Keywords: transient analysis, exact solution, inhomogeneity, elastic plate, Lamb's problem, inplane loading, reflection response, Cagniard method, method of images.

In this paper the exact solution of a inplane transient response for the linear inhomogeneous elastic half space with the weak inhomogeneity is considered, and finally the inplane transient response for a linear inhomogeneous elastic thick plate is mentioned. Linear inhomogeneity in which the phase velocity linearly changes along the thickness direction of the plate is assumed. Through these analyses and numerical results, much knowledge is obtained, and we can summarize these results as follows:

1. The potentials which expressed the displacement for the linear inhomogeneous elastic material can be determined.
2. The Lamb's problem for a linear inhomogeneous elastic half space with the weak inhomogeneity can be exactly solved using the Cagniard method.
3. The analytic solution of three stress components, one strain energy and two displacement components, and these transient responses were clarified using the accurate numerical calculation.
4. In the case of a weak inhomogeneity, it can be shown that the response for the inhomogeneous half space approached the response for the homogenous half space, and in the case of a strong inhomogeneity, the response could appropriately be expressed by a 100 homogeneous layered laminate model.
5. The fundamental ray analysis of the inplane transient waves for the elastic thick plate was clarified, and the wave propagation including Head wave and Rayleigh wave was discussed.
6. The method of images using the Snell's law which clarified the coupled reflection of P and S waves at the boundary of the plate was established, and was incorporated into the reflection response calculation algorithm.
7. The transient response of the inhomogeneous thick plate was obtained from the transient response of the homogeneous thick plate as a result of the wave propagation research for the inhomogeneity.
8. Since fundamentally closed analytic solutions were given, numerical calculation could be carried out using a 32 bits personal computer without depending on large computation,.
9. The appropriate moving image expression was possible for all responses, therefore, and the consistency of the analysis with the numerical results was confirmed.
10. The applicability of this exact procedure, will, in the future, be expected to be an accurate convenient computational approach.

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