In addressing the construction of beam solutions for elastic guided waves in plates, we propose expressions of the amplitude function for the Lamb wave and the shear horizontal plate wave. The amplitude function is governed by an in-plane amplitude equation describing plate waves that has a same form as the two-dimensional Helmholtz equation. Therefore, beam solutions can be constructed within the same framework as this Helmholtz equation. The paraxial approximation is then applied to this in-plane amplitude equation. On the basis of this approximation, Gaussian beam solutions for both the Lamb and shear horizontal plate waves are derived. The effects of beam width, wavenumber, and guided-wave mode upon the wavefields are examined through several numerical examples.