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

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

Started by ●December 6, 2007

Reply by ●December 6, 20072007-12-06

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

Reply by ●December 6, 20072007-12-06

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

Reply by ●December 6, 20072007-12-06

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. �����������������������������������������������������������������������

Reply by ●December 6, 20072007-12-06

>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, >JohnIn 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

Reply by ●December 6, 20072007-12-06

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 !! :)

Reply by ●December 6, 20072007-12-06

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.pngNo, 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. �����������������������������������������������������������������������

Reply by ●December 6, 20072007-12-06

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. �����������������������������������������������������������������������

Reply by ●December 6, 20072007-12-06

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. > > JerryYour 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 ).

Reply by ●December 7, 20072007-12-07

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. > > JerryYour 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