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

Firing rate demand for steam generator, Q. Firing Rate Demand for Steam Gen...

Q. Firing Rate Demand for Steam Generator? The firing rate for each Steam Generator is proportional to the feedwater flowing into that Steam Generator. Thus, the sum of the pa

Matlab program of location a point with respect a line, matlab program of l...

matlab program of location a point with respect a line

Frames, Frames: You have been introduced to the analysis of frames. By...

Frames: You have been introduced to the analysis of frames. By going through this unit, you may identify frames and differentiate them from trusses. The method for analyzing f

Define the abrasive jet cutting machines, Define the Abrasive Jet Cutting M...

Define the Abrasive Jet Cutting Machines 1. Abrasive jet cutting machines which are used to cut sheet materials or to remove materials of work piece from a surface by generatin

Spot and projection welding machines-rocker arm-press type, Spot and projec...

Spot and projection welding machines On the basis of the mechanical construction, resistance spot welding machines can be of four types viz. rocker arm, press, portable and mul

Numerical problems based on the cantilever beam, Numerical Problems Based o...

Numerical Problems Based on the Cantilever Beam: Draw the SF and BM diagram for the beam as shown in the figure given below. Also indicate principal values on diagrams.

Basic components of engine , Basic Components of Engine: The major compone...

Basic Components of Engine: The major components of the engine and their functions are briefly given hereunder. Figure shows a cut model of motorcycle engine, where various intern

Example of change of enthalpy, Example of change of enthalpy: Q 0.5...

Example of change of enthalpy: Q 0.5kg/s of a fluid flows in steady state process. The properties of fluid at the entrance are measured as p 1 = 1.4bar, density = 2.5k

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