The behavior depends on both the medium and the type of wave. In general, the speed of the wave (c) depnds on the frequency of the wave. The relationship between speed and frequency (or frequency and wave-number) is called dispersion relation. If the dispersion relation is not the trivial c=constant it is said that the medium is dispersive or that we have dispersion of that wave in the given medium. Othervise, there is no dispersion. Both transversal and longitudinal bulk waves in solids are technically dispersive but for low frequencies (including audio frequencies) the speed is practically constant, at least for typical crystalline solids. But there are other types of waves in solids like the surface waves called Lamb waves where the dispersion may be significant even at low frequencies. So, I suppose that the answer to your question is kind of "no" in general, it's not enough to specify the type of wave and the polarization (longitudinal or transversal). However, for some range of frequencies, which include ultrasound used in usual industrial nondestructive testing (5-50 MHz or about there), the dispersion is practically zero and you will find, for a given material, just one speed for each one of the two bulk waves (longitudinal and transversal or shear wave).
Edit: an example of table with sound velocities. I belive that "Extensional" refers to surface waves (Lamb waves). https://www.engineeringtoolbox.com/sound-speed-solids-d_713.html