Chapter 12, Problem 88GP

To determine

To Find: The fundamental frequency of the horn and the harmonic close to the F shape note.

Answer to Problem 88GP

  $50.4\text{Hz}$ & 7^th harmonic.

Explanation of Solution

Given:

The length of the alpenhorn, $\mathrm{l}=3.4\text{m}$

The frequency of F sharp from table 12-3, ${\mathrm{f}}_{\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{G}}=370\text{Hz}$

The speed of sound in the air, $\mathrm{V}=343\text{m/s}$

Formula used:

The fundamental frequency is calculated as:

  ${\mathrm{f}}_1=\frac{\mathrm{V}}{2\mathrm{L}}$

Here, $\mathrm{V}$ is the speed of the sound,

  $\mathrm{L}$ is the length of the alpenhorn.

Write the expression to finding the frequency at F note.

  ${\mathrm{f}}_{\mathrm{F}}=\mathrm{n}{\mathrm{f}}_1$

Here $\mathrm{n}$ is the number of harmonics.

Calculation:

Substitute the given values:

  ${\mathrm{f}}_1=\frac{343}{2(3.4)}=50.44\text{Hz}$

Write the expression for the frequency at F note.

  $\begin{array}{c}{\mathrm{f}}_{\mathrm{F}}=\mathrm{n}{\mathrm{f}}_1\\ 370=\mathrm{n}50.44\\ \mathrm{n}=\frac{370}{50.44}=7.33\end{array}$

Conclusion:

The fundamental frequency is $50.4\text{ Hz}$ & the sixth overtone or 7^th harmonic is close to F tone.