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# Fun and serious Low Frequency Oscillators effects (LFO)

Updated: Mar 10, 2023

f1(t) = A cos(2*pi*f1*t + phi1)

f2(t) = A cos(2*pi*f2*t + phi2)

If we add these two waves, we get the equation :

f(t) = A(cos(2*pi*f1*t + phi1) + cos(2*pi*f2*t + phi2))

Using the equation of the sum of cosine :

cos(a) + cos(b) = 2cos((a+b)/2)*cos((a-b)/2)

In the end :

f(t) = 2*A*cos(2*pi*(f1+f2)/2t + (phi1+phi2)/2)*cos(2*pi(f1-f2)t/2 + (phi1-phi2)/2)

This equation is the product of two sinusoidal signals, one of frequency equal to the average of the frequencies of the two waves, and the other of much lower frequency and proportional to the difference of the two frequencies. If we abstract from the variation of volume, we hear the sound produced by the amplitude variations of the first term of the equation, i.e. a frequency sound equal to the average of the two frequencies. This result corresponds well to the results of the oscillogram analysis. It is the second very slow oscillation is the one that creates the sensation of pulsation, what is called the sound envelope. The human ear does not detect frequency sounds below 20 hertz, so this signal is interpreted as a variation in the sound volume. It is noted that the sound is at least twice per cycle, so the perceived frequency of the pulse is double the frequency, i.e. f1-f2. We covered with the pulse many characteristics of a sound wave and widely used the investigation method for this analysis. The study of acoustic pulse is exciting both for the youngest, familiar with the effect of LFO, and the oldest who will put wave theory into practice. The use of FizziQ simplifies the experimentation process and allows students to quickly and easily conduct a real investigative approach.