Re: Sound Cancellation

["John S. Denker" <jsd@MONMOUTH.COM>]

"Polvani, Donald G." wrote:
> I arrange their phases so that the are exactly 180 deg out of phase.
> ... single frequency waves ... should
> maintain their 180 deg phase difference at all points in space.  I see no
> way that they will interfere destructively along some portion of the x axis
> and constructively along some other portion.

In this case we need to ask how you generated the waves.
There are various inequivalent possibilities, including:

1) Consider water waves.  Wave A is generated by a bar
that is shoved against the surface at location A and wiggled
up and down at the given frequency.  This radiates outgoing
waves in the +x and -x directions both.

Meanwhile, wave B is generated by an analogous bar at location
B.  It, too generates outgoing waves in both directions.

Do the superposition.  Work out the math.  If you arrange the
phases so that there is cancellation in the "outside" regions,
there will be a standing wave in the region between the bars.

The only slightly tricky thing about this is that the
power-requirements on the device that wiggles bar A will
be _reduced_ by the presence of the wave from bar B (and
vice versa).  So energy is conserved.




2) Consider light waves with a beamsplitter.

            |
            | /
            |/
        1---/----------->> 1 out  (+x direction)
        in /
          /


            |
            | /
            |/
            /----------->> 2 out  (+x direction)
           /|
          / |
            |
            2 in


So you see we have two different ways of creating a wave
in the +x direction.  If I arrange the phases just right,
I can achieve 100% cancellation of the two contributions.

But wait!  What about the wave going up the +y direction?
It turns out that according to the laws of reflection and
transmission, whatever phase corresponds to cancellation
in the +x direction leads to 100% construtive interference
in the +y direction.  Mother Nature has lots of tricky
ways of enforcing Her laws.

3) If you work out a bunch more examples, they will all
turn out similar to one of the foregoing examples.  If
there is destructive interference, either the load on
the source-device is reduced, or the energy is shifted
to some region of constructive interference, or some
combination.