Question

What is the order of the wavelength of ultrasonic in the air?
A) ${10^2}{\text{m}}$
B) ${10^1}{\text{m}}$
C) ${10^{ - 2}}{\text{m}}$
D) ${10^0}{\text{m}}$

Answer 1

Not all sound is audible to a human ear. Ultrasonic is one such category of sound and has frequencies exceeding the upper audible limit. The wavelength of any wave is found to be inversely proportional to its frequency. Sound, being a longitudinal wave will have its wavelength follow the same condition.

Formula Used: The velocity of a wave is given by, $v = \lambda f$ where $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.

Complete step by step answer:
Step 1: Find the expression for the wavelength of the sound wave.
An ultrasonic is a sound wave that has its frequency greater than the upper limit of the audible range that humans can hear. This upper limit is said to be $f = 20{\text{kHz}}$ .
The frequency of a wave is given by, $v = \lambda f$ where $f$ is the frequency of the wave and $\lambda$ is the wavelength of the wave.
The above relation connecting the frequency of a wave, its wavelength and its velocity in the air can also be expressed as $\lambda = \dfrac{v}{f}$ ------- (1).
Step 2: Find the order of the wavelength of ultrasonic by substituting the values for its frequency and velocity in air.
We have the frequency of the ultrasonic sound wave to be $f = 20{\text{kHz}}$ .
Now, the velocity of sound (audible or inaudible) in the air is $v = 320{\text{m/s}}$ .
Substituting the values for $f = 20{\text{kHz}}$ and $v = 320{\text{m/s}}$ in equation (1) we get, $\lambda = \dfrac{{320}}{{20 \times {{10}^3}}} = 0.016$
The above value can be expressed as $\lambda = 1.6 \times {10^{ - 2}}{\text{m}}$ .
∴ the order of the wavelength of ultrasonic is ${10^{ - 2}}{\text{m}}$ . The correct option is c.

Additional Information: Unlike light, the velocity of sound in the air is not constant. Different layers of the air have different temperatures and so the velocity will change. This is because the molecules at a higher temperature vibrate faster and lead to an increase in the speed of sound in these layers of air.

Note: When substituting values for physical quantities in an equation, make sure that the quantities are expressed in their S.I. units. The S.I. the unit of wavelength is metre, frequency is Hertz and that of velocity is metre per second. Necessary conversion of units must be done if this is not the case.