Abstract: The spectrum and spectral function of the mesoscale wave are studied by using nonstatic quasi-two-dimensional Boussinesq equations. It is supposed that basic flow is only the function of z. The equations are linearized with definite condition, and then the initial value and boundary value problems are changed into the eigenvalue problem of generalized matrix after assuming normal mode solution. The spectrum and spectral function of transversal disturbance in atmosphere are studied by using the numerical method when the vertical linear shear of basic flow exists and the stratification parameter is constant. The results of numerical calculation and theoretical analysis are discussed and compared. When the continuous spectrum areas of three waves do not overlap, the numerical solution is consistent with the analytic solution. But when continuous spectrum areas overlap each other, the structure of the continuous spectrum would be gotten by using the method of recombining spectral function. The basic principle of recombining is that, in the continuous spectrum overlapping area the spectral function of numerical solution is preprocessed, and then the running mean is calculated for the frequency consecutive spectrum and spectral function, and the running mean results are regarded as spectrum and spectral function after recombination. From the results of the spectrum and spectral function after recombination it can be seen that the structure of the disturbance is a hybrid wave of vortex wave and inertia-gravitational wave, that is, new wave mode appears. In the overlapping area of three waves of the continuous spectrum, the spectral function of the hybrid wave has the critical layer of vortex wave in the middle troposphere, which shows the character of vortex wave, and have the critical layers of inertia-gravitational waves along and against the basic flow propagation in the upper troposphere and the lower troposphere, respectively, which shows the character of inertia-gravitational wave.