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Pure mode wave propagation is important in applications ranging from imaging to avoiding parameter tradeoff in waveform inversion. We propose new artifact-free approximations for pure P- and S-waves in a transversely isotropic medium with vertical symmetry axis. Our approximations are very accurate compared to other known approximations as it is not based on weak anisotropy assumptions. As a result, the S-wave approximation can reproduce the group velocity triplications in strongly anisotropic media.