Speed of Sound Formula

Sound is a vibration or noise which travels through any medium. It travels in energy waves from one molecule to another, and when it enters a person's ear, it can be easily perceived. When an object vibrates, for example, it passes energy to the surrounding particles, causing them to vibrate as well. Owing to the lack of particles to serve as a conduit, sound cannot move through the vacuum. It can only flow through a medium like air, water, or solid.

[Image will be Uploaded Soon]

The distance travelled per unit time by a sound wave travelling in an elastic medium is known as the speed of sound. The speed of sound in a given medium depends on the elasticity and density of that medium. The higher the sound speed, the greater is the elasticity and the lower is the mass. As a result, the speed of sound is the maximum in solids and the minimum in gases.

Speed of Sound Formula

The speed of sound equation is expressed as,

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

Speed of sound = \[\frac{\text{(the coefficient ratio of specific heats} \times {\text{ the pressure of the gas)}}}{\text{(the density of the medium)}}\]

c = \[\frac{\gamma \times P}{\rho}\]

Where,

c: Speed of sound

P: Pressure 

ρ : Density

γ : Specific heat ratio

Speed of Sound in Air Formula

The speed of sound in air formula is,

γ = \[\sqrt{\frac{\gamma \times R \times T}{M}}\]

For air,  γ  = 1.4,  R = 8.31 J/mol, and M = 0.02897 kg/mol.

Speed of Sound in Solid Formula

The sound speed for pressure waves in rigid materials like metals is sometimes given for "long rods" of the material, which are simpler to measure.

The speed of pure pressure waves may be simplified in rods with diameters less than a wavelength and the speed of sound in solid formula is given by:

C\[_{solid}\] = \[\frac{E}{\rho}\]

Where E stands for Young's modulus. This expression is identical to that for shear waves, with the exception that Young's modulus replaces the shear modulus. The speed of sound for pressure waves in long rods will always be somewhat slower than the same speed in the homogeneous 3-dimensional solids, and the ratio of the speeds in the two types of objects is determined by the material's Poisson's ratio.

Speed of Sound in Water Formula

The only non-zero stiffness in a fluid is due to volumetric deformation (fluid does not sustain shear forces).

In conclusion, the sound speed in a fluid (water) is given by

C\[_{fluid}\] = \[\frac{K}{\rho}\]

Where,

K = bulk modulus of the fluid

ρ = Density of fluid

Speed of Sound in Gas Formula

The speed of sound in the fluid is,

C\[_{fluid}\] = \[\frac{K}{\rho}\]

K = bulk modulus of the fluid.

ρ = Density of fluid.

By adiabatic compression and rarefaction, sound waves move through a gas (expansion). The speed of sound in gas formula is given by,

γ =  \[\sqrt{\frac{\gamma \times R \times T}{M}}\]

γ = Adiabatic index

R = 8.314 J/mol - k universal gas constant

T = Absolute temperature

M = Molecular mass

Wavelength Sound Formula 

In simple terms, the wavelength is the distance between two consecutive crests or two consecutive troughs of a wave.

In addition, many different things move in similar forms of waves, like water, strings, air (sound waves), the earth or ground, and light. Furthermore, the wavelength of the wave is represented by the Greek symbol lambda (λ). Furthermore, the wave's wavelength is equal to the wave's velocity divided by its frequency. We also use meters (m) to measure wavelength in units.

The wavelength formula sound is given by,

Wavelength = \[\frac{\text{Speed of sound}}{Frequency}\]

λ = \[\frac{v}{f}\]

Where, 

ƛ = Wavelength

v = Speed of sound

f = Frequency

Solved Examples

Ex.1. The frequency of the middle C on the piano is 342 Hz. What will be the wavelength of the sound corresponding to the note of middle C if the speed of sound in air is assumed to be 445 m/s?

Answer: 

Given,

 v = 445 m/s

 f = 342 Hz

Let wavelength = λ

By using the wavelength formula sound we get, 

 v = f.λ

Rearranging the wavelength sound formula we get,

λ = \[\frac{v}{f}\]

λ = \[\frac{445}{342}\]

λ = 1.3011 m/s

The wavelength is 1.3011 m/s.

Ex.2. The sound wave with density 0.037 Kg/m\[^{3}\] and pressure of 4 kPa having the temp 50 degrees Celsius travels in the air. Find out the speed of the sound.

Solution:

Given,

Temperature T = 276 K

Density ρ = 0.037 Kg/m\[^{3}\] 

Pressure p = 4kPa = 4000 Pa

The specific heat in air = 1.4

The speed of sound equation is given by,

c = \[\sqrt{\gamma \times \frac{P}{\rho}}\]

c = \[\sqrt{1.4 \times \frac{4000}{0.037}}\]

c = \[\sqrt{1.4 \times 108, 108.1081}\]

c = \[\sqrt{151351.3513}\]

c = 389.0390 m/s