Sound waves in gases, Mechanical Engineering

Assignment Help:

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

1721_download (1).png 

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

1873_download (2).png 

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

1035_download (3).png 

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

1815_download (5).png 

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

2383_download (6).png 

Hence, the velocity of sound waves in a gaseous medium

448_download (7).png 

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

791_download (8).png 

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.

178_download (9).png

Related Discussions:- Sound waves in gases

Dynamic movement in polar coordinates, You are required to investigate the ...

You are required to investigate the dynamic movement of a mass m sliding along a rod as shown in the diagram below. The rod is rotating at a constant angular velocity θ. The mass

Erosion-corrosion in steam service, Q.What is Erosion-Corrosion in Steam Se...

Q.What is Erosion-Corrosion in Steam Service? Depending on the steam quality, Let-down valves are subject to erosion-corrosion when pressure drop exceeds 1000kPa. Hard facing i

Work-power-energy, Under the action of constant force, a 2kg block moves su...

Under the action of constant force, a 2kg block moves such that its position x as a function of time is given by X=t^3/3 ,where x in meters and t in sec , the workdone by force in

Safety and loss prevention, Q. Safety and Loss Prevention? Are there sp...

Q. Safety and Loss Prevention? Are there special venting requirements for flammables and toxic materials? Air conditioning, heating and ventilation: • are non-sparking fa

Machine and tool selection , Machine and Tool Selection The quality of...

Machine and Tool Selection The quality of a product largely depends on the type of machine, the type of process, machining conditions and the right selection of tools. The cur

Evaluate the motion - smooth pulley, Evaluate the motion - Smooth pulley: ...

Evaluate the motion - Smooth pulley: Block B is accelerated along the horizontal plane via mass A attached to it by a flexible inextensible massless rope passing over a smooth



Write Your Message!

Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd