Solve the Problem $$f ( t ) = \sin ( 10 t ) \sin ( t )$$

question 356

Essay

Solve the problem.
-A product of two oscillations with different frequencies such as
$$f ( t ) = \sin ( 10 t ) \sin ( t )$$
is important in acoustics. The result is an oscillation with "oscillating amplitude."
(i) Writethe product $\mathrm { f } ( \mathrm { t } )$ of the two oscillations as a sum of two cosines and call it $g ( \mathrm { t } )$ .
(ii) Usinga graphing utility, graph the function $g ( t )$ on the interval $0 \leq t \leq 2 \pi$ .
(iii) On the same system as your graph, graph $y = \sin t$ and $y = - \sin t$ .
(iv) Thelast two functions constitute an "envelope" for the function $g ( t )$ . For certain values of $t$ , the two cosine functions in $g ( t )$ cancel each other out and near-silence occurs; between these values, the two functions combine in varying degrees. The phenomenon is known (and heard) as "beats." For what values of $t$ do the functions cancel each other?

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