30. Twofold Re-reflections of Ultrasonic Waves from Obstacles in a Two-Dimensional Elastic Material, Taking into Account Any Laws of Their Reflections and Transformations

In this chapter, the problem of the diffraction of ultrasonic waves by cavity obstacles in an infinite two-dimensional elastic medium with double reflections is studied. A short pulse is introduced into the elastic medium with tonal filling with several periods of a plane high-frequency, monochromatic longitudinal or transverse elastic wave. Double re-reflections of waves with any possible reflections (longitudinal wave to longitudinal, transverse to transverse waves) and transformations (longitudinal waves to transverse, transverse waves to longitudinal) are considered. The integral representations of the displacements in the reflected waves are written out on the basis of the physical theory of Kirchhoff diffraction. An asymptotic estimate of multiple diffraction integrals using the multidimensional stationary phase method gives an explicit form of the geometrical optical approximation of displacements in doubly reflected and transformed waves.