JEE Main & Advanced Physics Wave Mechanics Beats

When two sound waves of slightly different frequencies, travelling in a medium along the same direction, superimpose on each other, the intensity of the resultant sound at a particular position rises and falls regularly with time. This phenomenon of regular variation in intensity of sound with time at a particular position is called beats.

(1) Persistence of hearing : The impression of sound heard by our ears persist in our mind for 1/10th of a second. If another sound is heard before 1/10 second is over, the impression of the two sound mix up and our mind cannot distinguish between the two.

So for the formation of distinct beats, frequencies of two sources of sound should be nearly equal (difference of frequencies less than 10)

(2) Equation of beats : If two waves of equal amplitudes 'a' and slightly different frequencies \[{{n}_{1}}\] and \[{{n}_{2}}\] travelling in a medium in the same direction are.

\[{{y}_{1}}=a\sin {{\omega }_{1}}\,t=a\sin 2\pi {{n}_{1}}\,t\,;\,{{y}_{2}}=a\sin \,{{\omega }_{2}}\,t=a\sin 2\pi {{n}_{2}}t\]

By the principle of super position : \[\overrightarrow{y\,}=\overrightarrow{{{y}_{1}}}+\overrightarrow{{{y}_{2}}}\]

\[y=A\sin \pi ({{n}_{1}}+{{n}_{2}})t\] where \[A=2a\cos \pi ({{n}_{1}}-{{n}_{2}})t=\] Amplitude of resultant wave.

(3) One beat : If the intensity of sound is maximum at time \[t=0,\] one beat is said to be formed when intensity becomes maximum again after becoming minimum once in between.

(4) Beat period : The time interval between two successive beats (i.e. two successive maxima of sound) is called beat period.

\[n={{n}_{1}}\tilde{\ }{{n}_{2}}\]

(5) Beat frequency : The number of beats produced per second is called beat frequency.

\[T=\frac{1}{\text{Beat frequency}}=\frac{1}{{{n}_{1}}\tilde{\ }{{n}_{2}}}\]