Reflected and head waves from a linear transition layer between two homogeneous media are studied in the frequency domain. A point source is located in the upper fluid of lower velocity, and the velocity structure is considered to be continuous throughout.
Exact solutions are derived by numerical evaluation of the contour integrals in the complex wave-number plane. These are compared with approximate solutions obtained by the saddle point method. It was found that the approximate solutions for head and reflected waves beyond the critical distance may be regarded as the asymptotic ones for large values of r/2h. Similarly, the approximate solutions for reflected waves inside the critical distance are the asymptotic ones for large values of H/2h, where H is the sum of vertical distances of the source and a receiver from the transition layer of thickness 2h, and r is the horizontal distance of a receiver from the source. At high frequencies, the spectral amplitudes of reflected waves inside the critical distance are proportional to ω−1, while head-wave amplitudes are proportional to ω−2/3, (ω being the angular frequency).
Numerical calculations also show that contributions from complex poles are significant near the critical distance if the transition layer is thick. In this region, the amplitude variations of the sum of the reflected and secondary waves are similar to those from a sharp discontinuity, although the maximum amplitude for a wave of narrow-frequency band width occurs at greater distances for thicker transition layers.