# symmetry-short-wave-beams.mp4

Chapter section: Wave Propagation and Interference

Caption (verbatim):

Interference and Rays

Image description (verbatim):

Waves through fixed openings becoming narrow beams as the wavelength decreases

## Before the animation

Chapter lines 1071–1073:

```math
x = x_0 + v_0t + \frac{1}{2}at^2
```

Chapter lines 1075–1075:

Such "physically valid" paths in the macroscopic world have a fascinating quality that can be leveraged to find the laws of motion that predict them. They are such that some quantity associated with possible paths, which is called **action**, is extremized at the valid path. In the next sections, we will explore the relationship between objects following paths and wave propagation. Just as an object following a definite path extremizes action, a wave, in the regime where it behaves like a ray, follows a path that extremizes accumulated phase. We will establish how contributions from many possible paths combine to produce this behavior, giving us a bridge between wave propagation and action extremization. We will then see how this stationary-path limit connects the quantum wave description we have alluded to with the well-defined paths that action extremization predicts for everyday macroscopic objects.

## After the animation

Chapter lines 1083–1083:

Thus far we have described the symmetry group of a wave with a single translation direction $x$ and wave number, $k$. If the wave is to propagate, we also require that the wave represent time translation. We also need to require that time translation commute with spatial translation, for otherwise, it would change the mode composition over time, and spatial translation would no longer be a symmetry of nature. A single-mode travelling wave is then given by:

Chapter lines 1085–1087:

```math
M_0e^{i\left[kx-\omega t\right]}.
```