Explain the production of beats and deduce analytically the expression for beat frequency
Answers
Beats
Beats is the phenomenon caused by two sound waves of nearly same frequencies and amplitudes travelling in the same direction.
For example:-
Tuning of musical instruments like piano, harmonium etc. Before we start playing on these musical instruments they are set against the standard frequency. If it is not set a striking noise will keep on coming till it is set.
Mathematically
Consider only the time dependent and not the position dependent part of the wave.
s1=a cos ω1t and s2=a cos ω2t; where amplitude and phase of the waves are same,but the frequencies are varying. Also considering ω1> ω2.
When these 2 waves superimpose s= s1+ s2=a[cos ω1t + cos ω2t]
By simplifying , 2a (cos(ω1 - ω2)/2)t cos(ω1 + ω2)/2)t)
=>ω1 - ω2 is very small as ω1> ω2.Let (ω1 - ω2 )=ωb
=>ω1 + ω2 is very large. Let (ω1 + ω2 )=ωa
s= 2a cos ωbt cos ωat
cosωat will vary rapidly with time and 2acosωbt will change slowly with time.
Therefore we can say 2acosωbt = constant. As a result 2acosωbt = amplitude as it has small angular variation.
kulsum hashmi
Explanation:
analytycally expression for the beat fre