1.

Three sound sources A, B and C have frequencies 400, 401 and 402 H_(Z), respectively. Cacluated the number of beats noted per second.

Answer» <html><body><p><br/></p>Solution :Let as make the following table. <br/> Beat time period for `A` and `B` is `1` s. It implies that `A` and `B` are in phase at time `t = <a href="https://interviewquestions.tuteehub.com/tag/0-251616" style="font-weight:bold;" target="_blank" title="Click to know more about 0">0</a>`, they are again in phase after `1` s. Same is the case with `B` and `C` . But beat time period for `A` and `C` is `0.5`s. <br/> Therefore , beat time period for all together `A, B` and `C` will be `1` s. Because if , at `t = 0, A, B` and `C` all are in phase then after `1` s. (`A` and `B`) and (`B` and `C`) will again be in phase for the first time while (`A` and `C`) will be in phase for the second time. Or we can that all `A, B` and `C` are again in phase after `1` s. <br/> `:.` Beat time period , `T_(b) = 1 s` <br/> or Beat <a href="https://interviewquestions.tuteehub.com/tag/frequency-465761" style="font-weight:bold;" target="_blank" title="Click to know more about FREQUENCY">FREQUENCY</a> , `f_(b) = (1)/(T_(b)) = H_(Z)` <br/> Suppose at time `t`, the equations of waves are <br/> `y_(1) = A_(1) sin 2 pif_(A)t``(omega = 2pi f)` <br/> `y_(2) = A_(2) sin 2pi f_(B) t` <br/>and `y_(3) = A_(3) sin 2pi f_(C) t` <br/> If they are beat time at some given instant of time `t`, then <br/> `2 pif_(A)t = 2 pi f_(B)t = 2pi f_(C)t` ...(i) <br/> Let `T_(b)` be the beat time period, i.e. after time `T_(b)` they all are again in phase. As `f_(C) gt f_(B) gt f_(A)` , so<br/> `2pi f_(C) (t + T_(b)) = 2 pi f_(A) (t + T_(b)) + 2 mpi` ...(ii) <br/>and `2pi f_(B) (t + T_(b)) = 2 pi f_(A) (t + T_(b)) + 2 npi` ....(<a href="https://interviewquestions.tuteehub.com/tag/iii-497983" style="font-weight:bold;" target="_blank" title="Click to know more about III">III</a>) <br/> Here, `m` and `n (lt m)` are positive <a href="https://interviewquestions.tuteehub.com/tag/integers-17173" style="font-weight:bold;" target="_blank" title="Click to know more about INTEGERS">INTEGERS</a>. <br/> From Eqs. (i) and (ii) <br/> `(f_(C) - f_(A)) T_(b) = m` ...(iv) <br/> Similary, From Eqs. (i) and (ii) <br/> `(f_(B) - f_(A)) T_(b) = n` ...(v) <br/> <a href="https://interviewquestions.tuteehub.com/tag/dividing-957391" style="font-weight:bold;" target="_blank" title="Click to know more about DIVIDING">DIVIDING</a> Eq. (iv)by Eq. (v), <br/> `(m)/(n) = (f_(C) - f_(A))/(f_(B) -f_(A))= (402 - 400)/(401 - 400)= (2)/(1)` <br/> Thus, letting `m = 2` and `n = 1` <br/> `T_(b)= (m)/(f_(C) - f_(A))` [from Eq. (iv)] <br/> = `(2)/(2) = 1 H_(Z)` <br/> `:.` Beat frequency , `f_(b) = (1)/(T_(b)) = 1 H_(Z)`</body></html>


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