Hamilton's principle/Applications/Continuum systems: the vibrating string

The vibrating string is the simplest system with infinitely many degrees of freedom, and it is where the variational method makes its most consequential change of scope: from a finite set of coordinates to a field. This page derives the wave equation from the action, obtains it independently as a limit of a discrete chain, and works the free end, where the boundary term that Boundary terms and natural boundary conditions studies is not zero.

The discrete chain

Fundamental modes with both ends fixed and with one end free.

Begin with $N$ beads of mass $m$ spaced $h$ apart on a massless string under tension $\tau$, displaced transversely by small amounts $u_j$. For small displacements the extension of the segment between beads $j$ and $j+1$ is

$\sqrt{h^2+(u_{j+1}-u_j{)}^2}-h\approx \frac{(u_{j+1}-u_j{)}^2}{2h}\mathrm{(}\mathrm{1}\mathrm{)}$

so the Lagrangian is

$L=\sum _j[\frac{1}{2}m{\dot{u}}_j^2-\frac{\tau }{2h}(u_{j+1}-u_j{)}^2]\mathrm{(}\mathrm{2}\mathrm{)}$

This is an ordinary finite system, and Several dependent variables applies. The equation for bead $j$ is

$m{\ddot{u}}_j=\frac{\tau }{h}(u_{j+1}-2u_j+u_{j-1})\mathrm{(}\mathrm{3}\mathrm{)}$

Taking the limit. Put $m=\rho h$ with $\rho$ the linear density held fixed, and let $h\to 0$. The bracket in (3) is $h^2{u}_{,xx}+O(h^4)$, so

$\rho h\ddot{u}=\frac{\tau }{h}\cdot h^2{u}_{,xx}⟹\rho u_{,tt}=\tau u_{,xx}\mathrm{(}\mathrm{4}\mathrm{)}$

Equally, the sum in (2) becomes an integral with $\sum _{j}h\to \int dx$ and $(u_{j+1}-u_j)/h\to u_{,x}$:

$L={\int }_0^{\ell }[\frac{1}{2}\rho u_{,t}^2-\frac{1}{2}\tau u_{,x}^2]dx\mathrm{(}\mathrm{5}\mathrm{)}$

The limit commutes with the variation. Varying then taking $h\to 0$ gives (4); taking the limit then varying gives (5) and, by Several independent variables, the same equation. That agreement is not automatic for every system, and checking it is the honest way to introduce a field Lagrangian rather than postulating one.

The field derivation

A bead chain approaching a continuous string.

The bracket in (5) is the Lagrangian density $ℒ$. Its Euler-Lagrange equation, from (7) of Several independent variables, with $\partial ℒ/\partial u=0$, $\partial ℒ/\partial u_{,t}=\rho u_{,t}$ and $\partial ℒ/\partial u_{,x}=-\tau u_{,x}$, is

$\rho u_{,tt}-\tau u_{,xx}=0\mathrm{(}\mathrm{6}\mathrm{)}$

the wave equation, with speed

$c=\sqrt{\tau /\rho }\mathrm{(}\mathrm{7}\mathrm{)}$

Conservation laws

$ℒ$ contains neither $t$ nor $x$ explicitly, so Noether's first theorem gives two divergence relations.

Time translation gives the energy density and its flux,

$ℰ=\frac{1}{2}\rho u_{,t}^2+\frac{1}{2}\tau u_{,x}^2,S=-\tau u_{,t}{u}_{,x}\mathrm{(}\mathrm{8}\mathrm{)}$

satisfying ${ℰ}_{,t}+S_{,x}=0$. Space translation gives the wave momentum similarly. Both are local statements — a density and a flux obeying a continuity equation — rather than the constants of the finite case, and that is the general form of Noether's result when the divergence has more than one component, as noted at the end of Several independent variables.

The free end

Fix the string at $x=0$ and leave $x=\ell$ free. The admissible variations no longer vanish at the right end, and the surface term of Boundary terms and natural boundary conditions survives. Its coefficient is $\partial ℒ/\partial u_{,x}=-\tau u_{,x}$, so stationarity requires

$\tau u_{,x}(\ell ,t)=0\mathrm{(}\mathrm{9}\mathrm{)}$

The string meets a free end with zero slope. This is the natural boundary condition, and it was not imposed — the variational problem produced it.

Its physical reading is the general one given on that page: the conjugate momentum vanishes at a free end. Here $-\tau u_{,x}$ is the transverse force one part of the string exerts on the next, so (9) says a free end transmits no transverse force.

The two conditions give different normal modes:

ends	conditions	modes	fundamental
both fixed	$u=0$ at both	$\sin (n\pi x/\ell )$	$c/2\ell$
one free	$u=0$, $u_{,x}=0$	$\sin ((2n-1)\pi x/2\ell )$	$c/4\ell$

A string free at one end has a fundamental an octave lower than the same string fixed at both, and its overtones are the odd harmonics only. That is why a closed organ pipe sounds an octave below an open one of the same length, and it is audible confirmation of a boundary term that a careless derivation would have discarded.

Separation and modes

Substituting $u=X(x)T(t)$ into (6) gives $X^{″}/X=T^{″}/(c^{2}T)=-k^2$, so each mode oscillates at $\omega =ck$ with the spatial profile fixed by the boundary conditions. The general solution is a superposition, and the coefficients are set by the initial data.

This is the field-theoretic version of the normal-mode decomposition of a coupled oscillator system, and the discrete chain of (2) makes the connection concrete: its $N$ normal modes become the string's infinite set as $h\to 0$. The chain's dispersion relation is

${\omega }^2=\frac{4\tau }{\rho h^2}{\sin }^2\left(\frac{kh}{2}\right)\mathrm{(}\mathrm{10}\mathrm{)}$

which reduces to $\omega =ck$ only for $kh\ll 1$. The continuum description is therefore an approximation valid for wavelengths long compared with the spacing — a fact with the same shape as the geometrical-optics approximation on the optical-mechanical analogy, and the same kind of failure at short wavelength.

What generalises

Nothing in the derivation used the specific form of (5) beyond its being quadratic. Replacing the density by a general $ℒ(u,u_{,\alpha })$ in several independent variables gives classical field theory, which is Classical field theory. The string is that subject with one space dimension and one field, and every structural feature — the Euler-Lagrange equation, the Noether currents, the natural boundary condition — appears here in its simplest instance.

Provenance: every equation on this page is derived here. No primary source is cited and none is claimed.

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Classical field theory generalises the construction.