Wave Propagation in an Elastic Medium

Just as a mass/spring system oscillates due to the inteiplay of an inertial force associated with the mass with a restoring force from the spring, an elastic wave 

Cc yright G 1997Acadenuc Press All rights of reproduction in any form reserved. 

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Lui, D.T., Longitudinal Wave Propagation in an Elastic-Plastic Medium, Lockheed Missiles and Space Company Technical Report LAC-MSC-2-60-64-39, Sunnyvale, CA, September 1964. 

Sound is an alteration in pressure, stress, particle displacement, and particle velocity, propagated in an elastic medium. Wave propagation is only longitudinal (i.e., compression) in gases and liquids but may also be transverse (i.e., shear) surface or other type in elastic media that can support such energy. The human bandwidth for sound ranges from 20 Hz to 20 kHz. Below 20 Hz is the domain of infrasounds while above 20 kHz is the region of ultrasounds. 

Newton (1686) first calculated the velocity of propagation of a compressional wave of permanent type in an elastic medium, and arrived at the general formula ... 

The analysis of propagating acoustic waves in an elastic medium allows its characterization by means of strain-stress relationships. The stress ay is defined as the ratio of an external force F parallel to a direction i (x,y or z) to a surface S perpendicular to the direction j. 

The vector form of the equations of motion (13.26) is called the Lame equation. The constants Cp and Cs have clear physical meaning. We will see below that equation (13.26) characterizes the propagation of two types of so called body waves in an elastic medium, compressional and shear waves, while the constants Cp and c are the velocities of those waves respectively. We will call them Lame velocities. 

Let us analyze the space and time structure of the elastic displacement field in detail. We will demonstrate that equation (13.26) describes the propagation of two types of body waves in an elastic medium, i.e., compressional and shear waves travelling at different velocities and featuring different physical properties. To this end, let us recall the well-known Helmholtz theorem according to which an arbitrary vector field, in particular an elastic displacement field U(r), may be represented as a sum of a potential, Up(r), and a solenoidal, Us(r), field (Zhdanov, 1988) ... 

In an elastic material medium a deformation (strain) caused by an external stress induces reactive forces that tend to recall the system to its initial state. When the medium is perturbed at a given time and place the perturbation propagates at a constant speed (or celerity) c that is characteristic of the medium. This propagating strain is called an elastic (or acoustic or mechanical) wave and corresponds to energy transport without matter transport. Under a periodic stress the particles of matter undergo a periodic motion around their equilibrium position and may be considered as harmonic oscillators. 

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

Ultrasound, like sound and infrasound, is made up of pressure waves, i.e. mechanical as opposed to electromagnetic waves. While the latter travel in vacuo, mechanical waves require an elastic medium to propagate.To generate ultrasound, one must do mechanical work on the propagation medium. Two possibilities are exploited magnetostriction and the piezoelectric properties of some materials. 

Mechanical waves can propagate only in an elastic media. If particle vibrations are agitated in a region of an elastic medium (soUd, liquid or gaseous), as a consequence of the... 

Noise Control Sound is a fluctuation of air pressure that can be detected by the human ear. Sound travels through any fluid (e.g., the air) as a compression/expansion wave. This wave travels radially outward in all directions from the sound source. The pressure wave induces an oscillating motion in the transmitting medium that is superimposed on any other net motion it may have. These waves are reflec ted, refracted, scattered, and absorbed as they encounter solid objects. Sound is transmitted through sohds in a complex array of types of elastic waves. Sound is charac terized by its amplitude, frequency, phase, and direction of propagation. 

For an elastic wave in a medium that is both isotropic and continuous the acoustic phonons have three different wave-polarisations, one longitudinal, with the atomic displacements in the direction of the wave propagation and two with transverse polarisation, the atomic displacements perpendicular to the propagation vector. In a crystal, the transverse modes are not necessarily degenerate except in specific symmetry directions and, because the atoms are located in discrete positions, the velocity of propagation will depend on its direction. 

In an infinitely extended solid medium elastic waves can propagate in two basic modes pressure (P) waves and shear (S) waves. However, if the medium is bounded, wave reflections occur at the boundary and more comphcated wave patterns emerge. 

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