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Question: Given that sound travels in air at \(340m{s^{ - 1}}\) , find the wavelength of the waves in air prod...

Given that sound travels in air at 340ms−1340m{s^{ - 1}} , find the wavelength of the waves in air produced by a 20kHz20kHz sound source. If the same source is put in a water tank, what would be the wavelength of the sound waves in water? Speed of sound in water =1,480ms−1 = 1,480m{s^{ - 1}}

Explanation

Solution

We are required to find the wavelengths of the sound in air and water. We are given with speed of sound and frequency of sound. Recall a formula which related the three: wavelength, speed and frequency of sound.

Complete step by step answer:
Lets first note down the given quantities from the question and find a relation between them and wavelength.
Speed of sound in air: 340ms−1340m{s^{ - 1}}
Frequency of sound: 20kHz20kHz
Speed of sound in water: 1,480ms−11,480m{s^{ - 1}}
Now, in order to find the wavelength, we will have to relate the given quantities with wavelength.
Remember that the frequency of sound remains the same, irrespective of the medium it travels through.
We have the relation:
v=f×λv = f \times \lambda
Where vvis the velocity of the sound in the medium
ff is the frequency of the sound wave
λ\lambda is the wavelength of the sound in a medium.
Using this formula and the given values from the question, we can easily calculate the wavelength of sound. The wavelength for air can be calculated in the following manner:
v=f×λv = f \times \lambda
⇒λ=vf\Rightarrow \lambda = \dfrac{v}{f}
Substituting the values of velocity of sound in air and frequency of sound, we have:
⇒λ=34020000\Rightarrow \lambda = \dfrac{{340}}{{20000}}
∴λ=0.017m\therefore \lambda = 0.017m
Therefore, the wavelength of sound in air is: 0.017m0.017m
Now the wavelength of sound in water will be given as:
v=f×λv = f \times \lambda
λ=vf\lambda = \dfrac{v}{f}
⇒λ=148020000\Rightarrow \lambda = \dfrac{{1480}}{{20000}}
∴λ=0.074m\therefore \lambda = 0.074m
The wavelength of sound in water is: 0.074m0.074m
Therefore, the wavelength of the sound in air and water is 0.017m0.017m and 0.074m0.074m respectively.

Note:
Irrespective of the medium, the frequency of sound in any medium remains constant.
Convert all units to SI units to get the final answer correct.
The speed of sound given in the problem is the average speed, in reality the speed of sound is not constant rather changing continuously.