Chapter 14, Problem 1OQ

A flute has a length of 58.0 cm. If the speed of sound in air is 343 m/s, what is the fundamental frequency of the flute, assuming it is a tube closed at one end and open at the other? (a) 148 Hz (b) 296 Hz (c) 444 Hz (d) 591 Hz (e) none of those answers

To determine

The fundamental frequency of the flute

Answer to Problem 1OQ

Option (C)

Explanation of Solution

The flute being a cube closed one end and open at the other end exhibits a standing wave pattern. The closed end of the flute will have a node and the open end of the flute will have an antinode.

Write the equation for the fundamental frequency of the flute using harmonic series.

    $f_1=\frac{v}{{\lambda }_1}$        (I)

Here, $v$ is the speed of sound in air and ${\lambda }_1$ is the wavelength of the sound waves corresponding to its fundamental frequency.

Write the equation for the wavelength of the sound waves corresponding to its fundamental frequency.

    ${\lambda }_1=4L$        (II)

Here, $L$ is the length of the flute.

Substitute equation (I) in equation (II).

    $f_1=\frac{v}{4L}$

Conclusion:

Substitute $343\text{m/s}$ for $v$ and $58\text{cm}$ for $L$

    $\begin{array}{c}{f}_1=\frac{343\text{m/s}}{4\left(58\text{cm}\right)}\\ =\frac{343\text{m/s}}{4\left(0.58\text{m}\right)}\\ =\frac{343\text{m/s}}{2.32\text{m}}\\ =148\text{Hz}\end{array}$

Thus, the flute has a fundamental frequency of $148\text{Hz}$. Therefore, only option (C) is correct and all other options are incorrect.