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## How Does Sound Propagate in a Gaseous Medium?

### Velocity of Sound in a Material Medium

The velocity with which sound wave will propagate in a material medium depends on two properties—density and elasticity—of the medium.

Density: If the density of a medium is low, the effect of the pressure which a vibrating body exerts on such a medium can transmit through a large distance within a short time. On the other hand, if the density of the medium is high, the effect of pressure transmits through a comparatively small distance in equal time. Hence, the lengths of compression and rarefaction produced in a lighter medium become comparative large and so the velocity of sound in this medium also becomes large.

Elasticity: A vibrating particle exerts force on a medium. This results in elastic stress in the medium. Due to elastic stress the compressed layer expands. The compressed layers of the medium which has large elasticity expand very rapidly and the expanded layers also contract very rapidly, i.e., transformation of compression into rarefaction and vice versa takes a very small time. As a result, velocity of sound in the medium increases.

From theoretical calculations Newton showed that the velocity of sound through a medium is given by,

c = \(\sqrt{\frac{E}{\rho}}\)

where E = modulus of elasticity of the medium

ρ = density of the medium

[As the symbol V will be used to represent volume in the next article, the velocity of sound is represented here by the symbol c.]

Velocity of sound in different media (at given temperature)

Medium | Velocity of sound (m ᐧ s^{-1}) |

Air (0°C) | 331 |

Air (20°C) | 343 |

Carbon dioxide (0°C) | 257 |

Oxygen (0°C) | 317 |

Hydrogen (0°C) | 1286 |

Distilled water (25°C) | 1496 |

Water (4°C) | 1436 |

Copper | 3970 |

Steel | 4700 – 5200 |

Glass | 4000 – 5000 |

Velocity of Sound in a Gaseous Medium

Air or any gas is a material medium. So the formula of velocity of sound described in the previous section is applicable for a gas. Since gas has no length or size, the bulk modulus is the only modulus of elasticity for it. Hence, if ρ is the density of the gas. k is the bulk modulus, then velocity of sound through the gas is given by,

c = \(\sqrt{\frac{k}{\rho}}\) ….. (1)

### Newton’s Calculation

Sound wave propagates through a gaseous medium by the mechanism of formation of alternate layers of compression and rarefaction. Newton assumed that propagation of sound through a gaseous medium is an isothermal process, i.e., successive compressions and rarefactions occur at such a rate that temperature of the gaseous medium remains constant during the propagation though pressure and volume of the gaseous medium may change.

Let the pressure and the volume of a particular amount of gas in a portion of the gaseous medium be p and V respectively [Fig.]

During propagation of sound, suppose the pressure of this portion increases to p + p_{1}, and p_{1} is very small with respect to p. As a result, its volume decreases to V – v, where v is very small with respect to V. So if temperature remains constant, according to Boyle’s law we can write,

pV = (p + p_{1})(V – v) = pV + p_{1}V – pv – p_{1}v

or, o = p_{1}V – pv

[neglecting the term p_{1}v which is very small]

or, \(\frac{p_1 V}{v}\) = p or, p = [llatex]\frac{p_1}{\frac{v}{V}}[/latex] = \(=\frac{\text { volume stress }}{\text { volume strain }}\)

So, bulk modulus, k \(=\frac{\text { volume stress }}{\text { volume strain }}\) = p

Proof of calculus: According to Boyle’s,

pV = constant

On differentiation we have,

pdV + Vdp = 0 or, p = \(\frac{-V d p}{d V}\) = \(-\frac{d p}{\frac{d V}{V}}\)

[the negative sign indicates that if pressure increases volume decreases]

∴ p \(=\frac{\text { volume stress }}{\text { volume strain }}\) = bulk modulus of the gas = k

So, from the equation (1) we get,

velocity of sound in a gaseous medium,

c = \(\sqrt{\frac{p}{\rho}}\) ……. (2)

To understand how far this formula is correct, we can calculate the velocity of sound in air with this formula.

We know that standard atmospheric pressure = 76 cmHg = 76 × 13.6 × 980 dyn ᐧ cm^{-2}

and density of air at STP according to Newton’s formula is

c = \(\sqrt{\frac{76 \times 13.6 \times 980}{0.001293}}\) = 28000 cm ᐧ s^{-1}

= 280 m ᐧ s^{-1} (approx.)

But it is found by different experiments that velocity of sound in air at 0°C is 332 m ᐧ s^{-1} (approx.). From this, it can be concluded that there is a defect in Newton’s formula for determining the velocity of sound in a gaseous medium.