"Beat frequency" phenomenon, but with 3 sine waves?

The phenomena of "beat frequency" involving 2 summed sine waves of close
but unequal frequency is simple to understand.  

When 2 sine waves of equal amplitude and slightly different frequency are
added together, the resulting wave has a frequency equal to the averaged
frequency of the 2 added waves, and a sinusoidal amplitude modulation of
frequency equal to the difference in frequency between the 2 added waves.

It's also described here with diagrams(in case my explanation doesn't make
sense):

http://en.wikipedia.org/wiki/Beat_frequency


However, when 3 sine waves are added together, all 3 sine waves being
close in frequency, the result is more complicated.  The amplitude
modulation pattern is more complicated, and the frequency of the resulting
wave appears to vary with time.

Are there a set of simple rules that describe this behavior (the amplitude
and frequency of the output wave as a function of time)?

I'm looking for a simple set of rules sort of like the ones i described at
the top of this post, which make calculating the frequency and amplitude of
2 summed sine waves very simple.  I would expect such a set of rules for 3
summed sine waves to be more complicated than the rules for 2 summed sine
waves.


Thanks, any help would be much appreciated

maxplanck wrote:
> The phenomena of "beat frequency" involving 2 summed sine waves of close
> but unequal frequency is simple to understand.  
> 
> When 2 sine waves of equal amplitude and slightly different frequency are
> added together, the resulting wave has a frequency equal to the averaged
> frequency of the 2 added waves, and a sinusoidal amplitude modulation of
> frequency equal to the difference in frequency between the 2 added waves.
> 
> It's also described here with diagrams(in case my explanation doesn't make
> sense):
> 
> http://en.wikipedia.org/wiki/Beat_frequency

It's not as simple as it looks. Compare the pictures in the link to a 
carrier amplitude modulated 100% by the single beat frequency. Note that 
the beat pattern envelope crosses zero at maximum slope, while the AM 
envelope has zero slope there. Note also that the beat "carrier" changes 
phase whenever the envelope crosses zero. Not so with AM.

The beat pattern amounts to double-sideband AM with suppressed carrier. 
It would not be immediately obvious how to demodulate it if you wanted to.

> However, when 3 sine waves are added together, all 3 sine waves being
> close in frequency, the result is more complicated.  The amplitude
> modulation pattern is more complicated, and the frequency of the resulting
> wave appears to vary with time.

A beats with B and C and B beats with C, for a total of three beat 
frequencies. With four notes, A beats with B, C, and D; B beats with C 
and D; and C beats with D, for a total of six. With for notes you get 
ten beats. If you like, you can compute the beats beating against one 
another, but many of the results will be frequencies already present.

> Are there a set of simple rules that describe this behavior (the amplitude
> and frequency of the output wave as a function of time)?

Not that I know of.  Have you examined the two-note case when the 
amplitudes are unequal? The envelope is then not sinusoidal. its 
amplitude can be constructed graphically with a crank diagram. 
Interestingly, the amplitude of a standing wave on a line with 
attenuation can be determined the same way.

> I'm looking for a simple set of rules sort of like the ones i described at
> the top of this post, which make calculating the frequency and amplitude of
> 2 summed sine waves very simple.  I would expect such a set of rules for 3
> summed sine waves to be more complicated than the rules for 2 summed sine
> waves.
> 
> 
> Thanks, any help would be much appreciated

Good luck.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
(snip)
> 
> Not that I know of.  Have you examined the two-note case when the 
> amplitudes are unequal? The envelope is then not sinusoidal. its 
> amplitude can be constructed graphically with a crank diagram. 
> Interestingly, the amplitude of a standing wave on a line with 
> attenuation can be determined the same way.

Jerry, you seem to be inferring that the envelope is sinusoidal when the
amplitudes are equal.  In fact the envelope is a cycloid when the 
amplitudes are equal and approaches a sinusoid only for very large 
differences in amplitude.

Regards,
John

John Monro wrote:
> Jerry Avins wrote:
> (snip)
>>
>> Not that I know of.  Have you examined the two-note case when the 
>> amplitudes are unequal? The envelope is then not sinusoidal. its 
>> amplitude can be constructed graphically with a crank diagram. 
>> Interestingly, the amplitude of a standing wave on a line with 
>> attenuation can be determined the same way.
> 
> Jerry, you seem to be inferring that the envelope is sinusoidal when the
> amplitudes are equal.  In fact the envelope is a cycloid when the 
> amplitudes are equal and approaches a sinusoid only for very large 
> differences in amplitude.

I think the envelope consists of overlapping sinusoids 180 out of phase; 
a positive and negative cosine wave, for example. A cycloid would have a 
much flatter top. I can't think of a simple demonstration off hand, but 
the crank diagram leads to the same conclusion.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

>Jerry Avins wrote:
>(snip)
>> 
>> Not that I know of.  Have you examined the two-note case when the 
>> amplitudes are unequal? The envelope is then not sinusoidal. its 
>> amplitude can be constructed graphically with a crank diagram. 
>> Interestingly, the amplitude of a standing wave on a line with 
>> attenuation can be determined the same way.
>
>Jerry, you seem to be inferring that the envelope is sinusoidal when the
>amplitudes are equal.  In fact the envelope is a cycloid when the 
>amplitudes are equal and approaches a sinusoid only for very large 
>differences in amplitude.
>
>Regards,
>John

In the two-note case, when the amplitudes are equal, the envelope is a
cosine wave according to this trig identity:

http://upload.wikimedia.org/math/2/2/e/22eb99ffde79056c36b047904a7aa0a8.png

On Dec 6, 4:30 pm, "maxplanck" <erik.bo...@comcast.net> wrote:
> >Jerry Avins wrote:
> >(snip)
>
> >> Not that I know of.  Have you examined the two-note case when the
> >> amplitudes are unequal? The envelope is then not sinusoidal. its
> >> amplitude can be constructed graphically with a crank diagram.
> >> Interestingly, the amplitude of a standing wave on a line with
> >> attenuation can be determined the same way.
>
> >Jerry, you seem to be inferring that the envelope is sinusoidal when the
> >amplitudes are equal.  In fact the envelope is a cycloid when the
> >amplitudes are equal and approaches a sinusoid only for very large
> >differences in amplitude.
>
> >Regards,
> >John
>
> In the two-note case, when the amplitudes are equal, the envelope is a
> cosine wave according to this trig identity:
>
> http://upload.wikimedia.org/math/2/2/e/22eb99ffde79056c36b047904a7aa0...

Note the this envelope has a cosine modulation with a
range of 1 to -1, which is 200% peak-to-peak.  Common
AM modulation only allows you to reduce the amplitude
from 100% to 0.  I haven't seen many volume controls
knobs that go below the OFF position to -10 (or -11 !! :)

maxplanck wrote:
>> Jerry Avins wrote:
>> (snip)
>>> Not that I know of.  Have you examined the two-note case when the 
>>> amplitudes are unequal? The envelope is then not sinusoidal. its 
>>> amplitude can be constructed graphically with a crank diagram. 
>>> Interestingly, the amplitude of a standing wave on a line with 
>>> attenuation can be determined the same way.
>> Jerry, you seem to be inferring that the envelope is sinusoidal when the
>> amplitudes are equal.  In fact the envelope is a cycloid when the 
>> amplitudes are equal and approaches a sinusoid only for very large 
>> differences in amplitude.
>>
>> Regards,
>> John
> 
> In the two-note case, when the amplitudes are equal, the envelope is a
> cosine wave according to this trig identity:
> 
> http://upload.wikimedia.org/math/2/2/e/22eb99ffde79056c36b047904a7aa0a8.png

No, it is a pair of cosine waves, one positive and one negative. The 
sine term in that formula rapidly oscillates between positive and 
negative, giving the envelope two parts. Look at the illustration  you 
provided in http://en.wikipedia.org/wiki/Beat_frequency.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Ron N. wrote:

   ...

> Note the this envelope has a cosine modulation with a
> range of 1 to -1, which is 200% peak-to-peak.  Common
> AM modulation only allows you to reduce the amplitude
> from 100% to 0.  I haven't seen many volume controls
> knobs that go below the OFF position to -10 (or -11 !! :)

Huh? sin(x) varies from +1 to -1. With ordinary AM, each sideband gets 
half the audio power. Still, if the carrier amplitude is +/-1, then the 
peak is +/-2, as with the beat case. There, when the two frequencies 
with amplitude +/-1 add constructively, the result is +/-2.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

Jerry Avins wrote:
> John Monro wrote:
>> Jerry Avins wrote:
>> (snip)
>>>
>>> Not that I know of.  Have you examined the two-note case when the 
>>> amplitudes are unequal? The envelope is then not sinusoidal. its 
>>> amplitude can be constructed graphically with a crank diagram. 
>>> Interestingly, the amplitude of a standing wave on a line with 
>>> attenuation can be determined the same way.
>>
>> Jerry, you seem to be inferring that the envelope is sinusoidal when the
>> amplitudes are equal.  In fact the envelope is a cycloid when the 
>> amplitudes are equal and approaches a sinusoid only for very large 
>> differences in amplitude.
> 
> I think the envelope consists of overlapping sinusoids 180 out of phase; 
> a positive and negative cosine wave, for example. A cycloid would have a 
> much flatter top. I can't think of a simple demonstration off hand, but 
> the crank diagram leads to the same conclusion.
> 
> Jerry

Your statement above is contradictory because if the envelope did in 
fact consist of overlapping sinusoids it would be no more sinusoidal 
than the output of a full-wave rectifier and I don't think anyone would 
describe that as sinusoidal.

When you do the maths for the crank diagram specified below you get:
	v = sqrt(2 - 2cos(theta))

On checking, I find that while the above expression is similar in some 
ways to a cycloid it is not in fact a cycloid (so that is something I 
have learnt today).  One thing though, it sure ain't a sinusoid!


(Crank diagram assumptions:
	Two arms AB and CD of unit length.
	Arm AB is fixed between points A (0, 0) and B (0, 1).
	Arm CD has one end (C) pivoting at point B.
	The other end (D) of arm CD is free to move about point B.
	The angle between AB and CD is theta.
	Length AD is the instantaneous amplitude v ).

Jerry Avins wrote:
> John Monro wrote:
>> Jerry Avins wrote:
>> (snip)
>>>
>>> Not that I know of.  Have you examined the two-note case when the 
>>> amplitudes are unequal? The envelope is then not sinusoidal. its 
>>> amplitude can be constructed graphically with a crank diagram. 
>>> Interestingly, the amplitude of a standing wave on a line with 
>>> attenuation can be determined the same way.
>>
>> Jerry, you seem to be inferring that the envelope is sinusoidal when the
>> amplitudes are equal.  In fact the envelope is a cycloid when the 
>> amplitudes are equal and approaches a sinusoid only for very large 
>> differences in amplitude.
> 
> I think the envelope consists of overlapping sinusoids 180 out of phase; 
> a positive and negative cosine wave, for example. A cycloid would have a 
> much flatter top. I can't think of a simple demonstration off hand, but 
> the crank diagram leads to the same conclusion.
> 
> Jerry



Your statement above is contradictory because if the envelope did in 
fact consist of overlapping sinusoids it would be no more sinusoidal 
than the output of a full-wave rectifier and I don't think anyone would 
describe that as sinusoidal.

When you do the maths for the crank diagram specified below you get:
     v = sqrt(2 - 2cos(theta))

On checking, I find that while the above expression is similar in some 
ways to a cycloid it is not in fact a cycloid (so that is something I 
have learnt today).  One thing though, it sure ain't a sinusoid!

I note the posting by maxplanck, and while it is true that the 
expression given contains sinusoidal terms, that does not make it 
sinusoidal.  The whole point of the discussion is whether the envelope 
of of sum of two sunusoidal waveforms is itself sinusoidal and it is not 
sufficient to show that the expression contains sinusoidal terms.  We 
knew that already! One needs to show that the expression reduces to 
A*sin(theta) and I don't believe this can be done.


(Crank diagram assumptions:
     Two arms AB and CD of unit length.
     Arm AB is fixed between points A (0, 0) and B (0, 1).
     Arm CD has one end (C) pivoting at point B.
     The other end (D) of arm CD is free to move about point B.
     The angle between AB and CD is theta.
     Length AD is the instantaneous amplitude v ).

Regards,
John