There is nothing like transverse, longitudinal. This should be interpreted as speed of transverse wave motion and speed of longitudinal wave motion separately. Just like in the event of tossing a coin, you get either head or tail, which is mutually exclusive, if the wave is transverse, it cannot be longitudinal and vice versa. 

In case of transverse wave, speed is given by v = √T/m or √η/ ρ

where T is the tension, m is mass per unit length; η is modulus of rigidity and ρ is the density of the material.

Similarly for longitudinal wave, speed is given by v = √B/ ρ or √Y/ ρ

where B is the Bulk modulus, Y is the Youngs modulus and ρ is the density of the material.