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A semi-infinite transversely isotropic linear elastic medium due to vertical moving point load is analysed. Using modified Lekhnitskii potential functions introduced by Eskandari Ghadi and Noorzad, the equation of motion are uncoupled and solved using the Fourier integral transform. This way, the displacements and stresses are obtained in Fourier space. These functions are transformed into the real space using the inverse integral transform. Finally, displacements and stresses due to point load are obtained by applying the Fourier series. These results are numerically evaluated and graphically shown. The excellent agreement between the existing results in literature and the obtained ones in this paper for isotropic material validates the accuracy of the numerical evaluation.