The mathematics of vibrations: From music to quantum mechanics

Why do different instruments have different sounds?
Can you hear the shape of a drum? How does a laser work?

Although these seem like very different questions, from a mathematical perspective they all boil down to understanding the theory of how objects vibrate. In this video, we discuss some fundamental properties of wave equations and their relationship to eigenvalues and eigenvectors of the Laplacian. We will focus on one dimensional strings whose endpoints are fixed and in the next video will discuss more complicated regions.

The purpose of this series of videos is to provide some motivation for some recent work done by Xuan Hien Nguyen, Malik Tuerkoen, Guofang Wei and me on the fundamental gap problem on Riemannian manifolds. As this video serves as an introduction to the topic, I did not mention our work, but plan to do so in future videos.


Chapters:
0:00 Introduction
0:51 The Wave Equation
2:34 The Laplacian and its eigenfunctions
5:30 The eigenvalues of a one-dimensional string
6:18 So why do different instruments sound different?
7:28 Hearing the overtones
8:17 Laplacians and lasers
11:43 Closing thoughts

Technical Notes:

At 0:17, the animation shows the continuity method for establishing log-convexity of an eigenfunction. The associated equation is called the barrier equation and the scrolling equation shows one example of a barrier. Another example is given in the thumbnail of this video. For more details, see the following preprint: https://arxiv.org/abs/2211.06403

From a mathematical perspective, the comment at 1:08 is imposing Dirichlet conditions on the string, and I will occasionally use this fact throughout the video.

At 4:31, I'm referring to the fact that on an interval with Dirichlet conditions, these trigonometric functions form a basis of L2, which can be used to construct arbitrary solutions to the wave equations. This does not work on the entire real line however.

At 6:55, the electric piano actually sounds a bit better in person. I had to adjust the levels so that my voice would be audible above the playing, which makes it sound much more hollow than it does. And for the recording on the grand piano, the footage you are seeing is a different take than the sound. I liked this camera angle better, but there was too much background noise to use the audio.

8:06: Victor Wooten has a wonderful version of Amazing Grace using natural harmonics, which I highly recommend.    • Victor Wooten Performs Amazing Grace ...  

At 8:20, I am purposefully being quite vague about the relevant Laplacian. In general, this will be a fairly complicated Schrodinger operator with a potential.

At 9:07, I'm giving an informal description of population inversion. The precise mechanism for how most of the electrons get to a higher energy state is a bit tangent to the main point of the video.

At 11:33, the book is Introduction to Quantum Mechanics Third Edition by David Griffiths and Darrell F Schroeter.


Acknowledgements:

The gifs of a sound sound wave at 0:22 and electricity at 9:00 were made using clips from Vecteezy.com

Thanks to the Iowa State physics demonstration lab for letting me borrow a laser and to the music department for allowing me to use one of their pianos to record.

All the music was played by myself except for the piece during the outro, which was recorded by Khurshed Rastomji.