Hello!

this is for the transverse vibration in flexible strings

Let us assume a vibrating string element of length dx

Condition- No bending rigidity, tension is constant

The force on the lower end of the element is T

and the force on the other end is T(θ+∂θ /∂xdx)

Balancing force along y direction

ρ dx y'' = Tsin( θ + θ'dx) - T sinθ

where ρ is the density of the string per unit length y'' is double differentiation of y wrt t and θ is the angle the string makes with the horizontal

θ is very small so sinθ=θ

so

ρ dx y'' = Tsin( θ + θ'dx) - T sinθ

=T (θ + θ'dx) - Tθ

on further simplification we get

ρ y'' = T ∂θ /∂x

Tan θ = ∂y/∂x

θ = ∂y/∂x

ρ y'' = T ∂θ /∂x = ρ d(∂θ /∂x)

So we get

(∂^2y/∂t^2) -c^2 (∂^2y/∂x^2)

c squared is shown as c^2

Hope this clears the doubt.