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# A piano tuner uses a 512-Hz tuning fork to tune a piano. He strikes the fork and hits a key on the piano and hears a beat frequency of 5 Hz. He tightens the string of the piano, and repeats the procedure. Once again he hears a beat frequency of 5 Hz. What happened?

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### 1 Concepts and Principles

1- The phenomenon of $\textbf{beating}$ is the periodic variation in intensity at a given point due to the superposition of two waves having slightly different frequencies. The beat frequency is:

$\begin{gather*} f_{\text{beat}}=|f_1-f_2|\tag{1} \end{gather*}$

where $f_1$ and $f_2$ are the frequencies of the individual waves.

2- $\textbf{Waves Under Boundary Conditions}$: the boundary conditions determine which standing-wave frequencies are allowed. For waves on a string, there must be nodes at both ends. The wavelengths and natural frequencies of normal modes are given by:

\begin{align*} f_n&=n\dfrac{v}{2L}=\dfrac{n}{2L}\sqrt{\dfrac{F_T}{\mu}}\;\;\quad\quad\quad\quad\quad \quad \quad \quad n=1,\;2,\;3,\;...\tag{2} \end{align*}

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