A HS teacher on a HS list asked, "Would a Transverse Wave and a Longitudinal Wave in the same material have the same speed?"
One respondent replied, "So the answer is that different types of waves have different speeds because they are caused by slightly different phenomena."
I wish to reply with a brief description appealing to simple notions for the benefit of an unusually inquisitive student.
The respondent gave, as an example, L and T waves in an aluminum rod, à la Paul Doherty at the Exploratorium.
ringing _Al_rod
My thought is to write (mechanical) waves require inertia and a restoring force similar to mechanical oscillators, e.g. pendulum and spring) In the L. (comprssion) case the inertia is the mass of the materia and the resoring force the elasticity of the material characterized by the uniaxial strain/ stress in a homogenous solid (Young's modulus). OTOH, for the T wave the inertia is the moment of inertia of an element of the bar's length and the restoring force the reactio to the shear force. Thus the T wave is is very different.