The axisymmetric motions in an infinitely long circular rod made of a transversely isotropic material are studied, and the frequency equation governing such motions is established. The relationship between frequency and wavelength contained in the equation is not explored in detail. Instead, exploration is limited to information that will be useful in a subsequent study in developing an approximate theory that will govern the same motions. By studying the distribution of cutoff frequencies on the frequency wave‐propagation constant plane, the conclusion is reached that the three lowest modes of motion strongly influence each other and are in turn influenced to a much lesser degree by the fourth and higher modes. Accordingly, numerical analysis is made of only the three lowest spectral lines.