## Related questions with answers

Two identical taut strings under the same tension $\boldsymbol{~F}$ produce a note of the same fundamental frequency $f_0$ - The tension in one of them is now increased by a very small amount $\Delta F$. (a) If they are played together in their fundamental, show that the frequency of the beat produced is $f_{\text {beat }}=f_0(\Delta F / 2 F)$. (b) Two identical violin strings, when in tune and stretched with the same tension, have a fundamental frequency of $440.0 \mathrm{~Hz}$. One of the strings is retuned by increasing its tension. When this is done, $1.5$ beats per second are heard when both strings are plucked simultaneously at their centres. By what percentage was the string tension changed?

Solution

Verified### Theoretical reminder

We know that the wavelength and frequency determine the speed of a wave. They are related by the following formula:

$\begin{equation} v = \lambda \cdot f \end{equation}$

The speed of a sound wave in guitar string is proportional to the force of tension in the string $F$, and inversely proportional to the linear density (mass per unit of length, unit: $\frac{\text{kg}}{\text{m}}$) of the string in the following way:

$\begin{equation*} V \sim \sqrt{\frac{F}{\mu}} = \sqrt{\frac{F \cdot L}{m}} \tag{2} \end{equation*}$

We know that when a standing wave is formed in a string the number of antinodes determines the harmonic of the wave. This gives us a result that the length of the string is equal to a integer number of wavelength halves, that is:

$\begin{equation*} L = \frac{n \cdot \lambda}{2} \end{equation*}$

Rearranging:

$\begin{align*} & 2 \cdot L = n \cdot \lambda \\ & \lambda = \frac{2 \, L}{n} \tag{3} \end{align*}$

From this we see that any wavelength lambda that fulfils this equation for an integer $n$ can produce a standing wave.

Beats are formed by the interfering of two frequencies that are played simultaneously.. They can be heard as a periodical change in loudness of the resulting sound. The frequency of a beat is the difference of the interfering frequencies:

$\begin{equation*} f_{\text{beat}} = \abs{f_1 - f_2} \tag{4} \end{equation*}$

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