🚀 go-pugleaf

In article <48870@seismo.CSS.GOV> black@beno.CSS.GOV (Mike Black) writes:
>I always though combining two samples was not possible in the time domain.
>If you add two opposite phase sine waves you'll get a null response.  It's
>necessary to do a Fourier Transform to the frequency domain, add the 
>amplitudes, and invert the transform.  This will give you twice the amplitude
>of the original two sine waves (this would seem to be the desired effect).
>Am I totally off-base in this rather long-held belief?
>Mike...

I'm afraid you are, Mike.  First of all, the Fourier transform is a linear
operator: F[a x(t) + b y(t)] = a F[x(t)] + b F[y(t)]. Thus adding in the 
frequency domain has the same result as adding in the time domain.  Linearity
holds for continuous, discrete, and mixed versions of the Fourier transform and
Fourier series.

Second, two opposite-phase sinusoids SHOULD cancel each other.  Have you
ever played a wind instrument?  While tuning, the rhythmic beating of the
blended sound of two horns is the result of the sinusoids going in and out of
phase. When the beating stops, you are in tune (same frequency).  If you are
more careful, you can play a long note to be both in tune and in phase with 
another player (w.r.t. a particular point in space), so that the volume of 
the summed sound waves of your two horns is near maximal.  Of course, there
are harmonics generated in the horn and the phase difference of the fundamental
depends on the position of the instruments and the listener and the room 
acoustics, but my basic tenent remains that the sound pressure levels 
generated by each instrument can be algenraically summed to give the sound 
which would be produced by the instruments playing together.

This brings up another question:  How linear is the compression of air?
Suppose you record each instrument in an orchestra individually from a
microphone at some fixed location, say on the conductor's podium.  You then
add all the recordings thus made to produce a recording of the full orchestra.
How does this compare to recording the orchestra all at one time, assuming a
noise-free environment and a perfectly duplicated performance?  If air 
compresses linearly, the two recordings should be the same.  But I could 
imagine a "saturation" effect, for instance in the air around the piccolo 
section, which might cause a change in the frequency response for some 
instruments when played together.  My guess is that this effect would
require many atmospheres of sound pressure, since PV=nRT (the natural gas law)
seems to work for any easily achievable pressure.  Thus the volume would be 
far beyond the injury threshold for anyone nearby.  But do I recall that 
at LOW pressures (as in the rarefaction waves that accompany compression
waves) gases behave nonlinearly?  Perhaps someone can enlighten me on the 
physical limits of sound production.

Tim Davis
tldavis@mit.edu