The string model: d'Alembert solutions, fixed-end modes and uniqueness for the adopted one-dimensional continuum

Adopt the one-dimensional string model: a uniform string of length $L>0$ (fixed ends) or of infinite extent, with linear mass density ${\rho }_{\mathrm{\ell }}>0$ and tension ${\tau }_s>0$, whose small transverse displacement $u:I\times J\to \mathbb{R}$ (where $I\subseteq \mathbb{R}$ is a nonempty time interval and $J=\mathbb{R}$ or $J=[0,L]$) satisfies

$u_{tt}=c^2{u}_{xx},c:=\sqrt{{\tau }_s\mathrm{/}{\rho }_{\mathrm{\ell }}},$

where $[L]=[x]=\mathrm{\ell }$, $[t]=\tau$, $[{\rho }_{\mathrm{\ell }}]=\mu {\mathrm{\ell }}^{-1}$, $[{\tau }_s]=\mu \mathrm{\ell }{\tau }^{-2}$, and $[u]=\mathrm{\ell }$. The initial displacement data $f$ and $F$ have dimension $\mathrm{\ell }$, while the initial velocity data $g$ and $G$ have dimension $\mathrm{\ell }{\tau }^{-1}$. Thus $[c]=\sqrt{[{\tau }_s]\mathrm{/}[{\rho }_{\mathrm{\ell }}]}=\mathrm{\ell }{\tau }^{-1}$, and $[{\omega }_n]={\tau }^{-1}$. These are the dimensions of Continuum body data, the fixed inertial frame and the dimensions of the fields, Dimensions and SI unit conventions of the non-relativistic classical mechanics framework. The identification of $c$ with the chain limit is that of The continuum string model as the proved limit of the mass-spring chain, with explicit convergence and dispersion error. Then:

(i) Cauchy problem on the line. For $f\in C^2(\mathbb{R})$ and $g\in C^1(\mathbb{R})$ the function

$u(t,x):=\frac{f(x-ct)+f(x+ct)}{2}+\frac{1}{2c}{\int }_{x-ct}^{x+ct}g(s)ds$

is of class $C^2$ on ${\mathbb{R}}^2$ ($C^k$ Euclidean maps and diffeomorphisms), satisfies $u_{tt}=c^2{u}_{xx}$ with $u(0,\cdot )=f$ and $u_t(0,\cdot )=g$, and is the unique $C^2$ field on ${\mathbb{R}}^2$ with those data.

(ii) Fixed ends. Let $F$ be odd and $2L$-periodic of class $C^5$ and let $G$ be odd and $2L$-periodic of class $C^4$ (Sine coefficients, sine series and the odd periodic extension) with finite $M_5=\sup \mathrm{\mid }{F}^{(5)}\mathrm{\mid }$, $M_4=\sup \mathrm{\mid }{G}^{(4)}\mathrm{\mid }$. Then the sine series $u$ of the smooth string theorem exists with absolutely and uniformly convergent termwise derivatives up to second order, satisfies $u_{tt}=c^2{u}_{xx}$, $u(t,0)=u(t,L)=0$, $u(0,\cdot )=F{\mathrm{\mid }}_{[0,L]}$, $u_t(0,\cdot )=G{\mathrm{\mid }}_{[0,L]}$, is the unique $C^2$ solution on $[0,T]\times [0,L]$ of that boundary-value problem, and for every $N\ge 1$ satisfies the tail bound $\mathrm{\mid }u-S_{N}u\mathrm{\mid }\le C(N+1{)}^{-4}$ with $C=2(2L^5{M}_5\mathrm{/}{\pi }^5+2L^5{M}_4\mathrm{/}(c{\pi }^5))$ on $[-T,T]\times [0,L]$, where $S_{N}u$ is the $N$-th partial sum of the series.

(iii) Reading and scope. The initial functions $f,g$ or $F,G$ are the model's initial transverse displacement and velocity; the solutions are exact for the model and are not approximations within it. The model is the linear small-transverse-displacement idealization of the string: bending stiffness, longitudinal motion, external forcing and large-amplitude geometric nonlinearity are excluded, and no measured tension, density or displacement is asserted.

• Energy reading. For the fixed-end model the uniqueness argument of [F2] uses the conserved energy $\frac{1}{2}{\int }_0^L(u_t^2+c^2{u}_x^2)dx$: a difference of two solutions with zero data has zero energy and therefore vanishes. Multiplication by the constant linear density ${\rho }_{\mathrm{\ell }}$ gives the model's mechanical energy for the string under the linear idealization; it is not a measured quantity on this page.

• Fixed-end data and the odd extension. The fixed-end problem is treated through odd $2L$-periodic extensions of the data (Sine coefficients, sine series and the odd periodic extension); the $C^5$ and $C^4$ smoothness of the extensions is what makes the coefficient tails and the termwise differentiated series converge, and it is a genuine restriction on the initial data of the theorem. Rough or merely continuous data require a weak formulation, which this linear model does not supply.