At 03:59 PM 1/26/00 -0500, Ludwik Kowalski wrote:
>
>It is good that stiffness waves are ignored in introductory courses.
>The tension and compression waves are already quite difficult for
>students to grasp.
I'm not sure what "introductory" means. Can you be more specific: Ninth
graders? Twelfth graders? College freshman physics majors who presumably
had at least a year of physics in high school?
I first ran into 'em as a college freshman. Didn't hurt me too much. :-)
-- Big waves on the surface of the ocean are dispersive.
-- Small "capillary waves" in a pot are dispersive, with a different
dispersion relation.
-- Radio waves in the ionosphere are dispersive.
-- Light waves in a prism are dispersive.
-- Shrödinger's wave equation is dispersive.
>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?
OK....
>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.
The foregoing paragraph seems to assume a totally nondispersive
situation. That way of looking at things will lead to trouble as soon as
there is dispersion.
In particular, it's bad luck to speak of "the" wave equation. There's lots
of wave equations.
>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.
Again: Shape-preserving travelling waves are associated with nondispersive
propagation.
>Here my dilemma. The stiffness wave satisfies the differential
>wave equation, as demonstrated by John (see below) .
Nope. If you re-read my derivation you'll see it explicitly applies to
***longitudinal*** waves driven by compression. It isn't even close to
right for transverse waves driven by stiffness.
>Therefore
>the initial shape must be preserved, as the wave travels to the
>left or to the right in a long stiff road.
Not true for stiffness waves.
>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.
Yes, that's what dispersion implies.
>John wrote:
> >
> > 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
^^^^^^^^^^^^^^^^^^^^^
>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.
Yup. Still true.
>But in the previous message John showed that stiffness waves do satisfy
>the second order equation.
Nope.
> How is flexion different from stiffness?
They describe the same physics. Flexiness is in some sense the reciprocal
of stiffness, but it's the same idea.
>It seems to me that Feynman deals with statics in this section, not with
>waves.
True, but the static physics tells you the forces, and the mass is obvious,
so that hard part of figuring out the dispersive wave equation is figuring
out how to set up the problem. In particular, the result that the flexy
spring constant depends on the *cube* of the length should tell you that
flexion is different from compression (which has a first-order dependence).
> > 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.
Yup. Still true.
> > 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