10.4 The Phenomenon of Beats of Sound

When two vibrating objects (Sources of audible Sound) vibrate with slightly different frequencies at a place far away - compared to the separation between the sources - from the sources the loudness (intensity) of sound waves and waves periodically, this phenomenon is called Beats of Sound. The beat frequency i.e., frequency of waxing and waning of sound is found to be equal to the positive difference between the two interfering frequencies.

The equations of two waves from the two sources with the same amplitude of waves; with sources very close to each other and to x- axis can be written as

Since the waves are traveling along the same direction i.e., X- axis, the superposition principle gives the resultant expression as,

[shifting the origin from sources to the point at distance 'x' does not alter the characteristics of interfering waves]

is the resultant amplitude of vibration at P, and is the frequency with which the particle at P vibrates.

Now, waxing and waning correspondent to maximum and minimum intentity respectively, i.e. on

Therefore frequency of waxings or wanings is N = | n₁ - n₂|

The Phenomenon of Beats can be illustrated graphically also, as shown below :

The above diagrams (a) and (b) shows the number of waves arriving from the two sources of frequencies n₁ and n₂ at the position of the Listener over the interval of 1 sec. The diagram (c) represents the effect of these superposed waves resulting into 4 maximas (waxings) and minimas (wanings) indicating that beat frequency is N = n₂ -n₁ = |n₁ - n₂| in general. These diagrams are only illustrative ones, since n₁, n₂ < 20 Hz and N < 20 Hz, the sounds are inaudible to humans.

Index

10.1 Intensity and Pitch
10.2 Doppler Effect
10.3 Superposition Principle
10.4 The Phenomenon of Beats of Sound
10.5 Standing or Stationary Waves
10.6 Forced Oscillations and Resonance

Chapter 11