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Unformatted text preview: ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM Content A simple example wave propagation An elastic wave is a deformation of the body that travels throughout the body in all directions. We can examine the deformation over a period of time by fixing our look on just one point in space. This is the case of fixing geophones or seismometers in the field, or Lagrangian description. We will begin by a simple case, assuming that we have (1) an isotropic medium, that is that the elastic properties or wave velocity, or not directionally dependent and that (2) our medium is contiunous. By examining a balance of forces across an elemental volume and relating the forces on the volume to an ideal elastic response of the volume using Hookes Law we will derive one form of the elastic wave equation. Let us begin by examining the balance of forces and mass (Newton's Second Law) for a very small elemental volume. The effect of traction forces and additional body forces ( f ) is to generate an acceleration ( u ) per unit volume of mass or density ( ): i j ij i f u + = , , (1) > To Acoustic Wave Equation where the doubledot above u ,the denotes the second partial derivative with respect to time ( 2 2 t u i ).The deformation in the body is achieved by displacing individual particles about their central resting point. Because we consider that the behavior is essentially elastic the particles will eventually come to rest at their original point of rest. Displacement for each point in space is described by a vector with a tail at that point. ) , , ( 3 2 1 u u u u = Each component of the displacement, i u depends on the location within the body and at what stage of the wave propagation we are considering. Density ( ) is a scalar property that depends on what point in 3D space we consider: ) , , ( 3 2 x x x = or, in other words ) ( ) , , ( 3 2 1 x x x x = Body forces all the forces external to the elastic medium except in the immediate vicinity of the elemental volume. For example commonly the effect of gravity is discarded as is the effect of the seismic source if the case is relatively distant from the cause, so that the homogeneous (partial differential) equation for motion states that the acceleration a particle of rock undergoes while under the influence of traction forces is proportional to the stress gradients across its surface, and that the acceleration is greater for smaller volume densities, i.e.: 3 3 2 2 1 1 x x x x u i i i j ij i + + =...
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 Fall '10
 Lorenzo

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