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Re: Sound Cancellation



Tim O'Donnell wrote:

When a crest of one wave meets a trough of another wave
or when a compression of one wave meets a rarefaction of
another wave there is cancellation. Each wave has
energy. The energy cannot be just cancelled - law of
conservation.

Exactly so.

Does the energy go into heating the air (medium)?

No. There may be heating, but it's a coincidence and
not the essential mechanism of energy conservation.

Also consider light waves: There's no medium to heat up.

==========

The way it really works is that if the waves interfere
destructively at one place (leading to the law 1 and 1
makes zero) they will interfere constructively at some
other place (leading to the law 1 and 1 makes 4). So
if we average over all locations, we get the law that
1 and 1 makes 2 on the average.

The foregoing is perfectly true, but somewhat nonspecific.
Here's a more-specific version:

A particularly interesting case concerns a plane wave
that encounters a pointlike scatterer. Suppose the plane
wave is propagating in the +x direction. Install a lot
of detectors covering a huge sphere miles and miles away.
Each detector is angle-sensitive, using a telescope to
focus on waves coming from the origin (x,y,z) = (0,0,0).
In the absence of the scatterer, the only detector that
picks up anything is the detector that is situated way
out on the +x axis, looking backward toward the origin.
The plane wave flows in from x=-infinity, flows through
the origin, and into that detector. (We have made some
slightly non-general choices about the size and angular
resolution of the detectors. More generally there might
be other nearby detectors that also respond to some degree.)

Now install the scatterer at the origin. All the detectors
will pick up the scattered wave. The scattered wave has
spherical symmetry, spreading outward from the origin.

Interestingly, according to the laws of scattering, the
phase of the scattered wave is such that it !!always!!
interferes destructively with the original plane wave.
The aformentioned detector that sits right on the +x axis
will notice this for sure. This is called the "shadow".
(Again, more generally, depending on how fine the resolution
is, and other details, nearby detectors may see the
shadow and/or diffraction pattern.)

If you carefully add up the energy in the spherical wave
and compare it to the energy lost by the plane wave, it
had better add up to conservation of energy. In fact the
scattering law automagically guarantees this. This is
called "the optical theorem". It's a really nifty theorem.
Don't let the goofy name fool you; it applies to acoustics
as well as optics.