Question

The audible range of a person is 20 to 20,000 Hz. If the speed of sound in air is $330m{s^{ - 1}}$ , calculate the longest wavelength of sound, which he can detect?
\[
  A.{\text{ }}\dfrac{{18000}}{{330}}m \\
  B.{\text{ }}\dfrac{{330}}{{1800}}m \\
  C.{\text{ }}\dfrac{{20}}{{330}}m \\
  D.{\text{ }}\dfrac{{330}}{{20}}m \\
 \]

Answer 1

Hint: In this question we will use the relation between the wavelength and the frequency given in terms of the speed of the travelling wave. So in order to get the longest wavelength, the frequency of sound must be minimum because wavelength is inversely proportional to the frequency.

Formula used- $V = \lambda \times f$

Complete step-by-step solution -

Given that
The speed of sound in air is $330m{s^{ - 1}}$
And Range of frequencies from 20Hz to 20,000Hz
As we know the relation between the wavelength and the frequency given in terms of the speed of the travelling wave is:
$V = \lambda \times f$
Where,
“V” is the velocity of sound
 “f” is the frequency
$\lambda $ is the wavelength
\[ \Rightarrow \lambda = \dfrac{V}{f}\]
As we can see that the wavelength is inversely proportional to the frequency, so the longest wavelength will exist for minimum frequency.
Substitute the value of velocity of sound and minimum frequency as 20 Hz.
\[
  \because V = \lambda \times f \\
   \Rightarrow \lambda = \dfrac{V}{f} \\
   \Rightarrow \lambda = \dfrac{{330m{s^{ - 1}}}}{{20Hz}} \\
   \Rightarrow \lambda = \dfrac{{330}}{{20}}m \\
 \]
Hence, the longest wavelength of sound, which the person can detect \[\dfrac{{330}}{{20}}m\]
So, the correct answer is option D.

Note- Wavelength is the distance between sound waves while frequency is the number of times in which the sound wave occurs. Students must remember this relation between the frequency and wavelength as this relation holds true even in the case of light. Frequency and wavelength are both directly and inversely related. For example, if two waves are traveling at the same speed, then they are inversely related. The wave with shorter wavelength will have a higher frequency while a longer wavelength will have a lower frequency.