Question

Beats are produced by the superposition of two waves of nearly equal frequencies. Of the following, the correct statement is

• A. all particles of the medium vibrate simple harmonically with frequency equal to the difference between the frequencies of component waves
• B. the frequency of beats changes with the location of the observer
• C. the frequency of beats changes with time
• D. amplitude of vibration of particle at any point changes simple harmonically with frequency equal to one half of the difference between the component waves

Answer 1

Answer: (D)

amplitude of vibration of particle at any point changes simple harmonically with frequency equal to one half of the difference between the component waves

$y_1=Asin{\omega }_{1}t,y=Asin{\omega }_{2}t$

$y_1+y_2=A(sin{\omega }_{1}t+sin{\omega }_{2}t)$

$=2Asin\left(\frac{{\omega }_1+{\omega }_2}{2}\right)+cos\left(\frac{{\omega }_1-{\omega }_2}{2}\right)t$

in case of beats

$|{\omega }_1-{\omega }_2|<<|{\omega }_1+{\omega }_2|$

$\therefore y_1+y_2=A^{1}sin{\omega }^{1}t$

${\omega }^1=\left(\frac{{\omega }_1+{\omega }_2}{2}\right)$

$A^1=2Acos\left(\frac{{\omega }_1-{\omega }_2}{2}\right)t$