Second neighbor influences

Figure: a row of identical masses with identical springs connecting neighbors and another set of springs connecting second neighbors

$$\begin{eqnarray*} A & = & \frac{k}{m} \left[ \begin{array}{ccccccccc} \ldots ... ...ots & . & . & . & . & . & . & . & \cdots \end{array} \right]. \end{eqnarray*}$$

The component by component equations for the eigenvectors of a pentadiagonal matrix would have the form

$$\begin{eqnarray*} a x_{i-2} + b x_{i-1} + c x_i + b x_{i+1} + a x_{i+2} & = & \lambda x_i \end{eqnarray*}$$

which would turn into recursion relations by writing

$$\begin{eqnarray*} x_{i+2} &=& -\alpha x_{i+1}+(\lambda - \beta) x_i-\alpha x_{i-1}-x_{i-2}, \end{eqnarray*}$$

and finally an equation with a wave matrix

$$\begin{eqnarray*}\left[ \begin{array}{c} x_{i+2} \\ x_{i+1} \\ x_{i} \\ x_{i-1... ... x_{i+1} \\ x_{i} \\ x_{i-1} \\ x_{i-2} \end{array} \right]. \end{eqnarray*}$$

The characteristic equation for the wave matrix reads

$$\begin{eqnarray*} \mu^4 + \alpha \mu^3 + (\lambda - \beta) \mu^2 + \alpha \mu + 1 & = & 0 \end{eqnarray*}$$

which in turn is a quadratic equation for $c = \mu + 1/\mu = 2\cosh(\phi)$.

$$\begin{eqnarray*} c^2 + \alpha c + (\lambda - \beta) & = & 0 \end{eqnarray*}$$