The network wave equation

Interpretation of results

In this section, we examine the eigenvalues of networks computed from our finite element discretization. While analysis of these values turns out to be difficult, we examine closed form solutions for a similar eigenvalue problem on networks from a paper by Joachim Von Below .

Numerical results

Unlike our original wave equation $u_{xx}=u_{tt}$ , our networks of strings are allowed three-dimensional freedom of motion. We can apply our network wave equation to a single string

where $v$ is the unit vector specifying orientation of our string and $P=k[(s-1)I-v{v}^T]$ . Assume without loss of generality that $v=\overrightarrow{k}={[0,0,1]}^T$ and that $k=1$ , $s=2$ . Then $P$ is simply a diagonal matrix and our equation $u_{xx}=P{u}_{tt}$ becomes

where $u_i,u_j,u_k$ are the the displacements of our string in the $i,j,k$ directions. Since each of the equations is independent of the others, we can solve for the eigenvalues and eigenfunctions of each one-dimensional wave equation separately

Then, if $\lambda$ is an eigenvalue of our one-dimensional wave equation, the eigenvalues ${\lambda }_i,{\lambda }_j$ and ${\lambda }_k$ for the three dimensional wave equation are

We can see this captured in Figures and - the eigenvalues of our discretization are the interleaved eigenvalues of three one dimensional wave equations. For general orientations, the result is the same.

We can still trace out the linear progression of the eigenvalues here. However, the eigenvalues of a network of strings turn out to be far more interesting and unpredictable.

presents the first few eigenvalues of a Y-shaped network of strings, similar to our tritar mentioned previously. Even among simple webs such as the tritar, the pattern of the progression of eigenvalues is not easily deduced.

We can observe a few parts at which the eigenvalue behavior mimics the three dimensional single string. At values around $1.5i$ and $3i$ , there are double eigenvalues reminiscent of our double eigenvalues in , but the pattern of the rest of the eigenvalues is much less coherent.

Figures to are FEM calculations of several eigenmodes of a more complex network. Note that as the number of legs and connections increase, the number of degrees of freedom for the movement of each leg (and thus the number of possible eigenmodes of the network) should increase as well. The eigenvalues for the more complex network exhibit a similarly nonlinear pattern in the progression of the eigenvalues of the tritar.

Closed form solutions

To better understand the nonlinear progression of our eigenvalues, we seek out closed form solutions for the eigenvalues of networks. Joachim Von Below's provides one such solution in his 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.