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# 6. Summary

[ "article:topic", "authorname:ucd7" ]

1. This section did not involve much in the way of new material; most of it was exploring the implications of the idea of superposition:

2. For mechanical waves this means the displacement of the medium $$\Delta y$$. For sound waves "displacement" can refer to the change in pressure $$\Delta P$$. For light waves it refers to the magnitude of the change in the electric or magnetic field.  Mathematically, superposing the displacement of two waves can be written as

$\Delta y_{total} (\text{at location } x \text{, time } t) = \Delta y_1 (\text{same location } x \text{, same time } t) + \Delta y_2 (\text{same location } x \text{, same time } t)$

3. We introduced

• Constructive Interference,  where the two waves added together maximally.  For harmonic waves this implies that they are out of phase by $$\Delta \Phi = 0, ±2 \pi, ±4 \pi, ±6 \pi, . . .$$
• Destructive interference,  where the waves cancelled each other maximally. For harmonic waves this implies that they are in phase, or out of phase by $$\Delta \Phi =± \pi, ±3 \pi, ±7 \pi, . . .$$
• Partial Interference:  any interference that is not completely constructive or destructive (where $$\Delta \Phi$$ is not an integer multiple of $$\pi$$).

4. Three terms contribute to the interference condition. We need to consider the combined effect of a possible path length difference $$\Delta x$$, the effect of the sources being in phase or not $$\phi$$ and that each wave might have a different frequency (which leads to an interference called beats).

5. The actual type of interference only depends on total phase difference $$\Delta \Phi$$.

6. We also introduced the phase chart, which is a useful tool in organizing the terms mentioned above.