Group Properties of the Acoustic Differential Equation

Group Properties of the Acoustic Differential Equation : Separation of Variables, Exact Solution

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Description

This text is an addition to the existing literature about the symmetrical properties of sound waves. The authors clarify the algebraic and analytical nature of the dynamic acoustic problem. Operator equations which are typical for linear systems and the more general Lie method are considered, which can be applied even to nonlinear problems. The information obtained allows the reader to construct different types of analytical solutions of the different acoustic equation. The acoustic differential equation describes sound waves in elastic media. If the media is non-homogeneous then the acoustic equation is generally very complicated and its exact solutions or analytical solutions may be considered as rare. This volume applies Lie algebra and Lie group techniques to separate independent variables and obtains exact analytical solutions. Special attention is paid to homogeneous and non-homogeneous media with different symmetry properties. The full wave acoustic equation is considered as well as the so-called phase acoustic equation which arises in the short-wave approximation.show more

Product details

  • Hardback | 166 pages
  • 170.7 x 255 x 15.2mm | 489.89g
  • Taylor & Francis Ltd
  • London, United Kingdom
  • English
  • 0748402802
  • 9780748402809

Table of contents

Part 1 Symmetry of the Main Equation: Symmetry operators of acoustic equations; Main operator equation; Non-homogeneous media with different symmetries; Non-homogeneous media with rotational symmetry; Lie algebra of infinitesimal operators for homogeneous media. Part 2 Separation of Variables - Exact solutions: General principles of the separation of variables in linear differential equations; Non-homogeneous media with translational symmetry; Non-homogeneous media with spherical and dilational symmetry; Use of group properties of acoustic equations to produce new solutions for homogeneous media. Part 3 Short-wave Approximation: Dimensionless form of the main equation; Acoustic trajectories are characteristic of phase acoustic equations; Contact symmetry of the phase equation; Separation of variables; Construction of Short-Wave Asymptotical Solutions. Part 4 Momentum Representation In Acoustics: Integral transformation of the main equation; Lie symmetry of the acoustic equation for linear media; Operator symmetry of the acoustic equation for quadratic media.show more