[sdiy] Harmonic content of the "sigmoid" half-sine wave

[Ian Fritz]

At 03:28 AM 6/4/2009, Forbes, William  ALGLSG-LXES wrote:

>Note:
>sin(a) + sin(b) = 2 * ( sin( (a+b)/2 ) * cos( (a-b)/2 ) )
>
>and:
>sin(a) * sin(b) = -( cos(a+b) + cos(a-b) )/2
>
>Thus what happens when we hear the beat frequency between two
>frequencies?
>
>If I play 19khz and 20kHz tone I don't hear the 1kHz until distortion
>causes
>the multiplication terms to be generated.
>My ears don't work at 19kHz so I hear nothing when the system is linear.
>But I can quite easily hear the 1kHz when the system is non-linear.
>
>Yet I can quite easily hear a beat frequency when tuning a guitar.

Easy to get confused by all this.

Your first formula describes the beats you hear when tuning your 
guitar.  Look carefully at the right hand side.  The factor with the 
(a+b)/2 frequency represents an audio frequency oscillation at the average 
frequency of the two pitches.  You can hear this because it is at audio 
frequency.  The second factor represents a slow amplitude modulation of the 
tone you hear. The amplitude modulation is the beating. The beat frequency 
is the (a-b)/2 frequency.

When you repeat this using 19kHz and 20kHz tones you hear nothing, simply 
because the average (a+b)/2 frequency is out of your hearing range.  The 
physics hasn't changed at all, and the waves will still beat on an 
oscilloscope trace, just as in the guitar tuning case.

Note that this is strictly a linear problem.  The Fourier spectrum contains 
just the two frequencies "a" and "b". (Left hand side of the 
equation.)  There is no difference frequency produced.  Beats and 
difference tones are different thing.

Hope this helps.  The hardest thing about teaching physics is overcoming 
preconceived misconceptions.

   Ian