A comprehensive study on transverse waves has been conducted in a functionally graded magneto-electro-elastic half-space with a corrugated boundary. The study considers two types of gradedness, namely linear and quadratic, to analyze the propagation of transverse waves. The variable-separable method is used to convert the partial differential equation into an ordinary differential equation. The WKB approximation technique is utilized to obtain solutions for the ordinary differential equations. By applying the obtained solutions along with the appropriate boundary conditions, dispersion relations in determinant form are derived for four special cases. To further analyze the results, analytical methods in conjunction with Mathematica 7 are utilized, providing numerical results for a particular model with available data. The numerical results are presented in graph format, enhancing the interpretation of the findings from a physical perspective.