Based on the first-order shear deformation theory, the free vibration and sound radiation of ABH-type functionally graded sandwich plates (ABH-type FG sandwich plate) under elastic boundary restraints are studied. The displacement field of the plate is described as an improved form of double Fourier cosine series and several closed-form auxiliary polynomial functions. The dynamic equation is solved by using the Rayleigh-Ritz method. The accuracy of the present method is validated and the effects of thickness variation coefficients and elastic boundary restraints on the vibr-acoustic characteristics are discussed. The research can provide an idea for the design of sandwich structures.