Transition from transient response to steady state for a layered medium.

The transition from transient response to steady-state for a layered medium subjected to antiplane loadings is studied. The steady-state formula for a layered medium is derived and the solutions for a layered half-space are then expressed explicitly in the form of wave number integrals. The transient response solutions for a layered half-space are obtained by the convolution of time harmonic loading function with transient response formula derived analytically from an effective matrix method. Two layered half-spaces with different ratios of wave velocities in the layer and half-space are considered and investigated by means of extensive numerical results to show their quite different transition behavior. The numerical results indicate that transient responses will approach steady state after certain characteristic times when the transient effects die away. The transition phenomena and characteristic times are investigated in detail through the responses from near field to far field as well as from low frequency to high frequency.