Speed of Sound Formula

The speed of sound is the pace at which sound waves travel through a material like air, water, or solids. In plain terms, it indicates how quickly sound travels through a substance.

When an object causes a disturbance by vibrating or making sound, such as clapping your hands or speaking, the ensuing sound waves spread outward in all directions. Speed of Sound Formula helps to calculate the rate at which sound is travelling. Learn more about sound wave, its speed and speed od sound formula in this post by Extramarks.

What is Sound Wave?

A sound wave is a mechanical wave that moves through a medium, such as air, water, or solids, by vibrating particles. When an object vibrates or moves back and forth, it disturbs the surrounding medium, causing particles to compress and expand in a predictable way. This disturbance travels outward from the source as a sound wave.

Sound waves are made up of compressions and rarefactions, with compressions representing high pressure and particle density and rarefactions representing low pressure and particle density. As the sound wave travels through the medium, it transfers energy from its source to the listener’s ears.

Speed of Sound

Speed of sound is the rate at which sound waves propagate through a medium, such as air, water, or solids. In simpler terms, it tells us how quickly sound travels through a substance.

Speed of Sound in Different Mediums

Air: The speed of sound in dry air at sea level and a temperature of 20°C (68°F) is roughly 343 meters per second (m/s) or 1235 kilometers per hour.

Water: Sound travels more faster in water than in air. It travels at 1482 m/s (5360 km/h) in seawater and somewhat faster in freshwater due to changes in density and temperature.

Steel: Sound travels even faster through things such as steel. Sound can travel at speeds of up to 5000 m/s (18,000 km/h) in steel, depending on the material’s characteristics.

Wood: Sound travels slower in wood than in steel, but faster than in air. It can vary from around 3000 to 4000 m/s.

Speed of Sound Formula

The general formula to calculate speed of sound is

v = λ x f

where,

• v is speed of sound
• λ is wavelength of sound
• f is frequency of souund

The another formula to calculate speed of sound is,

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

Mathematically,

c =(√γ×P/ρ)

Where,

• c Speed of sound
• P Pressure
• ρ Density
• γ Specific heat ratio

Here γ 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.

Speed of Sound Formula for Solid

The speed of sound formula for solid is

c = √(Y/ρ)

where,

• c is speed of sound
• Y is Young’s Modulus
• ρ is Density

Speed of Sound Formula for Fluid

The speed of sound formula for fluid is

c = √(B/ρ)

where,

• c is speed of sound
• Y is Bulk Modulus
• ρ is Density

Speed of Sound Formula for Gas

The speed of sound formula for gas is

c = √(γ x R x T)/M

where,

• γ is Adiabatic Cofficient
• R is Gas Constant
• T is Temperature
• M is Molecular mass of the gas

Solved Examples on Speed of Sound Formula

Example 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, ρ=0.034kg/m³

and, pressure P = 2k Pa

i.e. P = 2000 Pa

as we know that specific heat ratio in air is,

γ=1.4

Now the speed of sound formula is given by

c =(√γ×P/ρ)

c =(√1.4×2000/0.034)

c= (√82352.94)

c = 286.97

Therefore, speed of sound = 286.97 m/s.

Example 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, ρ = 0.05 KPa,

Speed of sound, c = 400 m/s

Also, γ = 1.4 at room temperature for gases.

Now the speed of sound formula is:

c =(√γ×P/ρ)

rearranging the formula,

P = c²×ρ/γ

P = 400²×0.051.4

P= 1600×0.051.4

P= 5714.28 Pa

Therefore, Pressure will be 5714.28 Pa.

Example 3: Calculate the wavelength of a sound wave traveling through air at a speed of 343 meters per second (m/s) with a frequency of 440 Hertz (Hz).

Solution:

We can use the formula for the speed of sound to find the wavelength ($𝜆$) of the sound wave. The formula for the speed of sound is:

$v=f\times \lambda$

• Speed of sound ($$) = 343 m/s
• Frequency ($$) = 440 Hz

We can rearrange the formula to solve for the wavelength

$\lambda \; =\; v/f$

$\lambda \; =\; 343\; m/s/440\; Hz\; =\; 0.779\; m$