Quantitative ultrasound techniques for assessment of bone quality have been attracting significant research attention. The axial transmission technique, which involves analysis of ultrasonic guided waves propagating along cortical bone, has been proposed for assessment of cortical bone quality. Because the frequency-dependent wavenumbers reflect the elastic parameters of the medium, high-resolution estimation of the wavenumbers is required at each frequency with low computational cost. We use an adaptive array signal processing method and propose a technique that can be used to estimate the numbers of propagation modes that exist at each frequency without the need for time-consuming calculations. An experimental study of 4-mm-thick copper and bone-mimicking plates showed that the proposed method estimated the wavenumbers accurately with estimation errors of less than 4% and a calculation time of less than 0.5 s when using a laptop computer.