Introduction to Sound Waves

Sound is a form of energy that is generated by mechanical vibrations. Therefore, sound waves require a medium for them to travel through. Sound cannot travel through a vacuum, and instead is propagated as longitudinal mechanical waves through solids, liquids, and gases.

Speed of Sound Waves in Solids, Liquids, and Gases

Newton’s Formula for Speed of Sound Waves

Newton demonstrated that the velocity of sound in a medium.

\(\sqrt{ \frac{E}{P} } = v\)

E = Modulus of Elasticity of the Medium

P - the density of the medium.

Check Out: Wave Motion

Speed of Sound Waves in Solids

\(\sqrt{ \frac{y}{P} } = v\)

Y = Young’s Modulus of the solid

P = density of the solid

Speed of Sound Waves in Liquid

$$(\sqrt{\frac{B}{P}} = v)$$

B – Bulk Modulus of the Liquid

P: Density of the Liquid

Speed of Sound Waves in Gases

Newton considered the propagation of sound waves through gases to be an isothermal process, where absorption and release of heat during compression and rarefaction are balanced, thus keeping the temperature constant. He then gave the expression for the velocity of sound in air as…

$$(\sqrt{\frac{P}{\rho}} = v)$$

P = 1.1013 x 10⁵ N/m²

The density of the air is ρ = 1.293 kg/m³

The speed of sound obtained was 280 m/s when the value of pressure and density were substituted.

There was a huge disparity between the speed of sound calculated using this formula and the values found through experimentation. As a result, Laplace provided a correction to the formula, which is now referred to as the Laplace Correction.

Laplace Correction

According to Laplace, the propagation of sound waves in gas takes place adiabatically. Therefore, the adiabatic bulk modulus of the gas (γP) must be used to calculate the speed of sound waves in the gas.

$$(\sqrt{\frac{\gamma P}{\rho}} = V)$$

γP: Adiabatic Bulk Modulus of the Gas

ρ – the density of the medium

For air, γ = 1.41

The speed of sound, after substituting the values, was calculated to be 331.6 m/s.

The results from the Newton – Laplace formula are in excellent agreement with experimental results.

Factors Affecting the Speed of Sound in Gases

• Effect of Pressure
• Effect of Temperature
• Effect of Density of the Gas
• Effect of Humidity
• Effect of Wind
• The Impact of Changing Frequency or Wavelength of a Sound Wave
• Effect of Amplitude

The Impact of Pressure

If the pressure is increased at a constant temperature, then according to the equation of state PV = RT, the volume, V, can be calculated by the molecular weight, M, divided by the density, ρ:

V = M/ρ.

And then we have

P(M/ρ) = RT

P/(ρ) = (R*T)/M

At constant temperature, if pressure changes then the density also changes accordingly.

P/ρ = constant

The speed of sound waves through a gas at constant temperature is not affected by a change in pressure.

Impact of Temperature

Velocity of Sound in a Gas

$$(\sqrt{\frac{\gamma P}{\rho }}=v)$$

But PV = RT for a gas and P = RT/V

$$(\sqrt{\frac{\gamma \cdot RT}{\rho \cdot V}})$$

$v\propto \surd T$

Therefore, the speed of sound is directly proportional to the square root of the absolute temperature.

Impact of Density

From the Velocity of Sound in the Gas

$$(\sqrt{\frac{\gamma P}{\rho}} = v)$$

The density of the gas is inversely proportional to the square root of the speed of sound.

Impact of Humidity

The density of water vapour is less than that of dry air. The presence of moisture decreases the effective density of air hence the sound wave travels faster in moist air or humid air than in dry air.

Effects of Wind

The resultant speed of sound is increased if the component of Vw of wind speed is in the direction of the sound wave, as wind adds its velocity vectorially to that of the sound wave.

$$V_{resultant} = V + V_w$$

V_W - Wind Speed

How Does Changing the Frequency or Wavelength of a Sound Wave Affect Its Effect?

Sound travels at the same speed in all directions, regardless of change in frequency or wavelength in a homogeneous isotropic medium.

$$V = \lambda f = \text{constant}$$

When the sound wave passes from one medium to another medium, the frequency remains constant but both the wavelength and velocity change.

Impact of Amplitude

From Velocity Relation

$$(\sqrt{\frac{\gamma P}{\rho}} = v)$$

Generally, the small amplitude does not affect the speed of sound in the gas; however, a very large amplitude may have an effect on the speed of the sound wave.

Relation between Speed of Sound in Gas and RMS Speed of Gas Molecules

From Velocity of Sound Wave

\begin{array}{l}\sqrt{\frac{\gamma PV}{P}}=\sqrt{\frac{\gamma PV}{M}}\end{array}

PV = nRT

n = 1

PV = RT

$$V_{rms}=\sqrt{\frac{3RT}{M}}$$

\begin{array}{l}{V_{rms}} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3}{P}}\sqrt{\frac{\gamma RT}{M}} = \sqrt{\frac{3}{P}},,V\end{array}

$$\sqrt{\frac{3}{P}} V_{rms} = V$$

Where, V – is the speed of sound waves through a gas.