path difference in terms of $\lambda$ first. If you find a path difference of $2.5\lambda$, you immediately know it is destructive interference because of the $.5$ remainder.

Common Mistake: Students often confuse 'zero displacement' with 'zero energy'. In destructive interference, the energy is not lost; it is simply redistributed to areas of constructive interference.