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.

**The wave equation**

The vibration of a string in one dimension can be understood through the standard wave equation, given by

where

or equivalently, the first order matrix equation

**Eigenvalues, eigenfunctions, and their significance**

We are especially interested in the eigenvalues

Since only trigonometric functions satisfy both our equation and our boundary conditions, our eigenfunctions take the form

These eigenfunctions constitute an infinite-dimensional basis for any solution to the wave equation, with

Intuitively, these correspond to the fundamental modes of a string - any vibration of the string can be decomposed into a linear combination of the fundamentals. The magnitude of each eigenvalue, likewise, is related to the frequency at which the corresponding fundamental mode vibrates - in other words, each eigenvalue is tied to a note in the progression of the Western scale. As we will see, this linear progression of the eigenvalues is lost when a single string is replaced by a network of strings, leading to more of a dissonant sound when a network is plucked.

**Finite element solution method**

In this report, we use the finite element method to numerically solve for solutions to the wave equation. The idea behind this method is based on picking a finite-dimensional set of

to the solution from the span of these basis functions via the solution to a matrix equation

We first rearrange our PDE into a more flexible form. Given a function

If we integrate the right hand side by parts and apply Dirichlet boundary conditions, we get

This form of the wave equation is called the equation's “weak form". Notice there is only one derivative with respect to

Let

Note that if we define a new “energy" inner product

for

where

Using the finite element method, we choose our basis functions to be piecewise linear “hat" functions. If we partition the space

for

Since the support of

We can solve for our coefficients

We can see the relation to the continuous system,

where

**Damping**

A closely related equation is the wave equation with viscous damping (resulting from a viscous medium in which the string vibrates, i.e. air). To simulate this effect, a velocity-dependent damping function

For the cases we consider here, we shall take

Thankfully, the finite element discretization of this equation doesn't involve much new work; all we do is reuse some of our calculations. If we make the substitution for

we get

Taking an inner product with

We usually refer to the matrix

As the damping factor grows from 0 to