The network wave equation

The motion of vibrating strings (such as musical instrument strings or, in this case, spiderwebs) can be described by the one dimensional wave equation on an interval $x\in [0,\ell ]$ , with $u(t,0)=u(t,\ell )=0$ , where $u$ is the displacement of the string and $\ell$ is the strings length. The eigenvalues derived from this model progress in a well-known linear fashion, similar to the Western scale, leading to a pleasant sound when the string is plucked. A network of connected strings can be expressed in a similar manner; however, the progression of eigenvalues is much less regular and depends largely on the topology of the network. We examine these eigenvalues and their associated eigenvectors using a finite element discretization of such networks, then compare these results to closed form eigensolutions based on Joachim Von Below's examination of networks of strings in“A Characteristic Equation Associated to an Eigenvalue Problem on $c^2$ -Networks", Linear Algebra and its Applications , Volume 71 (1985), p309-325.

Introduction

The purpose of the Physics of Strings seminar has traditionally been to study the motion of a vibrating string by analyzing its eigenfunctions and eigenvalues, equivalent to the string's fundamental modes and fundamental frequencies, respectively. The progression of these eigenvalues and eigenvectors tells us a great deal about the string; for example, given eigenvalues of a string, we can determine how quickly its vibrations decay, and whether the frequency of a vibration affects how quickly it's damped.

The properties of the string, likewise, can tell us something about the eigenvalues. Physical constants, such as the length of the string, are proportionally related to the eigenvalues. Given data on the vibration of a string, there are also methods for reverse-engineering the eigenvalues of that string. There are several models of a vibrating string, and the most detailed ones can reproduce eigenvalues that accurately match the reverse-engineered string eigenvalues. However, while much research has been done on several models of a single string, the behavior of networks of strings is less well understood.

We seek to mathematically model and investigate the motion of networks of strings, specifically by understanding eigenvalues and the corresponding modes of vibration. We study these behaviors within the context of the tritar (a guitar-like instrument based upon a Y-shaped network of 3 strings) and in the vibrations of more complex networks such as spiderwebs.