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From: robison@uiucdcsb.UUCP
Newsgroups: net.physics
Subject: Re: Query about Two-Source Interference
Message-ID: <10800011@uiucdcsb.UUCP>
Date: Fri, 9-Nov-84 13:25:00 EST
Article-I.D.: uiucdcsb.10800011
Posted: Fri Nov  9 13:25:00 1984
Date-Received: Sun, 11-Nov-84 20:57:13 EST
References: <714@sdcsla.UUCP>
Lines: 45
Nf-ID: #R:sdcsla:-71400:uiucdcsb:10800011:000:2280
Nf-From: uiucdcsb!robison    Nov  9 12:25:00 1984

You are right in suspecting that the ordinate axis should not be labeled 2Q.
Indeed, it should be labeled 4Q as shown:


E 4Q |            ***                       ***
n    |          **   **                   **   **
e    |        **       **               **       **
r 2Q |       *           *             *           *
g    |      *             *           *             *
y    |    **               **       **               **       etc.
     |  **                   **   **                   **   **
   0 |**-----------------------***-----------------------***--------
      0      1     2      3     4      5     6     7      8
	Position -->

The explanation is simple wave mechanics.  Whenever we add two waves, we add
their amplitudes.  The energy in a wave is proportional to the SQUARE of the
amplitude.  So when you add two coherent waves constructively, the total
total amplitude is double.  Therefore the total energy is quadrupled where
the two waves add constructively.  By the nature of interference, however,
there will also be corresponding places where the waves interfere destructively
and the total energy is zero.  If you integrate over your wall, you will find
that all the energy is conserved.

I just happen to have my prof's holography notes on hand, so I can give
the complete formula.  The energy of the summations of waves with energies A
and B is given by:

     A + B + 2*sqrt(A*B)*|d|*cos(a)

where d is the "normalized mutual coherence function" (wow - what mouth full!)
and a is the phase difference.  If two waves have d=0, the waves are incoherent
and the resulting energy is just the sum of the energies A and B.  If two waves
have d = 1, they are perfectly coherent and the total energy is dependent upon
the phase relation.  For 0<d<1, the waves are partially coherent, and the
energy is partially dependent upon the phase relation.

A good analogy is random variables.  The variance of a random variable
correspond to the energy of a wave.  The covariance corresponds to the
mutual coherence function.  Depending upon whether the variables are independent
(incoherent) or correlated (partially or perfectly coherent), the variance of
the sum can be greater than, equal to, or less than the sum of the variances.

Arch Robison @ uiucdcs