17.13:
Beats
When two sound waves of equal amplitude and slightly different frequencies interfere with each other, the displacement equation for the resultant wave at a particular position can be determined. Its intensity rises and falls alternately with time.
The total length of the time axis represents one second. When the two waves are in phase, their amplitudes add.
And, when the two waves are exactly out of phase, the total amplitude is zero.
This periodic variation in sound intensity with time forms beats.
The resultant wave looks like a single sinusoidal wave with varying amplitude, which goes through two maxima and two minima in one second. The frequency of this amplitude variation is two hertz.
The number of beats heard per second is called beat frequency and is equal to the difference in frequency of the sound waves.
The time interval between two successive beats is defined as the beat period, which is the inverse of the beat frequency.
17.13:
Beats
The study of music provides many examples of the superposition of waves and the constructive and destructive interference that occurs. Very few examples of music being performed consist of a single source playing a single frequency for an extended period of time. A single frequency of sound for an extended period might be monotonous to the point of irritation, similar to the unwanted drone of an aircraft engine or a loud fan. Music is pleasant and exciting due to mixing the changing frequencies of various instruments and voices.
An interesting phenomenon that occurs due to the constructive and destructive interference of two or more frequencies of sound is the phenomenon of beats. They look like a single sinusoidal wave with a varying amplitude that goes from a maximum to zero and back. The amplitude variation causes variations of loudness, called beats, and the frequency with which the loudness varies is called the beat frequency. It is defined as the difference between the two frequencies.
Piano tuners can use these beats to tune a piano. A tuning fork is struck, and a note is played on the piano. When the piano tuner tunes the string, the beats have a lower frequency as the frequency of the note played approaches the frequency of the tuning fork.
Beats between two tones can be heard up to a beat frequency of about 6 or 7 Hz. The engines on multi-engine propeller aircraft have to be synchronized so that the propeller sounds do not cause annoying beats, which are heard as loud throbbing sounds. On some planes, this is done electronically; on others, the pilot does it by ear, just like tuning a piano.
Suggested Reading
- Young, H.D. and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson; section 16.7; pages 531-533.
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/17-6-beats section 17.6; pages 880-881.