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

230_download.png 

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

Second law of thermodynamics, (a) Show that two statements of Second Law of...

(a) Show that two statements of Second Law of thermodynamics are similar to each other. (b) A house that was heated by electric resistance heater consumed 1200 KWh of electric e

How is varignons theorem useful in engineerin mechanics, how is varignons t...

how is varignons theorem useful in engineering mechanics

Vectors, Resolving vectors into I j k component

Resolving vectors into I j k component

Net hydrostatic moment on the cylinder, A long cylinder of radius 2 m hinge...

A long cylinder of radius 2 m hinged at point A is used as an automatic gate. When the water level reaches 15 m the gate automatically opens. Determine the magnitude of the net hyd

Chemistry, What is carbonation.? Discuss its stability s mathod of formatio...

What is carbonation.? Discuss its stability s mathod of formation

Manufacturing technology, Details of rolling process,Different rolling meth...

Details of rolling process,Different rolling methods,

What is floating foundation, What is floating foundation? Floating Foun...

What is floating foundation? Floating Foundation: Raft or mat is also recommended when the soil mass contains compressible layers or the soil stratum is variable. In such ca

Selection of metallic materials, Q. Selection of metallic materials? Th...

Q. Selection of metallic materials? This Engineering Practice has been written to assist those involved in the construction of new equipment and for the maintenance of existing

Determine of poissons ratio, a) Show that if E is assumed to be correctly c...

a) Show that if E is assumed to be correctly calculated, an error of 1% in the evaluation of C will involve an error of about 6% in the determine of Poisson's ratio when its correc

Evaluate the specific enthalpy - thermodynamics, Evaluate the specific enth...

Evaluate the specific enthalpy - thermodynamics: Steam is being generated in a boiler at a pressure of 15.25 bar. Calculate the specific enthalpy when ( i )  Steam i

Write Your Message!

Captcha
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