The network wave equation

As the damping factor grows from 0 to $6\pi$ , the eigenvalues shift further left in the complex plane. At factors of $\pi$ , the two smallest magnitude eigenvalues become completely real, with one moving left and one moving right in the complex plane. The physical significance of this lies in the fact that the real part of the eigenvalue furthest in the right half plane is determines proportionally how quickly the displacement $u$ decays. For large values of $a$ , the string becomes overdamped, floating in midair, while for smaller values of $a$ , the system oscillates before coming to rest. At $a=\pi$ , the damping is optimal for bringing the string to rest most quickly. This behavior is shown in .

Networks of strings

Unlike our simple one dimensional case, it is much more difficult to determine the closed form eigenvalues and eigenfunctions of a network of strings. To this end, we apply the finite element method to numerically simulate the behavior of a network wave equation.

Network wave equation

Let the $i$ th string in a network of strings be defined on an interval from $[0,{\ell }_i]$ , where ${\ell }_i$ is the length of that particular string. To generalize the wave equation to a network of strings in three dimensions, we reference Schmidt's system of equations for the planar displacement $u_i(x_i,t)$ of the $i$ th string, where $x_i\in [0,{\ell }_i]$ . We define the "stensor" matrix

where $k_i$ is stiffness, $s_i>1$ is prestress (string tension), and $v_i$ is a unit vector specifying 3-dimensional orientation of the $i$ th string. We characterize network movement by

where ${\rho }_i$ is the $i$ th strings density. $I$ is the 3-by-3 identity matrix. Our boundary conditions are Dirichlet at endpoints (displacement is fixed at 0) and a condition enforcing force balance laws and connectivity of each leg at the joint. We define an end of the first string to have position 0, and for the other endpoints, we consider them to be at position ${\ell }_k$ on their respective $k$ th string. Our Dirichlet conditions can be written as

If we define the set $S_i$ to be the set of integer indices of all strings incident to a joint at the end of the $i$ th string, the force-balance joint conditions connecting strings in the set $\{i,S_i\}$ can be described by

This network wave equation matrix $P_i$ can also be mathematically derived from the nonlinear model of Antman; the linear, one dimensional wave equation is derived by taking the orientation vector $v$ to be a standard basis vector.

The network wave equation is much more tractable for a concrete example. We begin by covering the network wave equation for the simplest net - a Y-shaped net called a“tritar", in honor of the guitar with Y-shaped strings (see http://www.tritare.com). For our simple case, then, we have the boundary conditions

with the force balance equation