Question

Calculate the wavelengths in air at 20?C for sounds in the maximum range of human hearing. The speed of sound in air is 343 m/s.

Part A

Calculate the wavelength of a 20-Hz wave.

Express your answer to two significant figures and include the appropriate units.

Part B

Calculate the wavelength of a 20,000-Hz wave.

Express your answer to two significant figures and include the appropriate units.

Part C

What is the wavelength of a 20-MHz ultrasonic wave?

Express your answer to two significant figures and include the appropriate units.

Answer 1

Concepts and reason

The main concepts required to solve this problem are the frequency, velocity and wavelength.

Initially, write the equation for the relation for the frequency, velocity and wavelength of a sound. Use this equation and calculate the wavelength for each of the part given in the question.

Fundamentals

Wavelength of the sound wave can be defined as the distance between the successive crests of a wave.

The equation for the relation of the frequency, wavelength and velocity is,

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

Here, $\lambda$ is the wavelength, v is the velocity and f is the frequency.

(A)

The equation for the relation of the frequency, wavelength and velocity is,

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

Substitute 343 m/s for v, and 20 Hz for f in above equation.

$\begin{array}{c}\\\lambda = \frac{{\left( {343{\rm{ m/s}}} \right)}}{{\left( {20{\rm{ Hz}}} \right)}}\\\\ = 17{\rm{ m}}\\\end{array}$

(B)

The equation for the relation of the frequency, wavelength and velocity is,

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

Substitute 343 m/s for v, and 20,000 Hz for f in above equation.

$\begin{array}{c}\\\lambda = \frac{{\left( {343{\rm{ m/s}}} \right)}}{{\left( {20,000{\rm{ Hz}}} \right)}}\\\\ = 1.7 \times {10^{ - 2}}{\rm{ m}}\\\end{array}$

(C)

The equation for the relation of the frequency, wavelength and velocity is,

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

Substitute 343 m/s for v, and 20 MHz for f in above equation.

$\begin{array}{c}\\\lambda = \frac{{\left( {343{\rm{ m/s}}} \right)}}{{\left( {20{\rm{ MHz}}\left( {\frac{{{{10}^6}{\rm{ Hz}}}}{{1{\rm{ MHz}}}}} \right)} \right)}}\\\\ = 1.7 \times {10^{ - 5}}{\rm{ m}}\\\end{array}$

Ans: Part A

The wavelength of a 20 Hz wave is 17 m.

Part B

The wavelength of a 20,000 Hz sound wave is $1.7 \times {10^{ - 2}}{\rm{ m}}$ .

Part C

The wavelength of a 20 MHz ultrasonic wave is $1.7 \times {10^{ - 5}}{\rm{ m}}$ .