In one-dimensional quadratic crystals with modulation of both the linear and nonlinear susceptibilities, the transmission properties of the beam, whose wave form is the square of hyperbolic secant, are investigated by split-step integral approach. Numerical calculations indicate that fundamental soliton is rapidly excited when only fundamental wave (FW) is launched, and it presents inherent oscillation. The second harmonic wave (SHW) excites soliton, but the corresponding dispersive wave is obvious. However, while both FW and SHW are launched, and their amplitudes and beam widths are appropriate, both excited solitons are stable. The periodicity of fundamental soliton oscillation is move evident, and the dispersive wave of SHW is extremely weak. Moreover, there is a critical value of the launched SHW amplitude for the excited soliton to reach the optimum state.