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Physics Formulas

Speed of Sound Formula

Sound is a vibration or disturbance which travels through any medium. It moves by transferring energy from one particle to another and can be heard easily when it reaches a person’s ear. For example, when an object vibrates then it transfers energy to the surrounding particles and as a result, makes them vibrate. Sound cannot travel through the vacuum due to the absence of particles to act as a medium. Sound travels only through a medium like water, air and solid. In this article, we will discuss sound waves and the speed of sound formula. Let us begin learning!

Speed of Sound Formula

What is a sound wave?

Sound is a wave that is transmitted through air and liquid as longitudinal waves but through solid as both longitudinal and transverse waves. The speed of the propagation of the sound wave is depending on the characteristics of the medium in which it propagates. Its speed does not depend on the characteristics of the wave or the force which generates it. Its propagation in a medium can be used to study some properties that medium.

The speed of sound is the distance traveled by a sound wave propagating through an elastic medium per unit time. The speed of sound in a given medium depends on the density and elasticity properties of that medium. According to physics, more is the speed of sound greater is the elasticity and smaller is the density. Therefore, the speed of sound is maximum in solids and minimum in solids.

Speed of Sound Formula

The speed of sound can be computed as,

speed of sound = the square root of (the coefficient ratio of specific heats × the pressure of the gas / the density of the medium).

Mathematically,

c =\( \sqrt(\frac{\gamma \times P}{\rho}) \) 

Where,

c Speed of sound
P Pressure
\( \rho\) Density
\( \gamma\) Specific heat ratio

Here \(\gamma\) is representing the adiabatic index and also known as the isentropic expansion factor. It is computed as the ratio of specific heats of a gas at a constant pressure to a gas at a constant volume.

Solved Examples

Q.1: The sound waves travel in the air with a density of 0.034 kg/m³ and pressure of 2k Pa with a temperature of 2°C. Calculate the speed of the sound.

Solution:

As given here:

Temperature, T = 2° C

Density, \(\rho = 0.034 kg/m³ \)

and, pressure P = 2k Pa

i.e. P = 2000 Pa

as we know that specific heat ratio in air is,

\(\gamma = 1.4 \)

Now the speed of sound formula is given by

c =\(\sqrt(\frac{\gamma \times P}{\rho})\)

c =\(\sqrt(\frac{1.4 \times 2000}{0.034})\)

c= \(\sqrt (82352.94)\)

c = 286.97

Therefore, speed of sound = 286.97 m/s.

Q. 2: Find out the pressure if sound travels through a medium having a density 0.05 KPa and speed of sound is 400 m/s.

Solution:

As given in the problem,

Density, \(\rho\) = 0.05 KPa,

Speed of sound, c = 400 m/s

Also, \(\gamma\) = 1.4 at room temperature for gases.

Now the speed of sound formula is:

c =\(\sqrt(\frac{\gamma \times P}{\rho})\)

rearranging the formula,

P = \(\frac{c^2 \times \rho}{\gamma}\)

P = \(\frac{400^2 \times 0.05}{1.4}\)

P= \(\frac {1600 \times 0.05}{1.4}\)

P= 5714.28 Pa

Therefore, Pressure will be 5714.28 Pa.

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