The tension in a stretched string is usually expressed in terms of it's modulus of elasticity
This is in fact an expression of Hooke's Law – Force is proportional to extension.
Suppose that a string of length l is stretched between two points A and B. A particle P is attached to the string at C between the points A and B.
Suppose the lengths AC and CB are in the ratio
then the natural length of AC is
and the extension of AC is
Similarly the natural length of CB is
and the extension of CB is
Suppose the particle P is pulled an extra distance x to the right.
The extension of AC is now
so
Similarly
It will experience a net force to the left, towards the equilibrium position C of magnitude
Now apply
with
which has the form of simple harmonic motion
with