General Lesson

Generally, we can find the complex wave amplitude at any point in a problem by summing the amplitudes for all waves which contribute at a given point (principle of superposition), where we determine the complex amplitudes for these waves by comparing the motions of the different waves at different points in space, including a factor for each comparison. Each of these comparisons may be thought of as fundamental event in the history of how the wave began at the reference point and ended up at the final observation point. Such fundamental events include propagation from point $a$ to point $b$, and reflection or transmission at boundaries.

This perspective allows us to summarize our general lesson as

Sum over histories:
The complex amplitude for wave motion at point $x$ equals the complex amplitude of the incoming wave at a reference point $a$, times the sum of the amplitudes for each possible history $h$ for how the incoming wave can get from $a$ to $x$. The amplitude associated with each history is the product of the complex amplitudes $\underline{a}(e)$ for each fundamental event $e$ in that history. Mathematically,

$$Q(x)=Q(a) \sum_{h} \left( \prod_{e \in h} \underline{a}(e) \right).$$ (12)

Here, $\underline{a}(e)=e^{ik\vert b-a\vert}$ for an event $e$ of propagation from point $a$ to point $b$, and $\underline{a}_e$ is the corresponding reflection or transmission amplitude for a scattering event.