the differentiable equation of wave in a vibrating string of mass per unit length m and tightened
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.
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