Physics of Beats and Interference

What phenomenon describes the periodic variation in loudness when two sound waves of slightly different frequencies interact?

What is the expression for the beat frequency derived from two sound waves with frequencies $n_1$ and $n_2$?

During the production of beats, when does the resultant amplitude reach its maximum value?

What role does the principle of superposition play in the beat phenomenon?

What expression shows the resultant wave's characteristics when two waves $y_1$ and $y_2$ are combined?

When does the intensity of sound reach its minimum during the beats phenomenon?

What is the time interval between successive maxima in the intensity of sound due to beats?

What happens to the resultant wave formed by two waves with frequencies $n_1$ and $n_2$ during complete destructive 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|.

This quiz explores the phenomenon of beats produced by the superposition of two sound waves with slightly different frequencies. It covers the expression for beats frequency and the concepts of constructive and destructive interference. Test your understanding of wave behavior and sound intensity variations.