Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Some elastodynamic theory
- 3 Wave motion in an unbounded elastic solid
- 4 Reciprocity in acoustics
- 5 Reciprocity in one-dimensional elastodynamics
- 6 Reciprocity in two- and three-dimensional elastodynamics
- 7 Wave motion guided by a carrier wave
- 8 Computation of surface waves by reciprocity considerations
- 9 Reciprocity considerations for an elastic layer
- 10 Forced motion of an elastic layer
- 11 Integral representations and integral equations
- 12 Scattering in waveguides and bounded bodies
- 13 Reciprocity for coupled acousto-elastic systems
- 14 Reciprocity for piezoelectric systems
- References
- Index of cited names
- Subject index
1 - Introduction
Published online by Cambridge University Press: 10 December 2009
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 Some elastodynamic theory
- 3 Wave motion in an unbounded elastic solid
- 4 Reciprocity in acoustics
- 5 Reciprocity in one-dimensional elastodynamics
- 6 Reciprocity in two- and three-dimensional elastodynamics
- 7 Wave motion guided by a carrier wave
- 8 Computation of surface waves by reciprocity considerations
- 9 Reciprocity considerations for an elastic layer
- 10 Forced motion of an elastic layer
- 11 Integral representations and integral equations
- 12 Scattering in waveguides and bounded bodies
- 13 Reciprocity for coupled acousto-elastic systems
- 14 Reciprocity for piezoelectric systems
- References
- Index of cited names
- Subject index
Summary
Reciprocity
Reciprocity is a good thing. Something is given and something else, equally or more valuable, is returned. So it is in reciprocity for states of deformation of elastic bodies. What is received in return is the main benefit from the reciprocal relationship. From a known solution to one loading case, some important aspect of, or the complete solution to, another loading case is returned. The return is, however, not always a complete solution, but sometimes an equation for computing such a solution.
For dynamic systems the concept of reciprocity goes back to the nineteenth century. A pertinent reciprocity theorem was first formulated by von Helmholtz (1860). Lord Rayleigh (1873, 1877), subsequently derived a quite general reciprocity relation for the time-harmonic motion of a linear dynamic system with a finite or infinite number of degrees of freedom. Rayleigh's formulation included the effects of dissipation. In a later work Lamb (1888) attributed the following general reciprocity theorem to von Helmholtz (1886):
Consider any natural motion of a conservative system between two configurations A and A′ through which it passes at times t and t′ respectively, and let t′ - t= τ. Let q1, q2 …, be the coordinates of the system, and p1, p2, … the component momenta, at time t, and let the values of the same quantities at time t′ be distinguished by accents. As the system is passing through the configuration A, let a small impulse δ pr of any type be given to it; and let the consequent alteration in any coordinate qs after the time τ be denoted by δ q′s.[…]
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- Reciprocity in Elastodynamics , pp. 1 - 12Publisher: Cambridge University PressPrint publication year: 2004