An advanced theoretical model is developed to investigate the dynamic response of sandwich structures featuring anisotropic laminated facings and a weak orthotropic core. The model examines various time-dependent external pressure pulses including sonic boom, rectangular, Heaviside, sine, tangential blast, Friedlander explosive, and underwater shock pulses enabling closed-form solutions for both eigenfrequencies and dynamic response via the Laplace transform and an extended Galerkin method. A comprehensive parametric study is performed, with numerical results presented over a broad range of material and geometric configurations, supported by an in-depth analysis of the findings.