Derive or solve wave equation

Result: the one dimensional wave equation is ∂²u/∂t² = c² ∂²u/∂x², where c² = T/ρ.

Assumptions. A uniform, perfectly flexible string of mass ρ per unit length is stretched to tension T between two fixed points. It makes small transverse vibrations in a plane; gravity is neglected; the slope is small, so the horizontal motion of particles can be ignored and the tension T is constant.

Derivation. Consider an element PQ of the string between x and x + δx. Let u(x, t) be the displacement. The tension T acts along the tangent at each end, at small angles α (at P) and β (at Q) to the x-axis.

 u ↑                  T ↗ β
   │             Q ●
   │          ╱
   │    P ●─╯   (string element PQ)
   │  T ↙ α
   └────┼─────────┼──────→ x
        x      x + δx

• Horizontal forces: T cos β − T cos α ≈ 0 (since the angles are small, cos ≈ 1). So T is constant.
• Vertical force on the element: T sin β − T sin α ≈ T(tan β − tan α), because for small angles sin ≈ tan.

Now tan α = (∂u/∂x)ₓ and tan β = (∂u/∂x)ₓ₊δₓ, so

net force = T[ (∂u/∂x)ₓ₊δₓ − (∂u/∂x)ₓ ]

By Newton's second law, mass × acceleration = force. The mass of the element is ρ δx:

ρ δx ∂²u/∂t² = T[ (∂u/∂x)ₓ₊δₓ − (∂u/∂x)ₓ ]

Divide by ρ δx and let δx → 0:

∂²u/∂t² = (T/ρ) ∂²u/∂x²

∂²u/∂t² = c² ∂²u/∂x², c² = T/ρ

Here c is the speed with which waves travel along the string.