The principle of superposition for waves

TL;DR

This paper presents a simple, conceptual method for calculating wave superposition using Gauss's arithmetic sequence sum, highlighting phase and amplitude modulation effects for waves with slightly different frequencies.

Contribution

It introduces an accessible teaching approach for wave superposition, emphasizing phase modulation, which is often overlooked in textbooks.

Findings

Superposition can be constructed using Gauss's method.

Waves with close frequencies produce phase and amplitude modulation.

The method simplifies understanding of wave interference phenomena.

Abstract

In this paper we will argue that the superposition of waves can be calculated and taught in a simple way. We show, using the Gauss's method to sum an arithmetic sequence, how we can construct the superposition of waves - with different frequencies - in a simple conceptual way that it is easy to teach. By this method we arrive to the usual result where we can express the superposition of waves as the product of factors, one of them with a cosine funtion where the argument is the average frequency. Most important, we will show that the superposition of waves with sligthly different frequencies produces a \emph{phase modulation} thogether with an amplitude moldulation. It is important to emphasize this result because, to the best of our knowledge, almost all textbooks only mention that there is an amplitude modulation.