# 11: Two and Three Dimensions

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

The concepts of space translation invariance and local interactions can be extended to systems with more than one space dimension in a straightforward way. But in two and three dimensions, these ideas alone are not enough to determine the normal modes of an arbitrary system. One needs extra tricks, or plain hard work.

## Preview

Here, we will only be able to discuss the very simplest sort of tricks, but at least we will be able to understand why the problems are more difficult.

1. We begin by explaining why the angular wave number, $$k$$, becomes a vector in two or three dimensions. We find the normal modes of systems with simple boundary conditions.
2. We then discuss scattering from planes in two- and three-dimensional space. We derive Snell’s law of refraction and discuss total internal reflection and tunneling.
3. We discuss the example of Chladni plates.
4. We give a two-dimensional example of a waveguide, in which the waves are constrained to propagate only in one direction.
5. We study water waves (in a simplified version of water).
6. We introduce the more advanced topic of spherical waves.

This page titled 11: Two and Three Dimensions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Howard Georgi via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.