Chapter 22, Problem 19SP

To determine

The wavelength of sound hearing limits that are $20\text{ Hz}$ and $20\text{ kHz}$, considering the speed of sound to be $340\text{m/s}$.

Answer to Problem 19SP

Solution:

$1.7\text{ cm}$ and $17\text{ m}$

Explanation of Solution

Given Data:

The frequency range of audible sound for human beings is $20\text{ Hz}$ to $20\text{ kHz}$.

The speed of sound is $340\text{ m/s}$.

Formula Used:

The formula for wavelength of wave is

$\lambda =\frac{v}{f}$

Here, $\lambda$ is the wavelength of the signal, $v$ is the speed of the wave, and $f$ is the frequency of the wave.

Explanation:

Recall the expression of the wavelength of the sound wave:

$\lambda =\frac{v}{f}$

Calculate the wavelength for the minimum frequency of audible sound wave:

Substitute $340\text{ m/s}$ for $v$ and $20\text{ Hz}$ for $f$

$\begin{array}{c}\lambda =\frac{340\text{ m/s}}{20\text{ Hz}}\\ =17\text{ m}\end{array}$

Again, calculate the wavelength for the maximum frequency of audible sound wave by substituting $340\text{ m/s}$ for $v$ and $20\text{ kHz}$ for $f$

$\begin{array}{c}\lambda =\frac{340\text{ m/s}}{20\text{ kHz}\left(\frac{{10}^3\text{ Hz}}{\text{1 kHz}}\right)}\\ =17\times {10}^{-3}\text{ m}\left(\frac{1\text{ cm}}{{10}^{-2}\text{ m}}\right)\\ =1.7\text{ cm}\end{array}$

Conclusion:

The wavelength range of the wavelength of audible sound is $1.7\text{ cm}$ to $17\text{ m}$.