Sound Waves in Gases

Sound waves are longitudinal pressure waves. Let us consider the motion of a plane sound wave moving along X-axis in a gas medium. In undisturbed position, the gas medium is described by its equilibrium pressure P₀ and density ρ₀. A mechanical disturbance deforms the equilibrium state of the gas. The gas particles are displaced longitudinally causing compressions and rarefactions. Consequently, the density and pressure of the gas changes. The pressure variation moves in the medium from one region to the other producing a pressure wave.

To obtain the wave equation, we consider the motion of a thin slab of the gas (of unit area), lying between position x and x + Δ x. Following the steps exactly as in the elastic rod, the average volume strain of this element of gas is given by

 

The volume strain is produced because the pressures along the X-axis on both sides of the thin element are different. The net pressure or stress on the gas element, within linear approximation, towards +ve X-axis is

 

Now (instead of Young's modulus), the elastic property of gas is defined in terms of its bulk modulus K as

 

The minus sign in the definition of K appears because volume strain is negative for positive stress.

 

That is, bulk modulus K is determined about equilibrium condition. Hence the net force on the gas element is

 

The equation of motion of the gas element (mass = ρ₀ Δ x), therefore, is

 

Hence, the velocity of sound waves in a gaseous medium

 

depends upon the equilibrium density ρ₀ and bulk modulus K of the gas. Note that bulk modulus K is also evaluated at equilibrium condition

 

The value of K therefore depends on how pressure of the gas changes with respect to volume during wave motion. It turns out that the temperature in a sound wave does not remain constant. The excess pressure causing the compression raises the temperature of gas there; the region of rarefaction cools slightly as the pressure falls. The time period of oscillation is so small that before heat could flow from one region to another, the region of compression turns into region of rarefaction and vice-versa. The sound motion therefore is an adiabatic process and gas obeys the equation.