craigm wrote:
Jim Kelley wrote:


David L. Wilson wrote:


"Jim Kelley" wrote in message
...
...


sin(a) + sin(b) = 2sin(.5(a+b))cos(.5(a-b))

A plot of the function reveals that cos(.5(a-b)) describes the envelope.


Ok.

The period of the 'enveloped' waveform (or the arcane, beat


modulated waveform) then can be seen to vary continuously and
repetitiously over time - from 1/a at one limit to 1/b at the other.


?



At a particular instant in time the period does in fact equal the average
of the two. But this is true only for an instant every 1/(a-b) seconds.


??

How do you come up with anything but a period of of the average of the
two for the enveloped waveform?

The error here is in assuming that the sin and cos terms in the
equivalent expression are representative of individual waves. They
are not. The resultant wave can only be accurately described as the
sum of the constituent waves sin(a) and sin(b), or as the function
2sin(.5(a+b))cos(.5(a-b)). That function, plotted against time
appears exactly as I have described. I have simply reported what is
readily observable.

jk



I would submit you plotted it wrong and/or misinterpreted the results.

Always a possibility, admitedly. However the superposition of two
waves each having a different frequency does not yield a resultant
waveform having a constant period. But you are certainly welcome to
try to demonstrate otherwise.

jk