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When an emperor penguin returns from a search for food, how can it find its mate among the thousands of penguins huddled together for warmth in the harsh Antarctic weather? It is not by sight, because penguins all look alike, even to a penguin. The answer lies in the way penguins vocalize. Most birds vocalize by using only one side of their two-sided vocal organ, called the syrinx. Emperor penguins, however, vocalize by using both sides simultaneously. Each side sets up acoustic standing waves in the bird's throat and mouth, much like in a pipe with two open ends. Suppose that the frequency of the first harmonic produced by side A is f_(A1) = 432 Hz and the frequency of the first harmonic produced by side B is f_(B1) = 371 Hz. What is the beat frequency between those two first-harmonic frequencies and between the two second-harmonic frequencies? |
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Answer» Solution :The beat frequency between two frequencies is their difference, as given by Eq. `(f_("beat") = f_1 - f_2)` For the two first-harmonic frequencies`f_(A1)`and`f_(B1)` , the beat frequency is `f_("beat.1") = f_(A1)- f_(B1) = 432 Hz - 371 Hz` Because the STANDING waves in the penguin are effectively in a pipe with two open ends, the resonant frequencies are given by Eq. (f = nv/2L), in which L is the (unknown) length of the effective pipe. The first-harmonic frequency is` f_1`= v/2L, and the SECOND-harmonic frequency is `f_2`= 2v/2L. Comparing these two frequencies, we see that, in GENERAL, `f_2 = 2f_1` For the penguin, the second harmonic of side A has frequency `f_(A2) = 2f_(A1)`, and the second harmonic of side B has frequency `f_(B2) = 2f_(B1)` . Using Eq.with frequencies `f_(A2)` and `f_(B2)` ,we find that the corresponding beat frequency associated with the second harmonics is `f_("beat.2")- f_(A2) - f_(B2)= 2f_(A1) - 2f_(B1)` ` =2 (432 Hz) - 2(371 Hz)` = 122 Hz |
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