Exercise 3: Exploring Beat Frequency
Objective
The goal of this exercise is to explore the phenomenon of beat frequency
using stereo sounds in Praat. You will combine sine waves with slightly
different frequencies, observe the formation of beats, and investigate how our
auditory system perceives these beats differently based on how the sound is
created. There is no submission required; focus on experimenting
and understanding. You are expected to know the basic concept of beat
frequencies for your final exam.
Understanding Beat Frequency
Beat frequencies occur when two sound waves with slightly different frequen-
cies interfere, creating a tone that fluctuates in volume between loud and
soft. This fluctuation is known as beats. The phenomenon arises from the
constructive and destructive interference between the two waves.
When two waves are expressed mathematically as:
x1 = A cos(ω1 t) and x2 = A cos(ω2 t),
their sum can be simplified using a trigonometric identity, leading to:
x = x1 + x2 = 2A cos(ωmod t) cos(ωavg t),
where:
ωavg = ω1 +ω2
2
: average angular frequency
ωmod = ω1 −ω2
2
: modulation angular frequency
1
For beat frequencies to occur, the two angular frequencies must be close in
value. Mathematically, this condition can be expressed as:
|ω1 − ω2 | ≪ ω1 + ω2
This implies that the average angular frequency, ωavg , is approximately equal
to ω1 or ω2 , while the modulation angular frequency, ωmod , is very small,
approaching zero. As a result, the cos(ωmod t) term oscillates very slowly.
Combined with the amplitude term 2A, the product 2A cos(ωmod t) provides
a slowly varying amplitude for the cos(ωavg t) function.
The beat frequency is found to be twice the modulation frequency, fmod ,
where fmod = ω2π
mod
. Thus, we have:
fbeat = 2 · fmod = f1 − f2
This means the rate of amplitude modulation (i.e., the number of beats per
second) corresponds to the absolute frequency difference between the two
waves.
The formation of beats is illustrated in the figure below. The upper
graph shows the signal at frequency f1 , the middle graph shows the signal at
frequency f2 , and the bottom graph represents the sum of the two signals,
with f1 and f2 very close to each other in value. The peaks and troughs of
the beats are separated by one beat period, as indicated in the figure.
2
Exercise Instructions
Follow the steps below to create and explore beat frequencies in Praat.
Step 1: Create Stereo Sound with Identical Channels
1. Open Praat and go to New → Sound → Create Sound from formula....
2. Use the following parameters in the pop-up window:
Name: stereoIdentical
Number of channels: Stereo
Start time (s): 0
End time (s): 2
Sampling frequency (Hz): 44100
Formula:
1/2 * sin(2*pi*200*x) + 1/2 * sin(2*pi*205*x)
3. Play the sound and examine the waveform. Since you just added two sine
waves with very close frequencies (200 Hz and 205 Hz), the resulting sound
has a beat frequency of 205 − 200 = 5 Hz. This beat frequency is perceived
as periodic variations in the intensity of the sound. Additionally, observe
that it produces a consistent beat pattern across both channels because the
sounds in both channels are identical.
Step 2: Create a Stereo Sound with Different Frequen-
cies in Each Channel
1. Go to New → Sound → Create Sound from formula....
2. Use the following parameters in the pop-up window:
Name: stereoDifferent
Number of channels: Stereo
Start time (s): 0
3
End time (s): 2
Sampling frequency (Hz): 44100
Formula:
if row = 1 then 1/2 * sin(2*pi*200*x) else 1/2 * sin(2*pi*205*x) endif
The conditional expression in the formula directs channel 1 to have a fre-
quency of 200 Hz and channel 2 to have 205 Hz.
Step 3: Listen with Headphones and Compare Percep-
tion
1. Listen to the sound using stereo speakers. You should hear beats caused
by the interaction of the two frequencies.
2. Use headphones and listen separately with the left ear and then with the
right ear. You will hear slightly different tones in each channel.
3. Listen with both ears using headphones. You will perceive beats, which
are not in the audio signal but are constructed in your brain. In contrast
to the beats in the previous step, which were present in the audio signals
reaching your ears, these beats are created solely by your brain.
4. Experiment with other values to create beats with different frequencies.
For example, try to create beats with 1 Hz and 15 Hz, and observe how
the beat frequency changes as you adjust the difference between the two
frequencies.
Conclusion
This exercise demonstrates the phenomenon of beat frequencies and how they
are perceived differently based on the listening context. When listening with
both ears, our brain integrates the two slightly different frequencies to create
a unified perception of beats. This activity highlights how auditory percep-
tion involves both the physical properties of sound waves and the processing
capabilities of the brain.
