Re: REFRACTION - REFLECTION

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

> Ok, OK, but how do you explain this to a student when SHE asks:
>
> "When a beam of light enters the surface of water at an angle of 
> 25 degrees with the normal why doesn't it continue at the same 
> angle after it enters the water?

If the light rays did not bend toward the perpendicular, according to
Snell's law, as they pass from the air into the water then the time it
would take for the rays to get from one fixed place on the air side of the
interface to another fixed place on the water side of the interface would
be longer than the minimal time required for the ray's transit between
these places.  The rays always choose the path which minimizes this transit
time.  This analogous as to why a cross country auto traveler takes the
bypass (rather than going straight through downtown) around a city which is
between the traveller's point of departure and destination when the
traveller approaches the city during rush hour.  Often people take a
superhighway path (with its high speed limits and lack of traffic lights)
between two places even though that path is a longer distance than taking
the, more direct, back roads (with their slow speed limits and frequent
stops) to get to their destination.  Fermat's principle says that the rays
(which represent the behaviour of waves whose wavelength is tiny compared
to the scale size of the inhomogeneities of the medium/a) are like hurried
travellers in that they take the path between the points of emission and
absorption which minimizes the time of transit.

>                                  Does Fermat's principle 
> explain why the wavefront must always perpendicular  to the direction 
> of travel? 

No.  It's the other way around.  The reason that the local direction of
travel is perpendicular to the local wave front is that the phase for
the local tangent plane-wave is a function of k(dot)r - [omega]*t.
For such a function the direction of the wave's motion (i.e. the
direction of maximally increasing phase with time) must be parallel to the
wave vector k, and for a fixed time the surface of constant phase (i.e.
the wavefront) must be perpendicular to the k vector.  When the small
wavelength limit is taken, in deriving Fermat's principle, the direction
of the ray's path is along k and is perpendicular to the local wavefront. 

Another way to see why the wavefronts are perpendicular to the wave motion
is to imagine a moving plane (representing, for instance, a plane of
constant wave phase for the local tangent plane wave) in space.  Any
general motion of this plane can be broken up so that the plane's velocity
vector has one component parallel to the plane and another component
perpendicular to it.  Motion parallel to the plane is just a parallel
sliding of the plane.  But such a motion is, by symmetry, equivalent to no
motion whatsoever. (This is like rotating a perfect circle about its
center.  Any such rotation is equivalent to no rotation at all.)  So, for
our moving plane we see that only the component of the velocity
perpendicular to the plane contributes to an actual motion of the plane.

> Herb Gottlieb from New York City 
> (Where wavefronts have learned that they must ALWAYS be
> perpendicular to the direction of wave propagation)

Fermat's principle says that the rays are so smart that they always know
how to take the minimal time path (and do).

> Better slack off on that a bit, Herb. It is only a requirement
> for waves with lightspeed group velocities. In birefringent
> materials (calcite, for example) the wavefronts need not be
> perpendicular to their directions of propagation.
>
> Leigh

Sorry to disagree with you here, Leigh, but the group velocities are
not relevant.  The entire dispersion relation [omega](k) is irrelevant
for monochromatic waves.  Refraction (and total internal reflection, etc.)
happens *separately* for each of the superposed monochromatic waves.  If
the media is/are dispersive, then the rays of the various frequencies may
follow different paths (such as through a prism) because the phase velocity
may be frequency dependent, however.  Birefringence is a complication which
only can occur for waves whose wave function is tranverse-vector-valued
(e.g. EM waves) where one transverse polarization has a different
dispersion relation function [omega](k) than the other orthogonal one.
Refraction effects can occur for waves which are not vector-valued.
Refraction can occur (and also wave motion remains perpendicular to the
wavefronts) for scalar-valued waves, longitudinal waves, spinor waves and
tensor waves as well.  The very concept of polarization doesn't even make
sense for a scalar or a longitudinal wave.  There is no such thing as
birefringence for acoustic waves propagating through the interface of two
immisible fluids (of differing density and/or bulk modulus), but there
*are* refraction effects as well as wave motions perpendicular to the 
local wavefronts. In short, refraction effects and the perpendicularity of
the wave fronts to the wave's motion depend on the *argument* of the
running wave function, but polarization effects depend on the *value* space
of that function.

David Bowman
dbowman@gtc.georgetown.ky.us