common periods of oscillations with rational frequency ratios
Finding the common period of multiple oscillations at distinct frequencies with rational ratios is sort of like finding a common denominator when adding fractions. for a sinusoidal function of frequencyLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbW9HRiQ2LlEifkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGRS1JI21pR0YkNiZRJyYjOTY5O0YnLyUnaXRhbGljR0YxRi9GMkYvRjI= , the period is 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 (since 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 changes the phase angle in the trig function by one period of the unit frequency cosine/sine function. Suppose we have two signals at frequencies LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjFGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGNUY3 and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUklbXN1YkdGJDYmLUkjbWlHRiQ2JlEnJiM5Njk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUrZXhlY3V0YWJsZUdGNC8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictRiM2Ji1JI21uR0YkNiVRIjJGJ0Y1RjcvRjNRJXRydWVGJ0Y1L0Y4USdpdGFsaWNGJy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnL0krbXNlbWFudGljc0dGJFEnYXRvbWljRidGNUY3 , with periods 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 and 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 (p , q with no common factors!)
then 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 because this is the smallest integer number of periods of each which will both fit into the same time interval.
Here is an example of two cases where the multiple of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbW5HRiQ2JVEiMkYnLyUrZXhlY3V0YWJsZUdRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUYjNiYtSSNtaUdGJDYjUSFGJy1GODYmUSUmcGk7RicvJSdpdGFsaWNHUSV0cnVlRidGLy9GM1EnaXRhbGljRidGL0YyRjdGL0Yy required is the lowest common denominator of the fractional frequencies. For three frequencies the analysis is more complicated, but similar reasoning.
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Two periods of 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 equal three periods of 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 namely 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 .
Thus we see two repetitions of the followingwaveform here:
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In contrast here the frequency ratio is irrational, and one sees beating, but it never quite repeats itself.
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Exercise for fun:
What is the common period of 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 ?
[Notice that 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 How are the lowest common denominator of the fractional frequencies and the corresponding numerators reflect in the common period calculation?]
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The common period is also the beating period when we look at the difference of two such functions with equal amplitudes. Here we see two beats.
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