Consider a string..
v
transverse =

(T/m), T = tension, m = mass/unit length
and v
longitudinal =

(Y/

), Y = Young's modulus,

= density
now m = 1.A.

, A = area of cross section
= A

and Y = (T/A)/strain
So v
transverse =

{T/(A

)}
and v
longitudinal =

{T/(A.strain.

)}
So v
transverse / v
longitudinal =

strain << 1
So vtransverse << vlongitudinal.
Got it frnd... :)