The continuum string model as the proved limit of the mass-spring chain, with explicit convergence and dispersion error

Let $L>0$, let ${\rho }_{\mathrm{\ell }}>0$ be a linear mass density and let ${\tau }_s>0$ be a tension scale, both in the units of Dimensions and SI unit conventions of the non-relativistic classical mechanics framework, and put $c:=\sqrt{{\tau }_s\mathrm{/}{\rho }_{\mathrm{\ell }}}$, of dimension $[c]=\mathrm{\ell }{\tau }^{-1}$ (Continuum body data, the fixed inertial frame and the dimensions of the fields). For an integer $N\ge 2$ put $a:=L\mathrm{/}N$ and consider the cyclic spring chain of The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class with parameter $c$ and spacing $a$, equivalently the chain of $N$ nodes of mass $m={\rho }_{\mathrm{\ell }}a$ and spring constant $k_s={\tau }_s\mathrm{/}a$; the two descriptions agree because $c^2\mathrm{/}{a}^2=k_s\mathrm{/}m$. Let $T_{\max }\ge 0$ and let $[0,T_{\max }]$ be the comparison interval. Let $u:[0,T_{\max }]\times \mathbb{R}\to \mathbb{R}$ be $C^2$ in $t$ and $C^4$ in $x$, $L$-periodic in $x$, and satisfy

$u_{tt}=c^2{u}_{xx}\text{on }[0,T_{\max }]\times \mathbb{R},$

with $M:=\sup \mathrm{\mid }{\mathrm{\partial }}_x^{4}u\mathrm{\mid }<+\mathrm{\infty }$ and $M_2:=\sup \mathrm{\mid }{\mathrm{\partial }}_x^{2}u\mathrm{\mid }<+\mathrm{\infty }$. Smoothness on this closed interval means that $u$ is the restriction of a map on an open time neighborhood of $[0,T_{\max }]\times \mathbb{R}$ with the indicated derivatives continuous up to the comparison interval, exactly as in Finite-time continuum limit of the cyclic spring chain: uniform $O(a^{3\mathrm{/}2})$ nodal and uniform interpolation convergence. All suprema are over the comparison interval and all $x\in \mathbb{R}$. A chain solution and its state $(q,q^{\mathrm{\prime }})$ are likewise restrictions from an open time interval containing $[0,T_{\max }]$; this convention includes $T_{\max }=0$. Let $q$ be the chain solution with the sampled initial data $q_j(0)=u(0,ja)$, $q_j^{\mathrm{\prime }}(0)=u_t(0,ja)$, put $e_j(t):=q_j(t)-u(t,ja)$ and $C:=c^2\sqrt{L}M\mathrm{/}12$. In the physical reading, $u$ is the transverse displacement of the one-dimensional string model: $[L]=[a]=[x]=\mathrm{\ell }$, $[t]=[T_{\max }]=\tau$, $[u]=[q_j]=[e_j]=\mathrm{\ell }$, $[{\rho }_{\mathrm{\ell }}]=\mu {\mathrm{\ell }}^{-1}$, $[{\tau }_s]=\mu \mathrm{\ell }{\tau }^{-2}$, and $[c]=\mathrm{\ell }{\tau }^{-1}$; consequently $[M]={\mathrm{\ell }}^{-3}$ and $[M_2]={\mathrm{\ell }}^{-1}$. In clause (iii), $A$ is a displacement amplitude, $[A]=\mathrm{\ell }$, $[\kappa ]={\mathrm{\ell }}^{-1}$ and $[\omega ]={\tau }^{-1}$. Then:

(i) $q$ exists and is unique on $[0,T_{\max }]$, and for every $t\in [0,T_{\max }]$

$\mathrm{\parallel }e(t){\mathrm{\parallel }}_a\le \frac{C{a}^2{t}^2}{2},\mathrm{\parallel }{e}^{\mathrm{\prime }}(t){\mathrm{\parallel }}_a\le C{a}^{2}t,{\max }_j\mathrm{\mid }{e}_j(t)\mathrm{\mid }\le \frac{C{a}^{3\mathrm{/}2}{t}^2}{2},$

in the discrete norm of The cyclic spring chain, its discrete energy, and the finite-regularity periodic comparison class; consequently the nodal displacements converge to the sampled continuum motion uniformly on $[0,T_{\max }]$ as $a\to 0$ with $L,T_{\max }$ fixed.

(ii) With $I_{a}q$ the piecewise-linear interpolation of the chain data on the circle of circumference $L$,

${\sup }_{x\in [0,L]}\mathrm{\mid }{I}_{a}q(t,x)-u(t,x)\mathrm{\mid }\le {\max }_j\mathrm{\mid }{e}_j(t)\mathrm{\mid }+\frac{M_2{a}^2}{8}(0\le t\le T_{\max }),$

so the interpolated chain displacement converges uniformly to the continuum solution as $a\to 0$.

(iii) For $A\mathrm{\ne }0$ and cyclic wave numbers $\kappa Na\in 2\pi \mathbb{Z}$, the travelling ansatz $q_j(t)=A\cos (\kappa ja-\omega t)$ solves the chain equation if and only if ${\omega }^2=(4c^2\mathrm{/}{a}^2){\sin }^2(\kappa a\mathrm{/}2)$, and for these solutions with $\mathrm{\mid }\kappa a\mathrm{\mid }\le 1$ one has $\mathrm{\mid }{\omega }^2-c^2{\kappa }^2\mathrm{\mid }\le c^2{\kappa }^4{a}^2\mathrm{/}12$: the continuum dispersion relation is the long-wave limit with an explicit second-order error in the spacing for fixed wave number.

The result justifies the continuum string model with speed $c$ as the proved limit of the discrete mechanical chain in this regime; it asserts no exactness of the string model for any material and no measured constant.

• What the limit does and does not say. The theorem compares the chain to a solution of the continuum wave equation and quantifies the difference; it does not derive the wave equation from Newton's laws alone, because the mass and stiffness scaling $m={\rho }_{\mathrm{\ell }}a$, $k_s={\tau }_s\mathrm{/}a$ is part of the adopted correspondence between discrete and continuum data, and the smooth periodic comparison class selects the regime. The string model is an idealization of the discrete mechanical system, with the stated error.

• Exact solutions of the string model. The chain-limit comparison presupposes a smooth solution $u$; the companion theorem of this page supplies such solutions from data (d'Alembert formula, fixed-end modes and the sine-series solution), so the two items together give a complete conditional statement about the string model and its discrete counterpart.