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Stiffness waves (was SLINKY)

It is good that stiffness waves are ignored in introductory courses.
The tension and compression waves are already quite difficult for
students to grasp. These questions below are for my own education
only. I never had a chance to teach advanced mechanics, acoustic
or e&m. My only encounter with dispersion was in optics
(accepting it as an experimental fact rather than as a theoretical
topic) and in courses I took as student nearly 4 decades ago.
How to associate dispersion with waves. Let me elaborate?

Thinking about one-dimensional waves we say that any
function of x and t in which these two independent variables
appear in the form of a (x-c*t) argument is a mathematical
description of a traveling wave. Thus any function f(x-c*t) is
a description of a possible wave. It satisfies the differential
wave equation. And vice versa, any function which satisfies
the equation must have the f(x-c*t) form, where c is a positive
or negative constant called speed. If we plot that function
versus x, for different moments of time, we see "a moving
shape". The shape is preserved, if it is an equilateral triangle
at t=0 then it must also be the equilateral triangle (same
base length) at t>0, far away.

I am assuming the dissipation of energy is negligible, so that
"far away" really means about ten characteristic distances
(such as initial shape's length), or more. In my model only
energy dissipation can be responsible for changes of shape.

Here my dilemma. The stiffness wave satisfies the differential
wave equation, as demonstrated by John (see below) . Therefore
the initial shape must be preserved, as the wave travels to the
left or to the right in a long stiff road. I am assuming that the
rod is uniform (mu and s do not change along the x axis).
This conflicts (in my mind) with what John later said about
dispersion. Dispersion implies that the shape is not preserved
(each harmonic travels with different speed) and that changes
in shape are not caused by energy dissipation. Any comment?
Also see an additional question in the body of John's second
message below.
Ludwik Kowalski

John wrote:

1) Model the macroscopic spring as a 1-dimensional lattice of tiny
lumps
connected by tiny springs. Let the lattice constant be delta_x.

2) Each lump has a rest position. Let the displacement field y(x)
represent the displacement of each lump from its rest position. In
this
longitudinal case the displacement-vector "y" is in the same direction
as
the location-vector "x" but please ignore that; it will just confuse
you. Instead, please treat them as independent quantities just as you

would in the transverse case (where y would be perpendicular to x).

3) Let the force field F(x) represent the force on the lump at (x) due
to
the spring ON THE LEFT. Therefore -F(x + delta_x) is the force due to

the spring on the right.

4) Observe that a uniform translation of the whole lattice
(represented by
y(x) = const) produces no force on the lumps.

More generally, we have that the extension of the tiny spring is
extension = (dy/dx) delta_x
and the force produced thereby is (using Hooke's law)
F(x) = -(dy/dx) delta_x / (s delta_x)
where
s = compliance per unit length.

5) Observe that a uniform compression of the whole lattice produces no

*net* force on the lumps. For each lump, the force from the spring on

the left is just cancelled by the force from the spring on the right.

So in fact the *net* force on a lump depends on the sum of the left
and
right contributions:
net force = (d/dx) (dy/dx) delta_x delta_x / (s delta_x)

6) The mass of each lump is just
m = rho delta_x.

7) Therefore the acceleration of each lump is
a(x) = net force / m = (d/dx) (dy/dx) 1 / (rho s)
where we note that delta_x has dropped out of the result, as it
should. This leads immediately to a wave equation
(d/dt) (dy/dt) - (d/dx) (dy/dx) / (rho s) = 0
in which we recognize the wavespeed
c = 1 / sqrt(rho s)
QED.

Later John Denker wrote:

.... the physics of transverse waves driven by stiffness
is different physics, it requires a more sophisticated analysis,
and leads to more complicated results.

Here's the deal: for case (a) or (b), we get a wave equation of the
form
(d/dt) (dy/dt) - (d/dx) (dy/dx) c^2 = 0
which admits running-wave solutions of the form
f(x,t) = F(x+-ct)
for any F(), and which has the simplest possible dispersion relation
omega = +- c k

IN CONTRAST... the physics of flexure is more complicated. See _The
Feynman Lectures on Physics_ volume II section 38-4 for an
introduction.

But in the previous message John showed that stiffness waves do satisfy
the
second order equation. How is flexion different from stiffness? It seems

to me that Feynman deals with statics in this section, not with waves.

There are, alas, some people on this list who rely on intuition and
seem
allergic to doing calculations or experiments. Some of them may
consider
it intuitively obvious that flexy waves must be analogous to
compressional
waves. Well, too bad, it doesn't happen to be true.

Take a long coil spring and paint the first foot red. Load it (in
extension) with a suitable weight. Measure the extension of the red
section. Take a spring that is identical except for being twice as
long,
and again paint the first foot red. Load it with the identical
weight. Observe that the extension of the first foot is the same as
before.

Now do the same thing except use a flexional spring, clamped
horizontal at
the near end and loaded with a weight at the far end. Paint the first
foot
red. Repeat with a double-length spring, and observe that the same
weight
produces *more* deflection of the first foot. The force is the same,
but
it's got more leverage and produces more flexion.

For flexy waves we have an equation of the form
(d/dt)^2 y - const * (d/dx)^4 y = 0
and yes, that's a fourth-order spatial derivative. This is a linear
differential equation (linear in y) so we retain the superposition
principle (which means this is much simpler than, say, fluid dynamics,

where the equations are nonlinear). But it is highly anharmonic, i.e.

highly dispersive. Because of the dispersion, there cannot be
running-wave
solutions that retain their shape. The dispersion relation is
omega = const * k^2