Physics of Beats and Interference

Production of Beats

• Beats are produced when two sound waves with slightly different frequencies superpose.
• The intensity of the sound varies periodically with time.
• This variation in loudness is the phenomenon of beats.
• Beats can be considered as superposition of waves, forming standing waves in time at a single point.
• The two waves are out of phase, causing constructive and destructive interference.

Expression for Beats Frequency

• Consider two sound waves with the same amplitude but different frequencies (n₁ and n₂).
• Let the displacement of each wave at a point (x=0) be represented as:
  • y₁ = a sin(2πn₁t)
  • y₂ = a sin(2πn₂t)
• According to the superposition principle, the resultant wave is:
  • y = y₁ + y₂ = 2a sin[2π(n₁+n₂)/2]t cos[2π(n₁−n₂)/2]t
• Simplifying the equation:
  • y = 2a cos[π(n₁−n₂)/2]t sin[2π(n₁+n₂)/2]t = A sin(2πnt)
• A = 2a cos[π(n₁-n₂)/2]t
• The resultant wave has a frequency (n), which is the mean of n₁ and n₂

Maximum Intensity (Waxing)

• The amplitude (A) is maximum when cos[π(n₁-n₂)/2]t = ±1
• This occurs when [π(n₁−n₂)/2]t = 0, π, 2π, 3π, etc
• The corresponding time intervals are 0, 2/(n₁-n₂), 4/(n₁-n₂), etc.
• The time interval between successive maxima is 1/(n₁-n₂).

Minimum Intensity (Waning)

• The amplitude is minimum (0) when cos[π(n₁-n₂)/2]t = 0
• This happens when [π(n₁−n₂)/2]t = π/2, 3π/2, 5π/2, etc
• The time interval between successive minima is also 1/(n₁-n₂).
• Beat frequency (N) in waxing or waning is equal to |n₁-n₂|.