Question

Deduce the relation between wavelength, frequency and speed of sound.

Answer 1

Hint: Sound is a wave and hence, we can derive the relation between the wavelength, frequency and speed of a wave by using the proper definitions of the terms. The wavelength is essentially the distance travelled by the wave in one time period of its cycle whereas the frequency is the number of cycles per second of the wave.
Formula used:
$\text{Speed = }\dfrac{\text{Distance}}{\text{time}}$
$\text{Frequency of wave = }\dfrac{1}{\text{Time period of wave}}$
$\text{Wavelength = Distance travelled by the wave in one time period}$

Complete step by step answer:
As explained in the hint, we can solve this problem by using the definitions of the wavelength and the frequency of a wave to get the relation with the speed using the definition of speed.
$\text{Wavelength = Distance travelled by the wave in one time period}$ --(1)

The wavelength of a body is essentially the distance travelled by a point on the wave (and hence, the whole wave) in one time period.
Also, the frequency is the number of cycles completed by the wave in each second. It is the inverse of the time period (the time taken by the wave to complete one cycle). Hence, mathematically,

$\text{Frequency of wave = }\dfrac{1}{\text{Time period of wave}}$--(2)

The speed of a body is the distance travelled by the body in unit time. Hence,

$\text{Speed = }\dfrac{\text{Distance}}{\text{time}}$ --(3)

Now, the speed of the wave can be said to be the ratio of the distance travelled by the wave in one time period to the time span of one time period. Hence, using (3), we can say that the speed of the wave can be written as

$\text{Speed of wave = }\dfrac{\text{Wavelength}}{\text{Time period}}$ --(4)

Putting (2) in (4), we get,
$\text{Speed of wave = Wavelength}\times \text{Frequency}$.

Hence, we have got the required relation between the speed, wavelength and frequency of the sound wave.

Note: Students should know that the frequency of sound depends upon its source and does not change when sound travels from one medium to another. However, when travelling from one medium to another, the wavelength of the sound wave changes and hence the speed of the wave changes too. Often students get confused and think that the frequency changes too. Students should be careful of this concept as many questions are based upon this principle.