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Question

# Tovi is playing a flute and Dawn is playing a violin. At the beginning of the performance, they tuned their instruments by ear, so the difference between the fundamental frequencies of the A notes played on each instrument was less than 1 Hz. At the end of the performance, they measured the A note fundamental frequencies of their instruments using a device that measures sound frequencies with uncertainty 0.05 Hz. Dawn's frequency was 442.6 Hz and Tovi's was 439.3 Hz. (a) Can you say for sure that the A note frequency of at least one musical instrument changed during the performance? (b) Can you say for sure that the A note frequency of both musical instruments changed during the performance? (c) What do Dawn and Tovi hear when they simultaneously play the A note at the end of the performance? Give a quantitative description.

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### 1 Concepts and Principles

$\textbf{Beat Frequency}$: A beat is a wave that results from the superposition of two waves of about the same frequency. The beat (the net wave) has a frequency equal to the average of the two frequencies and has variable amplitude. The frequency with which the amplitude of the net wave changes in called the beat frequency $f_{\text{beat}}$; it equals the difference in the frequencies of the two waves:

$\begin{gather*} f_{\text{beat}}=\abs{f_1-f_2} \end{gather*}$

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