Wave Phenomena and Wave Equations

TL;DR

The paper addresses how to derive the wave equation for different wave phenomena from experimentally determined dispersion relations using a plane-wave form $f(t,\vec{x}) = e^{-i\omega t + i\sum_l k_l x_l}$. It demonstrates a general mapping $g(\omega,k_1,k_2,k_3)=0$ to a differential operator by substituting $i\,\partial_t$ for $\omega$ and $-i\,\partial_{x_l}$ for $k_l$, and illustrates this with light, water waves, and matter waves. For deep-water gravity waves the dispersion $\omega^2 = gk$ leads to a 1D wave equation featuring an imaginary unit and a two-real-component description of surface motion, while for matter waves the dispersion $\omega = \frac{\hbar}{2m}k^2 + C$ yields the Schrödinger equation $i\hbar\partial_t\Psi = -\frac{\hbar^2}{2m}\Delta\Psi + V\Psi$ upon the usual operator substitutions. The discussion emphasizes that the dispersion relation fixes only the mathematical structure of the wave equation and not the physical interpretation of the wave function, highlighting interpretational questions for coherent matter waves and the two-dimensional character seen in water waves.

Abstract

For any kind of wave phenomenon one can find ways to derive the respective dispersion relation from experimental observations and measurements. This dispersion relation determines the structure of the wave equation and thus characterizes the dynamics of the respective wave. Different wave phenomena are thus governed by different differential equations. Here we want to emphasize the experimental approach to matter waves, but before doing so we will discuss and test the procedure for other types of waves, in particular water waves.

Figures (3)

• Figure 1: Wave pattern behind a duck travelling on water
• Figure 2: Propagation of deep-water surface waves is the result of a circular motion of the surface molecules. The dotted lines represent the surface. Formally, two different solutions are possible. (a) Variant 1: the individual molecules in the crest are moving in the opposite direction to the crest as indicated by the red arrows. The same is true - to a lesser extent - for the molecules close to the surface (details of the collective motion not shown). (b) Variant 2: the individual molecules in the crest are moving along with the crest. Only (b) is physically relevant. See text for details.
• Figure 3: Sketch of an apparatus for studying the diffraction of electrons after impinging on graphite. The angle $\alpha$ is proportional to the wavelength, as we know from analogous scenarios in optics. This angle changes when the electron acceleration voltage $U_0$ is varied