Derivation of wave equation

Linear Methods of Applied Mathematics

Evans M. Harrell II and James V. Herod*

#### *(c) Copyright 1994,1995,1996 by Evans M. Harrell
II and James V. Herod. All rights reserved.

version of 8 August 1996

The physics and mathematics of the vibrating string were studied by Jean le Rond
d'Alembert, and later by Joseph Louis
Lagrange, Leonhard
Euler, and Daniel
Bernoulli, who gave a satisfactory
discussion of the physics of the vibrating string. I have not been able to locate a detailed
discussion of Bernoulli's derivation of the wave equation, but it is likely that he based it on
an energy principle, somewhat as follows.
First let us recall that in classical mechanics, Newton's equations for the motion of a particle
are equivalent to the vanishing of the first variation of the
Lagrangian action integral,

The kinetic energy is usually of the form

while the potential energy PE may be a function of position, U(**q**). (I
call the position **q** to avoid confusion with an x used below). The
action integral is calculated along a trajectory C = (**q**(t)) beginning at
position and time coordinates (**q**_{0}, t_{0}) and ending at coordinates
(**q**_{1}, t_{1}). The trajectory chosen by physics is one which is stationary
with respect to variations of the path. In other words, if we replace C with
C() = (**q**(t) +
**h**(t)),

where **h**(t) is a smooth function with **h**(t_{0}) = **h**(t_{1}) = 0,
and we calculate *L*(
), then
(You may hear that the physical trajectory *minimizes* the action, but
this is only a necessary condition for minimum, and the physical trajectory is
not always an actual minimum.) If we expand *L* in powers of
and
retain only the first-order term, the stationary condition becomes
and if we integrate by parts, it is:
(The boundary terms vanish because of the conditions **h**(t_{0}) =
**h**(t_{1}) = 0, which served to fix the beginning and end of the trajectory.)
If, now, we let the three components of **h** range over a complete set such
as sin(n
t/L), with L = t_{1}-t_{0}, we see that the only possibility is for
which is Newton's law.

The big advantage of Lagrangian mechanics is that it allows us relatively
easily to find the equations of motion of an extended body, such as a string.
Suppose now that we have a taut string along the x-axis between positions a and
b, but displaced laterally by an amount u(t,x). If the density is
,
then the kinetic energy of a small bit of string at position x is

so the total kinetic energy is
It is plausible that the microscopic force transmitted by the string to a point
x is proportional to the amount by which the string is stretched at x, i.e., it
depends on the arc length element ds at x. As we know,
Thus we may assume that the differential potential energy depends on u only
through u_{x}. The total potential energy would be
of the form
The Lagrangian action integral will be the integral of KE - PE with respect to
time. Notice that it is u and not x which corresponds to the **q** used
above; the "trajectory" of the string is specified by the function u(t,x), and
a variation would entail replacing u(t,x) with
u(t,x) +
h(t,x),

where h(t_{0},x) = h(t_{1},x) = 0. If the ends of the string are fixed, we would
also have
h(t,a) = h(t,b) = 0. The action is a double integral,

and if we use Taylor's theorem to keep only the first-order term in
Lagrange's condition reads:
This general formula may be useful in deriving realistic wave equations for
non-homogeneous strings, but let us simplify at this stage by assuming that
and T := 2 F' are constants. If we integrate the first integrand by
parts in the t variable and the second by parts in the x variable, we now find
that:
If h is arbitrary enough to run through a complete set, we must conclude that

u_{tt} = c^{2} u_{xx},
(WE)

with c^{2} = T/
a positive constant with dimensions of
velocity^{2}.
Judging from guitar strings, for which a 1 meter taut string gives a musical note in the mid range of the musical scale, typical values of c for thin metal strings are on the order of 1000 m./sec. The wave equation (WE) also describes one-dimensional acoustic waves (c ~ 344 m/sec. in air at room temperature or 330 m./sec. at 0 C) and light waves (c ~ 300,000,000 m./sec.), although the physical derivation in these cases is very different.

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