FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

- 3A8 -
3. PROPERTIES OF THE ELASTIC WAVE EQUATION
We have already noted that the Elastic Wave Equation is hyperbolic (in both
second and first order form) and therefore has travelling wave solutions. In
the interior of the space domain disturbances are propagated with speeds /p/p
and /(\+2|i)/p which are often denoted by c2 and el, respectively.
Compressional waves are disturbances with displacements parallel to the
propagation direction and travel with speed cl. Shear waves are disturbances
perpendicular to the direction of travel and have speed c2. In ultrasonic
testing, we are particularly interested in waves which are localized in space,
have short wavelengths and are fast-moving relative to the length and time
scales of the problem.
The energy in a perfectly elastic material is the sum of its Kinetic and
Elastic Strain Energies. The Kinetic Energy Density Is given by
and the Elastic Strain Energy Density by
1/2 (\+2(i)(U + V ) 2 + |i/2 /(U +V ) 2 - 4U V )
x y V y x x y/
The total energy is found by integrating these densities over the space region
of interest. We note that the energy density can be expressed as an algebraic
combination of the components of the solution vector In the first order form,
but that it involves partial derivatives of the second order form solution.
Energy is conserved in a perfectly elastic material, that Is, the rate of
change of energy in a space region with respect to time is equal to the total
of the energy flux through the boundary of the region. This conservation
property is fundamental and should be modelled accurately.
4. FINITE DIFFERENCE APPROXIMATION
Finite difference approximations of the Elastic Wave Equation were first
developed In the 1960's when their main application was seismology. A major
contributor to this field was Alterman (see [3]). Most of the work with finite
difference approximations in NDT simulation has concentrated on approximating
the second order form of the equations, but two exceptions are mentioned
above. General surveys of these methods in NDT simulation are given by Aboudi
[4] and Bond [5]. A general description of finite difference approximations of
hyperbolic PDE's is given in chapter 4 of Mitchell and Griffiths [6J.
Schemes of implicit type, that is, schemes in which the approximation formula
involves coupling between the points at the latest time level, can be rejected
almost immediately. The reason for this is the prohibitive cost of solving the