Beats Produced by two Tuning Forks

# Objective

To demonstrate the phenomenon of beats produced by two tuning forks of slightly different frequencies.

# Theory

## Superposition theorem

When two waves (Y1 and Y2 ) meet in a given space and time, resultant waves are obtained by simply adding displacements at each location. This is known as the superposition theorem.

Figure :Superposition

### Interference

Superposition is the basis of interference. In this case, we are dealing with waves of same frequency.

When we add waves with equal amplitude and equal frequency at each point, we get maximum amplitude. This is constructive interference. In the case of sound waves, you will hear sound with maximum loudness during constructive interference of two sounds.

In destructive interference, the amplitude of the resultant wave results in 0. In the case of sound waves, you will not hear sound during the destructive interference of waves.

Figure: Constructive and destructive interference

## Beats

If we strike two tuning forks with slightly different frequencies, we hear the resultant sound as the intensity of the sound changes periodically. Here, constructive interference and destructive interference of sound waves happen periodically.

Figure: Beats

When two sounds of nearly different frequencies reach our ears simultaneously, the frequency of the combined sound will be the average of frequencies of 2 sound waves, and we hear periodic variations in the sound. These periodic variations of sound are called beats.

Beats are observable when the difference between the frequencies of sources is smaller. Consider two harmonic sound waves of angular frequencies of ω1 and ω2. The difference between the ω1 and ω2 is small.

ω1 is slightly greater than ω2

S1 – Displacement of first wave

S2 – Displacement of second wave

S - Displacement of resultant wave

Displacements of two sound waves respectively, are

S1= a cos (ω1t)

S2=a cos (ω2t)

When two sounds are heard simultaneously, the resultant displacement is by the principle of superposition.

S=S1+S2= a (cos (ω1t)+ cos (ω2t))

= 2 a cos((ω1- ω2)t/2)   cos((ω1 + ω2)t/2)

ωb= (ω1 - ω2)/2

ωa= (ω1 + ω2)/2

S = 2 a cos((ωb)t) cos((ωa)t)

ω1 =2* π * ν

ω2 =2* π * ν

ωa =2* π * νa

ωb =2* π * ν

ωa – Angular frequency of the resultant wave

The intensity of the resultant wave increases and decreases at the angular frequency of ωa.

ν1, ν2 are frequencies sound waves.

The frequency of resultant wave is νa.  Increasing and decreasing loudness happens at the frequency of  νb. It is called beat frequency.

νb =   ν1-   ν

Example

Sound waves of 11 Hz and 9 Hz give the beats of frequency 2 Hz.

Superposition of two harmonic waves, one of frequency 11 Hz (a), and the other of frequency 9 Hz (b), giving rise to beats of frequency 2 Hz as shown in (c).

# Learning Outcomes

• Students learn how beats are produced using tuning forks.
• Students understand the phenomena of beat.
• Students grasp the correlation between the variance in tuning fork frequencies and the resulting  frequency of beats.