Magnetoacoustic resonance between transverse phonons and magnons is studied analytically on the basis of a set of linearized equations previously deduced from a fully nonlinear, rotationally invariant, thermodynamically admissible theory of uniaxial antiferromagnets of the easy‐axis type. The present paper concerns the case where the bias magnetic field is relatively weak and aligned with the axis of symmetry. Various types of dispersion are exhibited depending on whether disturbances propagate along the symmetry axis in the positive or negative direction or orthogonally to this axis. In the first situation, two magnetoacoustic regions are placed in evidence (forward travelling modes), which correspond to the interaction between, or the entanglement of, either left‐circularly polarized transverse phonons and the upper magnon branch or right‐circularly polarized transverse phonons and the lower magnon branch. The resulting dispersion is thoroughly discussed in an analytical manner in terms of a small parameter which represents globally magnetoelastic couplings. The accompanying magnetoacoustic Faraday effects are also exhibited. In the second situation, longitudinal and slow‐transverse phonons propagate independently of magnons, while fast transverse phonons couple by resonance with both upper and lower magnon branches, which provides a direct analogy with resonance phenomena in purely mechanical theories, such as vibrations in rods.