Often, we hear sound when talking to people, listening to music, playing musical instruments, etc. Have you wondered how fast sound can travel and how we calculate its speed theoretically? In this article, let us find answers to all these questions.
The velocity of sound is given by
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Newtons formula for the speed of sound
Newton worked on the propagation of sound waves through the air. He assumed that this process of propagation is isothermal. Absorption and release of heat during compression and rarefaction will be balanced. Thus, the temperature remains constant throughout the process.
According to Boyle’s law
PV = Constant
Where,
P is pressure
V is the volume of gas.
On differentiating above equation we get-
PdV+VdP=0
⇒ PdV = -VdP
⇒ P = B
Where,
The velocity of the sound wave can be written as –
Thus substituting B =P we get-
Speed of sound in air
At Normal Temperature and Pressure, the velocity of sound in air is given by –
Where atmospheric pressure P = 1.1013×105 N/m2
The density of air (𝜌)= 1.293 kg/m3
The value got here does not match with the experimental value. That is 332 m/s. Which implies that some correction should be done to Newton’s equation.
Laplace Correction for Newton’s Formula
He corrected Newton’s formula by assuming that, there is no heat exchange takes place as the compression and rarefaction takes place very fast. Thus, the temperature does not remain constant and the propagation of the sound wave in air is an adiabatic process.
For an adiabatic process
PV𝛾 = Constant
Where,
𝛾 is adiabatic index
Cp specific heat for constant pressure
Cv specific heat for a constant volume.
Differentiating both the sides we get-
Dividing both the sides by V𝛾-1
The velocity of sound is given by
Substituting B= 𝛾P in above equation we get-
Velocity of sound formula
Velocity of sound
Calculate the velocity of sound wave using Laplace correction to Newton’s formula at Normal Temperature and Pressure.
The velocity of the sound formula is given by-
Where,
Adiabatic index 𝛾 – 1.4
Where atmospheric pressure P = 1.1013×105 N/m2
The density of air (𝜌)= 1.293 kg/m3
Substituting the values in the equation we get-
Which has a very good match with the experimental value
Physics Related Topics:
| Bulk modulus |
| Adiabatic Process |
| Kinetic Theory of Gases |
| Wave Speed Motion |
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