Re: [Phys-l] Wave speed

[John Denker <jsd@av8n.com>]

On 07/13/2009 11:02 AM, David Abineri wrote:
> I have taught in my High School Physics classes that the speed of a 
> mechanical wave is determined by the medium. 

Well, for sure the wave speed _depends_ on the medium, but
it is not solely "determined" by the medium.

In the real world, propagation often depends on various factors
including the wavelength, the polarization, and symmetry of the 
situation.  The speed of a long wave in shallow water depends 
on the depth of the channel (in addition to wavelength, symmetry, 
et cetera).

The general term for all this is _dispersion_.  

  By symmetry I mean the difference between planar, cylindrical,
  and spherical waves.  The wave equation in polar coordinates
  is dispersive even when plane waves in the same medium would
  be non-dispersive.

The _dispersion relation_ gives wavelength as a function of
frequency (and/or vice versa).
  http://en.wikipedia.org/wiki/Dispersion_relation

If you're good with scaling arguments, you can construct some
useful dispersion relations without having to solve the equations
of motion in any detail.
  http://www.av8n.com/physics/dimensional-analysis.htm#sec-synthesis

The phase velocity and group velocity can be calculated from
the dispersion relation.

Familiar examples of dispersive propagation of light include
the spectra produced by rainbows and lead-glass prisms.

A familiar example of dispersion in polar coordinates is the
sound of an explosion.  Up close it goes "snap!" but farther
away it goes "boooom" because the different frequency components
have been spread out in time.  (There is also some frequency-
dependent attenuation going on, but let's not worry about that 
at the moment.)

The ripples in ye olde beloved classroom ripple tank are highly
dispersive.  Using two generators you can easily demonstrate one 
wave overtaking and passing through another.

Also it is as easy as π to use a ripple tank (with a single
source) to measure the dispersion relation, i.e. frequency
versus wavelength.

The propagation of electromagnetic waves in a waveguide is highly
dispersive, especially when the frequency is not too far above
cutoff.  This is a nice pedagogical example, because the wave
equation for this case is easy to derive from first principles.
(This is in contrast to water waves, which are much harder to
analyze exactly.)

The QM wave equation for the motion of a particle is also highly
dispersive.  In fact it is the same equation as mentioned in the 
previous paragraph, if we don't worry about polarization;  the 
mass of the particle corresponds to the cutoff frequency.  Look
up "massive scalar Klein-Gordon equation".

The propagation of EM waves through the ionosphere (or any other 
plasma) is highly dispersive.  By this mechanism distant lightning
produces the the _whistlers_ you sometimes hear on an AM radio.
This phenomenon is put to good practical use by instruments that
determine (roughly) the distance to the lightning, which is of
interest to pilots.  (You want to stay away from thunderstorms,
because the turbulent air currents that produce the lightning may
be strong enough to pull the wings off your airplane.)

The propagation of transversely-polarized sound on a stiff rod is
highly dispersive.  The Green function for bending a stiff rod 
features the fourth derivative (in contrast to bending a supple
string under tension, which features only the second derivative).
There is enough stiffness in piano wire to be non-negligible, even 
relative to the tremendous tension.  The stiffness makes the piano 
significantly anharmonic.

Dispersion should not be confused with nonlinearity.

================

There are really very few examples of NON-dispersive propagation.
 -- Plane waves of sound in air.
 -- Plane waves of electromagnetism in vacuum.
 -- Transverse waves on a supple string under tension.
 -- a few more.

In other cases, if you see a wave, you should assume it is dispersive.
The non-dispersive case is quite an unusual special case.