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Re: two sine tones simultaneously within one critical band

On October 6, 2005, Bob Masta wrote:

>Not to put too fine a point on this, but it's only 
>"physics" if there is some nonlinearity in the system.  
>Two sine waves, added linearly, contain no extra "beat" 
>components... even though beats seem to be clearly visible
>in the waveform.  What really happens is that there are 
>peaks in the total waveform which occur at the beat frequency. 
>If such a wave passes through a nonlinear system, then beat 
>frequencies are generated as real tones.   One might suppose 
>that given the known nonlinearities in the peripheral auditory 
>system, this might be the explanation of the beat phenomenon.  
>However, I seem to recall that there is evidence against this 
>explanation... I just don't recall what it was, nor how
>compelling it was.  Anyone?

Sound pressure of first sine-tone:

p_1(t) = p_0 * sin(99 * 2pi * t);

sound pressure of second sine-tone:

p_2(t) = p_0 * sin(101 * 2pi * t).

[ * = multiplication sign; t = time in seconds.]

Total sound pressure:

p(t) = p_1(t) + p_2(t) = 2p_0 * cos(t) * sin(100 * t).

That last formula implies a 100-hertz sine-tone 
amplitude-modulated so that there are two beats per second.

The 1-mm-long basilar membrane piece strongly excited by a
soft 99-hertz sine-tone and that strongly excited by
a soft 101-hertz sine-tone overlap almost completely.

Reinhart Frosch.

Reinhart Frosch,
Dr. phil. nat.,
Sommerhaldenstr. 5B,
CH-5200 Brugg.
Phone: 0041 56 441 77 72.
Mobile: 0041 79 754 30 32.
E-mail: reinifrosch@xxxxxxxxxx