Fundamental Physics of Ultrasound

Shutilov

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000673173
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

Based on lectures by the author, this volume is designed as a textbook on general ultrasonics. The text provides coverage of the propagation of ultrasonic waves in media with different elastic properties and under conditions close to those encountered in scientific and practical applications of ultrasound.

XI. Propagation of Ultrasound in Crystals

§ 1. General acoustic equations for crystals

A crystal is an anisotropic medium, so that its equation of motion in the linear approximation must be expressed in the general form (I.11). The stress σ_jk can be expressed in the same approximation in terms of the strain with the help of relation (I.13c) :

σik=ciklj∊lj,XI.1

in which the elastic moduli c_iklj must be written with four indices (i, k, l, j = 1, 2, 3). Since, according to the definition (I.2), ∊_lj = (1/2)(δu/δx_l + δu_l/δx_j) for small deformations, the substitution of relation (XI.1) into the equation of motion (I.11) reduces the latter to an equation for one unknown — the displacement vector:

ρ0∂2ui∂t2=-ciklj(∂2uj∂xk∂xl+∂2ul∂xk∂xj).XI.2

If in the expression c_ikljδ²u_l/δx_kδx_j we interchange the dummy indices l and j, we obtain c_ikjl, δ²u_j/δx_kδx_l The tensor c_ikjl, however, is symmetric relative to the second pair of indices; both terms in parentheses in Eq. (XI.2) are therefore equal and the equations of motion for the components of the displacement vector u assume the form

ρ0∂2ui∂t2=ciklj∂2uj∂xk∂xl.XI.3

This expression, as before, contains three equations for the components of ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Foreword to the English Edition
• Preface
• List of Symbols
• I. Basic Equations of the Theory of Elasticity
• II. Propagation of Ultrasonic Waves in Liquids and Gases
• III. Sinusoidal Plane Waves with Infinitesimal Amplitude
• IV. Finite-Amplitude Plane Waves
• V. Steady Forces Arising in an Ultrasonic Field
• VI. Ultrasonic Cavitation
• VII. Reflection, Refraction and Scattering of Ultrasonic Waves
• VIII. Transmission of Plane Waves Through Layers. Electro Acoustical Analogies. Radiation of Plane Waves
• IX. Spherical Waves
• X. Propagation of Ultrasound in an Isotropic Solid
• XI. Propagation of Ultrasound in Crystals
• Problems
• Solutions
• References
• Bibliography
• Subject Index