View source for 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
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
studies is not zero.

== The discrete chain ==

[[File:OPL-HP-F-28.png|thumb|right|400px|Fundamental modes with both ends fixed and with one end free.]]

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

:<math>\sqrt{h^2 + (u_{j+1}-u_j)^2} - h
\approx \frac{(u_{j+1}-u_j)^2}{2h}
\qquad\qquad\mathrm{(1)}</math>

so the Lagrangian is

:<math>L = \sum_j \left[ \tfrac{1}{2}m\dot u_j^2
- \frac{\tau}{2h}\bigl( u_{j+1} - u_j \bigr)^2 \right]
\qquad\qquad\mathrm{(2)}</math>

This is an ordinary finite system, and
[[Hamilton's principle/Mathematics/Several dependent variables]] applies. The
equation for bead <math>j</math> is

:<math>m\ddot u_j = \frac{\tau}{h}\bigl( u_{j+1} - 2u_j + u_{j-1} \bigr)
\qquad\qquad\mathrm{(3)}</math>

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

:<math>\rho h\, \ddot u = \frac{\tau}{h}\cdot h^2 u_{,xx}
\qquad \Longrightarrow \qquad
\rho\, u_{,tt} = \tau\, u_{,xx}
\qquad\qquad\mathrm{(4)}</math>

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

:<math>L = \int_0^\ell \left[ \tfrac{1}{2}\rho\, u_{,t}^2
- \tfrac{1}{2}\tau\, u_{,x}^2 \right] dx
\qquad\qquad\mathrm{(5)}</math>

'''The limit commutes with the variation.''' Varying then taking <math>h\to 0</math>
gives (4); taking the limit then varying gives (5) and, by
[[Hamilton's principle/Mathematics/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 ==

[[File:OPL-HP-F-31.png|thumb|right|400px|A bead chain approaching a continuous string.]]

The bracket in (5) is the '''Lagrangian density''' <math>\mathcal{L}</math>. Its
Euler-Lagrange equation, from (7) of
[[Hamilton's principle/Mathematics/Several independent variables]], with
<math>\partial\mathcal{L}/\partial u = 0</math>,
<math>\partial\mathcal{L}/\partial u_{,t} = \rho u_{,t}</math> and
<math>\partial\mathcal{L}/\partial u_{,x} = -\tau u_{,x}</math>, is

:<math>\rho\, u_{,tt} - \tau\, u_{,xx} = 0
\qquad\qquad\mathrm{(6)}</math>

the wave equation, with speed

:<math>c = \sqrt{\tau/\rho}
\qquad\qquad\mathrm{(7)}</math>

== Conservation laws ==

<math>\mathcal{L}</math> contains neither <math>t</math> nor <math>x</math>
explicitly, so
[[Hamilton's principle/Mathematics/Noether's first theorem]] gives two divergence
relations.

Time translation gives the energy density and its flux,

:<math>\mathcal{E} = \tfrac{1}{2}\rho\, u_{,t}^2 + \tfrac{1}{2}\tau\, u_{,x}^2,
\qquad
S = -\tau\, u_{,t}u_{,x}
\qquad\qquad\mathrm{(8)}</math>

satisfying <math>\mathcal{E}_{,t} + S_{,x} = 0</math>. 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
[[Hamilton's principle/Mathematics/Several independent variables]].

== The free end ==

Fix the string at <math>x = 0</math> and leave <math>x = \ell</math> free. The
admissible variations no longer vanish at the right end, and the surface term of
[[Hamilton's principle/Mathematics/Boundary terms and natural boundary conditions]]
survives. Its coefficient is
<math>\partial\mathcal{L}/\partial u_{,x} = -\tau u_{,x}</math>, so stationarity
requires

:<math>\tau\, u_{,x}(\ell, t) = 0
\qquad\qquad\mathrm{(9)}</math>

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 <math>-\tau u_{,x}</math> 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:

{| class="wikitable"
! ends !! conditions !! modes !! fundamental
|-
| both fixed || <math>u = 0</math> at both || <math>\sin(n\pi x/\ell)</math> || <math>c/2\ell</math>
|-
| one free || <math>u = 0</math>, <math>u_{,x} = 0</math> || <math>\sin((2n-1)\pi x/2\ell)</math> || <math>c/4\ell</math>
|}

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 <math>u = X(x)\,T(t)</math> into (6) gives
<math>X'' /X = T''/(c^2 T) = -k^2</math>, so each mode oscillates at
<math>\omega = ck</math> 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 <math>N</math> normal modes become the string's infinite set as
<math>h \to 0</math>. The chain's dispersion relation is

:<math>\omega^2 = \frac{4\tau}{\rho h^2}\sin^2\!\left( \frac{kh}{2} \right)
\qquad\qquad\mathrm{(10)}</math>

which reduces to <math>\omega = ck</math> only for <math>kh \ll 1</math>. 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
[[Hamilton's principle/Applications/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
<math>\mathcal{L}(u, u_{,\alpha})</math> in several independent variables gives
classical field theory, which is
[[Hamilton's principle/Applications/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.

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

== Next ==

[[Hamilton's principle/Applications/Classical field theory]] generalises the
construction.

== References ==

<references />