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We hear multiple sounds daily. Whether it is a tuning fork or the melody of a piano, different sounds are created by the interference of two sound waves. The waves thus created by superimposition have different frequencies and alternating softness and loudness. Such fluctuations are beats that are of great interest in physics.
What are Beats?
Beats in physics are the pulsations created by the combination of two waves having different frequencies. The consecutive playing of piano keys demonstrates this.
Applications of Beat Frequency
Beat frequency is defined as the difference in the frequencies of any two waves. It is represented by w. It is denoted as:
w = w1 – w2
where w1 is the frequency of the first wave and w2 is the frequency of the second wave.
Further, w = 2 π v
And v (beat) = v1 – v2
Some of the main applications of beat frequency include the following:
It helps determine the unknown frequency of a sound note.
Some popular applications like echocardiogram and doppler ultrasonography operate on beat frequencies.
It is used to match the frequency of different musical instruments by artists.
Doppler RADAR helps determine the speed and velocity of the aeroplane.
Mathematics and Physics of Beat Tones
It is important to understand the law of superimposition, which states that two tones sounding seamless are superimposed by adding their amplitudes. The maxima and minima of the waves don’t remain constant when a pure note is created but keep changing over time. When these waves are located 180-degrees apart, the maxima of one wave cancels the minima of the other wave. When the waves are in phase, the perceived volume is raised.
According to the sum-to-product trigonometric identities, it is observed that the envelope of the maxima and minima forms a wave that has a frequency which is half of the difference between the original wave frequencies. Considering the two sine waves of unit amplitude, the mathematical equation is-
cos (2 π f1 t) + cos (2 π f2 t) = 2 cos ( 2 π t (f1 + f2) / 2) cos ( 2 π t (f1 – f2) / 2)
When the original frequencies of two waves are close, the value of (f1 – f2) / 2 is too low to be termed an audible pitch. Hence, the frequency of the envelope is estimated to have twice the frequency of the modulating cosine, and the beat frequency becomes-
fbeat = f1 – f2
The physical interpretation can be given by:
cos ( 2 π t (f1 – f2) / 2) = 1 when the phase of two waves interfere constructively and are in phase. The value of cos ( 2 π t (f1 – f2) / 2) = 0 when the two waves are out of phase. When beats occur in different volume sounds or complex sounds, the calculation of beat frequency becomes difficult.
Binaural Beats
Binaural beats are two different types of beats with different frequencies that are heard by the human ear. The human brain processes the difference in the binaural beats to understand their frequencies. These can be potentially dangerous to the ears as they can cause noise-induced hearing loss. In this condition, the eardrum ruptures and the middle bone of the ear gets damaged.
Conclusion
The process of two waves overlapping in space with corresponding overlap in time has specific importance in physics. It becomes easy to understand the different applications of the beats, especially in the music industry. Students can go through the details and learn the mathematics and physics of the beat tones.
