10.6 Networks of uniform strings  (Page 2/3)

The boundary of the network are the nodes that are associated with only one string. Also tension at a point in the string is equal to the following

One should interpret this tension as the force that the segment of string i from $\widehat{s}$ to 1 would exert upon the segment of string i from 0 to $\widehat{s}$ .

To solve the network wave equation we must also have initial conditions, $u_i(s,0)=f_i\left(s\right)$ and ${\displaystyle \frac{\partial u_i}{\partial t}}(s,0)=g_i\left(s\right)$ . However since we are concerned with determining eigenvalues and eigenmodes, initial conditions are not necessary. In this context an eigenvalue/eigenmode of the network wave equation is a solution whose shape is independent of time that is the shape is invariant. The amplitude may change but the shape is the same. More concretely we are assuming that solutions have the form

where we have used u _i with two parameters to denote the solution of the wave equation and with one parameter is an eigensolution to the wave equation. The use of complex numbers is a bit of mathematical trickery and to get at something which has physical meaning we would work with the real part of u _i . For the form we have assumed our eigenmodes are oscillatory, such assumption is reasonable considering that the wave equation conserves energy.

To compute eigenvalues for the network wave equation we must determine solutions, λ and u _i which satisfy the following:

as well as the boundary conditions given above. From a first course in differential equations one might suspect the general solution for each stringwould involve a combination of sines and cosines. Such suspicions would be correct. For any given string we have two coupled second order differential equations for the x and y components of the displacement of the string. By appealing to one's physical intuition or to the eigenvectors of the P _i matrix one can decouple these equations yielding equations for the transverse and longitudinal displacements.The general solution is as follows

where the first term accounts for the longitudinal displacement and the second term accounts for the transverse displacement. In light of the general solution, the eigenvalue problemamounts to determining the λ for which there exist $a_{ij}$ that will satisfy all the boundary conditions. To answer this question one must have a total of $4N$ equations.

In the examples to follow, we let the longituidnal stiffness be one, $\sigma =1$ and we observe how the eigenvalues of different string networks vary as we alter the transvere stiffness. The eigenvalues should exhibit three different kinds of behavior. The eigenvalues whose modes consist solely longitudinal displacements should not vary at all as the longiutdinal stiffness is constant. Eigenvalues whose modes consist solely of transverse displacements should vary as ${\displaystyle \frac{1}{\sqrt{k}}}$ . We would like to understand the behavior of eigenvalues whose modes involve both transverse and longitudinal displacements. An understanding of these mixed modes may prove useful in solving inverse problems, deducing connectivity of network based on the behavior of the egienvalues.