The velocity of the mechanical wave depends on elasticity and inertia of the medium.
Velocity of a transverse wave along a stretched string
Let consider a string filled at one of its ends and tension be applied at the other end. When the string is plucked at a point, it begins to vibrate. Consider a transverse wave proceeding from left to right in the form of a pulse when the string is plucked at a point as shown in Figure.
EF is the displaced position of the string at an instant of time. It forms an arc of a circle with O as centre and R as the radius. The arc EF subtends an angle 2θ at θ.
If m is the mass per unit length of the string and dx is the length of the arc EF, then the mass of the portion of the string is m dx.
Centripetal force = (m.dx.v2)/R
This force is along CO. To find the resultant of the tension T at the points E and F, we resolve T into two components Tcosθ and T sinθ.
T cosθ components acting perpendicular to CO are of equal in magnitude but opposite in direction, they cancel each other.
T sinθ components act parallel to CO. Therefore the resultant of the tensions acting at E and F is 2 T sin θ. It is directed along CO. If B is small, sinθ = θ and the resultant force due to tension is 2Tθ.
resultant force = 2Tθ
we know, 2θ = dx/R
so, 2T (dx/2R) = T (dx/R)
For the are EF to be an equilibrium,
(m.dx.v2)/R = T (dx/R)
So, v2 = T/m
Or, v = √( T/m)