Re: two sine tones simultaneously within one critical band

[Bob Masta <audio@xxxxxxxxxxx>]

On 7 Oct 2005 at 12:40, Reinhart Frosch wrote:

> 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:

I am completely at a loss to understand how you arrived
at your last formula.  It appears that you have not simply
added the pressure waves, but multiplied them.  This is
not what happens in air at normal sound levels, where 
there is essentially no nonlinearity.  In air the two original
tones are linearly summed and spectral analysis of
the waveform output from a (linear) microphone shows
that only those components are present.  The beat
tones are only in the head of the listener.


Best regards,

Bob Masta

audioATdaqartaDOTcom