Numerical integration method for very high frequencies in the evaluation of Green’s functions for layered media: transient wave propagation phenomena

Resumo

This paper is the sequel of a numerical scheme prepared for numerical integration of improper integrals,
containing an infinite number of singularities and a decaying tail that oscillates indefinitely. This kind of integrals
is common in many engineering unbounded media problems that involve computing Green’s functions, which
are usually solved with transformed integrals approaches. This work considers the Green’s function for case of
the time-harmonic response of a multilayered transversely isotropic half-space under external excitations. This
response is obtained with the aid the Hankel transform and presented in terms of an exact stiffness matrix method.
The solution, in the Hankel transformed domain, was integrated numerically to obtain the corresponding physical
domain displacements for very high frequencies. The Fast Fourier Transform (FFT) algorithm was employed in
the previously synthesized frequency domain solutions, which results in the determination of transient responses
for very small time-steps. The results show that three wave fronts are generated. The displacement velocity of
these wave fronts can be associated to compression, shear and Rayleigh waves. This work contributes to our
understanding of the numerical evaluation of time-harmonic and transient responses of soil media and is important
in many fields of geotechnical engineering and other unbounded media problems.