Re: [Phys-L] More on Gravity Waves

[John Denker <jsd@av8n.com>]

Alas on Tuesday, April 26, 2016 1:15 AM, I wrote:

> > The thing that's longitudinal isn't a wave.
> > The thing that's a wave isn't longitudinal.

On 05/02/2016 02:34 PM, Moses Fayngold wrote:

> This is true for known vacuum waves. Acoustic waves may be longitudinal. 

Oh, I was even more wrong than that.  In fact:

-- The E-field for electromagnetic *plane* waves is strictly transverse.
-- For spherical waves, not so much.

The same goes for gravitational waves.  The proof that they are
transverse is only for plane waves.

There is one *term* in the spherical wave solution that is
strictly transverse ... but that term by itself is not
strictly a solution to the Maxwell equations.  Take your
pick:  it's either a spherical solution or it's purely
transverse, not both.

In the far field *limit* the transverse term dominates ... but
in that limit the waves are indistinguishable from plane waves,
so that doesn't tell us anything we didn't already know.

Consider a monochromatic spherical wave emanating from a tiny
dipole antenna.  It has a sinusoidal dependence on time and
space.

The equation for the wave is:
    E	=	   (1/4πє0) k^3	 (n×p)×n    	e^ikr (1/kr)	  (1a)
 	 	−i (1/4πє0) k^3	[3n(n·p) − p]	e^ikr (1/kr)^2	  (1b)
 	 	+  (1/4πє0) k^3	[3n(n·p) − p]	e^ikr (1/kr)^3	  (1c)

where:
	p =	 electric dipole moment    
	 	 including the e^iωt time dependence      
	r =	 radial coordinate                
	n =	 unit vector in the dr direction

Reference: Jackson equation 9.18.

The symmetry of the second and third terms is such that they rotate
from being transverse at the equator to longitudinal at the poles.
For moderately large kr, these terms are small, but they're never
strictly zero, so the wave is never strictly transverse, except at
the equator.

The second term is 90 degrees out of phase with the other two.  That
means that twice per cycle you can find shells where the first and
third terms vanish.  On such shells, the second term is the leading
term, and it's never transverse, except on the equator.   For 
moderately large kr, you can argue that the second term is small,
but it's not super-small.

Near the poles, the first term is always zero, and the remaining
field is always longitudinal.  For large kr the field at the
poles dies off like (1/kr)^2 ... so it extends farther than the
electrostatic dipole field.

It must be emphasized that almost none of this can be figured
out by looking at the electrostatic equations and generalizing
by guessing.  You have to do the math.

By the same token, there's a lot going on in spherical waves
that cannot be figured out just by looking at plane waves and
generalizing by guessing.

Here is a page from about a dozen years ago, inventorying various
ways of visualizing the EM field:
  http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/guidedtour/Tour.htm

Section VI talks about animations, but you have to click on the 
word Figure in the figure captions to get to the actual animations.

Of particular relevance to the current discussion are:
  http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/QuarterWaveAntenna/MicroWaveDLICS_640.mpg
  http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/AntennaPattern/uWaveAntenna_640.mpg
  http://web.mit.edu/8.02t/www/802TEAL3D/visualizations/light/dipoleRadiationReversing/SmPointDipole_640.mpg

Apparently these are real computed solutions, not mere
artists' impressions.