Exact dispersion relation ${\omega }^2=(4c^2\mathrm{/}{a}^2){\sin }^2(\kappa a\mathrm{/}2)$ for the cyclic chain and its continuum error bound

Let $N\ge 2$, $a>0$, $c>0$, let $\kappa ,\omega \in \mathbb{R}$ satisfy $\kappa Na\in 2\pi \mathbb{Z}$, and let $A\in \mathbb{R}\setminus \{0\}$. Consider the travelling cosine ansatz for the cyclic chain of The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class:

$q_j(t):=A\cos (\kappa ja-\omega t)(j\in \mathbb{Z}\mathrm{/}N)\mathrm{.}$

(a) The curve $q:\mathbb{R}\to {\mathbb{R}}^N$ solves $q_j^{\mathrm{\prime }\mathrm{\prime }}=\frac{c^2}{a^2}(q_{j+1}-2q_j+q_{j-1})$ for all $t\in \mathbb{R}$ and all $j$ if and only if

$$\begin{array}{cccc}& {\omega }^2=\frac{4c^2}{a^2}{\sin }^2\text{\hspace{0.17em}⁣}\left(\frac{\kappa a}{2}\right)\mathrm{.}& & \text{(DR)}\end{array}$$

(b) If (DR) holds and $\mathrm{\mid }\kappa a\mathrm{\mid }\le 1$, then

$\mid {\omega }^2-c^2{\kappa }^2\mid \le \frac{c^2{\kappa }^4{a}^2}{12},$

so the continuum dispersion ${\omega }^2=c^2{\kappa }^2$ is recovered with an explicit error bound of second order in the spacing $a$ and fourth order in the wave number $\kappa$.

• The exact relation (DR) is bounded between $0$ and $4c^2\mathrm{/}{a}^2$; the long-wavelength branch has ${\omega }^2\to c^2{\kappa }^2$ as $\kappa a\to 0$, and the bound of (b) makes that limit quantitative. The bound is stated for $\mathrm{\mid }\kappa a\mathrm{\mid }\le 1$ only; no claim is made for short waves.

• The ansatz is a real travelling wave of constant amplitude; it is not asserted that every chain solution is a superposition of such waves, and no stability or energy estimate for the ansatz is claimed here. The continuum comparison of this item is strictly local in wave number.

The scalar remainder calculation in steps 1.2 and 3.1 proves $\mathrm{\mid }\sin z-z\mathrm{\mid }\le \mathrm{\mid }z{\mathrm{\mid }}^3\mathrm{/}6$ and $\mathrm{\mid }4{\sin }^{2}z-4z^2\mathrm{\mid }\le 4\mathrm{\mid }z{\mathrm{\mid }}^4\mathrm{/}3$ for every real $z$: those inequalities use only $\mathrm{\mid }\sin z\mathrm{\mid }\le \mathrm{\mid }z\mathrm{\mid }$, not the additional long-wave restriction of clause (b). They can therefore be used directly to check a specified compatible mode outside that restricted clause.