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

Find out nature of position vector, Find out nature of position vector: ...

Find out nature of position vector: The position vector of a specific at any instant t with respect to the axes x, y and z is specified by r (t ) = (4 t 2   + 5) i¯ + (3 t

Different types of constrained motions, (a) Illustrate with examples the di...

(a) Illustrate with examples the different types of constrained motions ? (b) Describe in detail the following Mechanism : (i) Single slider crank chain mechanism (ii) Fou

Hazardous and safe areas in plant layout, Q. Hazardous and Safe Areas in pl...

Q. Hazardous and Safe Areas in plant layout? Plants implicitly involve identifiable hazards in respect to the environment, location and presence of potentially explosive materi

Beam reactions, i would like to know how much it would cost to for my assig...

i would like to know how much it would cost to for my assignment without the actual graphs but the dimentions and without the assignment being converted into a word document as lon

Define milling cutter - geometry of cutting tool, Define Milling Cutter - G...

Define Milling Cutter - Geometry of Cutting Tool Out of large verity of milling cutter used in milling cutter used in milling practice, two types of cutter chosen to describe t

#power plant, Components of power plant &it''s functions

Components of power plant &it''s functions

Engine, maximum rpm at engine

maximum rpm at engine

Moments, what is resultant of coplanar non concurrent forces

what is resultant of coplanar non concurrent forces

Evaluate the net work done by the gas, An ideal gas is heated at fixed volu...

An ideal gas is heated at fixed volume until its temperature is 3 times the original temperature. It is then expanded isothermally till it reaches original pressure. The gas is t

Features of the scheme entrepreneurship development, Features of the Schem:...

Features of the Schem:  Features of the scheme are as follows : DIC is the primary registering centre. Registration is voluntary and not compulsory. Two types of

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