In this blog, we will study the acoustic phenomenon of rhythmic pulses. This effect is very easy to create with the FizziQ application and rich in discoveries in the context of the investigation process. A rhythmic pulses appears when two pure sounds of very close frequencies are simultaneously emitted. We then clearly hear the pulsation that results from the periodic interference of the two sounds. This is an example of rhythmic pulse. Rhythmic pulses have been used historically to tune musical instruments, but your students will surely recognize this effect often used today by electronic music composers of the Future Bass genre, such as Flume or San Holo. This effect is called the LFO, or Low Frequency Oscillator. To study the pulse phenomenon with the FizziQ app, ask your students to use the frequency synthesizer on the Tools tab to generate a 600 hertz frequency sound on the first track and a 660 hertz sound on the second. Then ask them to gradually reduce the frequency of the second way to 600 hertz. When the frequency of the second way is less than 620 hertz, they will begin to perceive the phenomenon of rhythmic pulses. The following questions will then be addressed. How to describe the acoustic phenomenon that appears when both frequencies are very close? When does this phenomenon appear? How to define the frequency of pulse? How to slow down or accelerate the frequency of pulse? Is the pulse an illusion of our brain or a measurable physical phenomenon? The rhythmic pulse is characterized by a phenomenon of pulsation of the sound volume. By experimenting with the synthesizer, students will find that the pulse becomes perceptible when the difference between the two frequencies is less than 20 hertz. Students will also discover that the lower the frequency difference between the two sounds, the lower the frequency of the pulse. Finally, they will be able to ensure that the pulse is not an illusion of our brain by doing the following experiment: by using two laptops each generating a single frequency, they will check that we hear the pulse if we place the laptops next to each other. On the other hand, if you place one laptop near the right ear and the other near the left ear, the phenomenon disappears. It is therefore a physical phenomenon of interaction between the two waves and not a auditory illusion. Once the general analysis of the phenomenon is completed, and having found that the frequency of the pulse depends on the frequencies two sounds, students will be asked to look for an empirical relationship between these quantities. By timing the period of the pulse, can they empirically determine this relationship? To calculate this relationship more precisely, can they use a measuring instrument from the FizziQ application to accurately measure the period of the pulses? Is the determined relationship dependent on the absolute level of frequencies or only on their difference? Students will easily determine that the frequency of the pulse is equal to the difference in the frequency. They will be able to use a stopwatch and measure the period or better record the sound volume in the Measurements tab and analyze the sequence in their experience book. They will be able to redo the measurement for different combinations and thus check that the frequency of the pulse does not depend on the absolute level of the frequencies but only on their difference. These analyses may be summarized by the student in the experience book in the form of graphs or a summary table to which students will add text to explain their hypotheses and conclusions. To better visualize the pulse phenomenon, you can then ask students to use the oscillogram to study the amplitude of the sound signal. It is preferable to use two pure sounds with a frequency deviation of the order of 15 to 20 hertz. The scale of the oscilloscope can be adjusted using the scale button at the top left. By studying the phenomenon with the oscilloscope, will students be asked to characterize the curve followed by the amplitude of the signal? We often talk about the envelope phenomenon, what is the envelope of this curve? Can students measure the period of the envelope? Can they also measure the frequency of the signal that is enveloped? What can they deduce from this analysis? The oscillogram allows students to visualize the phenomenon very precisely. They will determine that the envelope makes it possible to modulate a signal by increasing and reducing the sound volume of this signal on a regular basis. They will thus be able to better interpret the LFO phenomenon that is very familiar to them. On the other hand, they will also find that the modulated signal has a frequency equal to the average of the two signals. It is difficult to go further in analysis for younger students, but for senior students in the context of the grand oral, or in the first year of university, the next step is then to explain the results of these experiments by theory. Consider two frequency waves f1 and f2 of the same maximum amplitude A. The equations of these two waves are:

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.

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