
Consider an elastic string under tension which is at rest along the

dimension. Let

,

, and

denote the unit vectors in
the

,

, and

directions, respectively. When a wave is
present, a point

originally at

along the string is
displaced to some point

specified by the
displacement
vector
Note that typical derivations of the
wave equation consider only the
displacement

in the

direction. This more general treatment
is adapted from [
122]. An alternative clear
development is given in [
391].
The displacement of a neighboring point originally at

along the string can be specified as
Let

denote string tension along

when the string is at rest, and

denote the vector tension at the point

in the present displaced
scenario under analysis. The net vector
force acting on the infinitesimal
string element between points

and

is given by the vector sum of
the force

at

and the force

at

, that is,

. If the string
has stiffness, the two forces will in general not be tangent to the string
at these points. The
mass of the infinitesimal string element is

,
where

denotes the mass per unit length of the string at rest. Applying
Newton's second law gives
 |
(B.31) |
where

has been canceled on both sides of the equation. Note
that no approximations have been made so far.
The next step is to express the force

in terms of the tension

of the string at rest, the elastic constant of the string, and
geometrical factors. The displaced string element

is the
vector
having magnitude
 |
(B.34) |
Let's now assume the string is perfectly flexible (zero stiffness) so
that the direction of the
force vector

is given by the unit
vector

tangent to the string. (To accommodate
stiffness, it would be necessary to include a force component at right
angles to the string which depends on the curvature and stiffness of
the string.) The magnitude of

at any position is the rest
tension

plus the incremental tension needed to stretch it the
fractional amount
If

is the constant
cross-sectional area of the string, and

is the
Young's modulus (stress/strain--the ``
spring constant''
for solids--see §
B.5.1), then
so that
![$\displaystyle \mathbf{K}= \left[K+ SY\left(\frac{ds}{dx} - 1\right)\right]\frac{d{\bf s}}{ds}$](http://www.dsprelated.com/josimages_new/pasp/img3016.png) |
(B.35) |
where no
geometrical limitations have yet been placed on the
magnitude of

and

, other than to prevent the
string from being stretched beyond its elastic limit.
The four equations (
B.31) through (
B.35) can be combined
into a single
vector wave equation that expresses the
propagation of waves on the string having three
displacement
components. This
differential equation is
nonlinear, so that
superposition no longer holds. Furthermore, the three
displacement components of the wave are
coupled together at all
points along the string, so that the
wave equation is no longer
separable into three independent
1D wave equations.
To obtain a linear, separable
wave equation, it is necessary to assume
that the
strains

,

, and

be
small compared with unity. This is the same assumption
(

) necessary to derive the
usual
wave equation for
transverse vibrations only in the

-

plane.
When (
B.35) is expanded into a
Taylor series in the strains,
and when only the first-order terms are retained, we obtain
 |
(B.36) |
This is the
linearized wave equation for the string, based only
on the assumptions of elasticity of the string, and strain magnitudes
much less than unity. Using this linearized equation for the force

, it is found that (
B.31) separates into the three wave
equations
where

is the
longitudinal wave velocity, and

is the
transverse wave velocity.
In summary, the two
transverse wave components and the longitudinal
component may be considered
independent (
i.e., ``superposition''
holds with respect to vibrations in these three dimensions of
vibration) provided powers higher than 1 of the strains (relative
displacement) can be neglected,
i.e.,

and
The physical forward
momentum carried by a
transverse wave
along a string is conveyed by a secondary
longitudinal wave
[
391].
A less simplified
wave equation which
supports
longitudinal wave momentum is given by [
391, Eqns. 38ab]
where

and

denote longitudinal and
transverse
displacement, respectively, and the commonly used ``dot'' and
``prime'' notation for partial derivatives has been introduced,
e.g.,
(See also Eq.

(
C.1).) We see that the term

in the first equation above
provides a mechanism for
transverse waves to ``drive'' the generation
of longitudinal waves. This coupling cannot be neglected if momentum
effects are desired.
Physically, the rising edge of a transverse wave generates a
longitudinal displacement in the direction of wave travel that
propagates ahead at a much higher speed (typically an order of
magnitude faster). The falling edge of the transverse wave then
cancels this forward displacement as it passes by. See
[
391] for further details (including computer
simulations).
Next Section: Properties of GasesPrevious Section: Properties of Elastic Solids