Wave propagation phenomena in two magnetoelastic, initially stressed, conducting, homogeneous, and isotropic media are discussed. The coefficients of reflection and refraction are obtained in cases of the incidentPwave and incident shear wave incident obliquely at a plane interface between two welded, magnetoelastic media. Energy relations in both cases are also derived. Total internal reflection for an incident shear‐horizontal (SH) wave is analyzed. The equation of caustic for the SH wave is obtained. The velocities of the SH waves in both media are calculated. Analysis of the results are carried out by drawing graphs between the initial pressure and critical angle and the initial pressure and velocity of the SH wave. The equation of caustic is also analyzed by determining the values of distances of the points on the caustic for different values of the angles of incidence. Taking the initial stress and magnetic field to zero the results obtained are in compliance with those results in elastic homogeneous isotropic media.